Paper Title: Freiman's $3k-4$ Theorem for Function Fields.


§ ABSTRACT 
Freiman's 3k-4 Theorem states that if a subset A of k integers has a Minkowski sum A+A of size at most 3k-4, then it must be contained in a short arithmetic progression. We prove a function field analogue that is also a generalisation: it states that if K is a perfect field and if S⊃ K is a vector space of dimension k inside an extension F/K in which K is algebraically closed, and if the K-vector space generated by all products of pairs of elements of S has dimension at most 3k-4, then K(S) is a function field of small genus, and S is of small codimension inside a Riemann-Roch space of K(S).


A method for verifying the generalized Riemann hypothesis
    [
    
=========================================================





§ INTRODUCTION


We are interested in analogues of addition theorems that occur
in field extensions F/ of a base field . If S and T are
finite-dimensional -vector subspaces of F, we denote by ST the
-linear span of the set of all products st, s∈ S, t∈ T.
The general purpose of this area of research is to characterise
subspaces S and T whose product ST has unusually small
dimension: it is naturally inspired by one of the goals of additive
combinatorics which is to characterise subsets A,B of elements of a
group that have sumsets A+B of small cardinality, where A+B
denotes the set of elements a+b, a∈ A, b∈ B.

This line of research was arguably triggered by Hou, Leung and Xiang
when they proved <cit.> an extension field analogue of Kneser's
classical addition Theorem for Abelian groups <cit.>. The
extension field version essentially states that, given a field extension F/, if S and T are
-linear spaces of finite dimension contained in F, then whenever ST< S + T -1, it must be the case that the product space
ST is stabilised by a non-trivial (different from ) subfield of
F. This theorem was initially proved for separable extensions in
<cit.>, and was shown to be true for all extensions in
<cit.>.  Versions of Kneser's Theorem for products of vector
spaces were also shown to hold in other algebras than field extensions
<cit.>.  Many applications of the theory of products of
spaces in the algebra _q^n with componentwise multiplication are
discussed in <cit.> : the latter theory is concerned in particular
with the existence and the construction of products of linear codes
of small dimension, and since many of these codes are constructed by
evaluating functions drawn from a function field over _q, this
makes the study of the dimension of products of spaces in function
fields relevant to this topic as well.

If F/ does not contain a finite extension of , and if
S,T are -subspaces of F such that ST≠ F, then Kneser's
Theorem for extension fields implies that ST≥ S +
T -1. A study of the limiting case when ST= S +  T
-1 was undertaken in <cit.>. A particularly simple case occurs
when  is algebraically closed. One can then easily uncover the
structure of the spaces S and T by considering sets A and B of
discrete valuations of the field elements in S and T and arguing
that their Minkowski sum A+B must satisfy |A+B|=|A|+|B|-1, from
which addition in  tells us that A and B must be arithmetic
progressions with the same difference: It can then be inferred that the spaces S and T
must have bases in geometric progression.  
Switching to the symmetric case S=T for the sake of
simplicity, it becomes then tempting to ask for the structure of
spaces S such that

    S^2 = 2 S -1 +γ
 for increasing values of γ. It was suggested in
<cit.> that one should draw inspiration from what is often known
as Freiman's “3k-4” Theorem <cit.>, (called Freiman's 3k-3
Theorem in <cit.>). Recall that it states that, if A
is a set of integers,
  
    |A+A|=2|A|-1+γ

  implies, when γ≤ |A|-3, that A is included in an
  arithmetic progression of length |A|+γ. Natural analogues of
  such sets A are spaces S that are of codimension at most
  γ inside a space with a basis in geometric
  progression. However, the point was made in <cit.> that other
  examples of spaces satisfying (<ref>) are Riemann-Roch
  spaces Ł(D) of an algebraic curve of genus γ.  Note that a
  space with a basis in geometric progression is a particular instance of a
  Riemann-Roch space in a genus 0 function field.  In the present article
  we show that Riemann-Roch spaces play the role of arithmetic progressions in
  the extension field setting and that the following analogue of
  Freiman's 3k-4 Theorem holds:
  
   
    Let  be a perfect field and F
    be an extension field of  in which  is algebraically
    closed (i.e. any element of F that is algebraic over
     is in ). Let S be a K-subspace of finite dimension in
    F such that ⊂ S and F = K(S). If
    
    S^2 = 2 S -1 +γ  with  
        0≤γ≤ S -3,

    then F has transcendence degree 1 over K and its genus g
    satisfies g≤γ. Furthermore, there exists a Riemann-Roch
    space Ł(D) of F that contains S and such that
    Ł(D) - _ S ≤γ -g.


Theorem <ref> was first conjectured in <cit.> where the
statement was proved for γ≤ 1. Moreover, it was also proved that, under the conditions of the theorem, F must always be of transcendence degree 1 over K. Later on, the conjecture was proved
for γ = 2 under the additional assumption that F =  (x) <cit.>.

The integer γ in (<ref>)
was called the combinatorial genus of the space S because
of the way it relates to the geometric genus g of the associated
function field.  We note that Theorem <ref> is not only a
linear analogue of Freiman's 3k-4 Theorem, but can also be seen a
generalisation since it easily recaptures the original additive
statement. Indeed, if A is a set of integers satisfying
(<ref>) with γ≤ |A|-3, then, supposing without loss
of generality that A⊂ [0,n=max A] and gcd(A)=1, we
may form the vector space S generated by x^a, a∈ A, inside the
rational function field extension (x). We have that S^2 is
generated by x^a, a∈ A+A, and Theorem <ref> therefore
applies. The relevant Riemann-Roch space Ł(D) in the conclusion of
the Theorem can only be Ł(nP_∞), where P_∞ is the place
at infinity and n = max A, and that S is of codimension at most
γ inside Ł(nP_∞) means exactly that A equals the arithmetic
progression [0,n] with at most γ missing elements. 

The way we prove Theorem <ref> borrows from Lev and
Smeliansky's elegant proof of Freiman's additive 3k-4 Theorem
<cit.> (see <cit.> for a concise description). The
idea was to project the set of integers under study onto a finite
quotient /n and apply Kneser's Theorem in this finite group. Our
starting point will be to view F = K(S) as a finite extension
of a transcendental extension K(x) of K, for a suitably chosen element
x∈ S. The goal will then be to apply the extension field version
of Kneser's Theorem to K(S)/K(x) and extract non-trivial information
from it: there will be significant differences with the additive situation
however. In the next section we give an overview of the paper and the
structure of the proof of Theorem <ref>, which will draw upon
a number of algebraic and combinatorial techniques.

 

§ OVERVIEW



The additive case.
To describe the main steps of the proof of Theorem <ref>, it
is useful to have in mind the template for Lev and Smeliansky's proof
of the original 3k-4 Theorem. One starts with a set of integers A
satisfying (<ref>) 

    |A+A|=2|A|-1+γ

with min A = 0, max A=n and
gcd(A)=1: the goal is to prove that |A|≥
n+1-γ. To this end one reduces the set A to the set
A of integers modulo n, so that we have
|A|=|A|-1. Since the sets A and A+n have only n as a
common element, and since both reduce to A modulo n, we
have that

    |A+A|≤ |A|+γ≤ 2|A|-2.

Now Kneser's Theorem tells us that there exists a subgroup H of /n
such that A+A=A+A+H and |A+A|≥ 2|A|-|H|
(with (<ref>) therefore implying that |H|>1). It could happen that
A + A = /n, in which case we are done: indeed, the left-hand
inequality of (<ref>) entails precisely that
|A| ≥ n - γ and hence |A| ≥ n+1 -
γ. 
The rest of the proof consists of showing that H≠/n cannot happen.
This is achieved by partitioning A into sets that represent cosets modulo H:
by studying the behaviour of sums of these sets, one finds elements of A+A that are equal modulo H and counting them contradicts the hypothesis γ≤ |A|-3 in (<ref>). The argument is detailed concisely in <cit.>.

The function field setting.
It will be shown that under the hypothesis of the Theorem,
 the field F must have transcendence degree 1 so that it is a function field. It is then natural enough to consider the smallest Riemann-Roch space Ł(D) containing S as the analogue of the interval [0,n] in the additive case. The divisor D is therefore the least sum of all the places P, counted with suitable multiplicities, for which there exist functions in S with poles at P. We then identify a function x in S whose pole divisor is exactly D. Next, we augment the space S by considering its linear span S=K(x)S over K(x). The choice of x will ensure that S∩ xS=Kx so that _K(S+xS)=2 S -1. This guarantees that

    _K(x)S^2 ≤_K(x)S +γ≤ 2_K(x)S -2

similarly to (<ref>). Viewing now F as an algebraic extension of K(x) (rather than K), 
we can apply Kneser's Theorem for field extensions <cit.> which says that there exists a field L,
K(x)⊂ L⊂ F, such that S^2=LS^2 and _K(x)S^2≥ 2_K(x)S-_K(x)L. Since 2_K(x)S-_K(x)L≤ 2_K(x)S -2 we have that the field L must be strictly larger than K(x).
It will not be difficult to obtain that if L=F, then, somewhat similarly to the additive case, the conclusion of Theorem <ref> holds. This is a consequence of Clifford's Theorem and is detailed in Section <ref>.

It will remain therefore to show that the situation L≠ F cannot occur. At this point, similarities with the additive case break down rather seriously and we are confronted with new challenges. 
Following the additive template would be easier if we could prove the existence of an L-adapted basis of S, i.e. the existence of elements s_i∈ S such that LS decomposes as the direct sum of the Ls_i, and S decomposes as the direct sum of S_i=S∩ Ls_i. We could then study the dimensions of the products S_iS_j without fearing the appearance of unexpected linear relations between these subspaces.
We do not know how to argue for the existence of such a decomposition of S however, but we did manage to obtain a partial result in this direction. We showed that the dimension over K of L∩ S is a non-trivial quantity: this is the object of Section <ref>. The way this is achieved is by studying the evaluations of functions of S on suitably chosen places, and showing that some of these functions must have block-wise constant evaluations, exactly as elements of L do. Galois Theory arguments then enable us to show that these evaluations must come from functions of S that also fall into L. A shorter proof is provided in the case when K has characteristic 0.

Section <ref> switches from the algebraic techniques developed in Section <ref> to a much more combinatorial approach. It uses the result of Section <ref> on S∩ L to decompose S into a direct sum of spaces that is much cruder than what would be provided by a hypothetical L-adapted basis, but which turns out to be sufficient to give an evaluation of _KS^2 that only just contradicts the hypothesis (<ref>) of Theorem <ref>.

A geometric comment.
Using K(x)S as we did may not seem like the most natural approach. 
Since the additive case resorted to a quotient of the ring of integers, a tempting analogue, 
that may seem closer in spirit to the additive proof, would be to consider a subring of F containing S and reduce it modulo some ideal.
In Section <ref> this is exactly what we do: we choose a place O of K(x) which splits totally into a sum Q of places in F and we study S modulo Q. However, we have not seen how to derive a proof of Theorem <ref> using only this approach, even though there exist Kneser-type theorems for these quotients <cit.>.

However, we believe that using (x)S definitely makes sense from a geometric point of view. 
Indeed, the extension F/(x) corresponds to some curve covering
𝒴→𝒳, where 𝒳, 𝒴
have respective function fields (x) and F. The space (x)S can be
interpreted as the restriction of S to the generic fibre of the
covering 𝒴→𝒳, while reducing
S modulo Q corresponds to a restriction to some special fibre.
In the end, proving the result required us to restrict S both to the generic fibre and to some special fibre.

Organisation of the article
In Section <ref> we provide the setup for the proof of Theorem <ref>, introduce the subextension fields of K, K(x)⊂ L⊂ F, and prove that the conclusion of the Theorem holds if L=F. In Section <ref> we study the dimension over K of S∩ L. In Section <ref> we conclude the proof by showing that L≠ F cannot happen.
Both Sections <ref> and <ref> rely on the results of Section <ref>:
Section <ref> relies upon the result of Section <ref> on the dimension of S∩ L, but it is possible to put off its proof and read Section <ref> before Section <ref>, if one so wishes. 
The proof of Theorem <ref> developed through Sections <ref> to <ref> uses the hypothesis that the ground field K is algebraically closed: in Section <ref> we show how to extend the result to perfect fields, yielding Theorem <ref> in full generality.

 


§ SETTING UP THE PROOF OF THEOREM <REF>: THE SUBFIELDS K(X) AND L



Background and Notation
The arithmetic of function fields
Number theoretic background is fairly basic and most of it can be found in Chapters 1 and 3 of <cit.>, from which we borrow notation.

Given a field K,
the function fields over K that we will encounter (see Section <ref> below) will be exclusively function fields in one variable, i.e. finite extensions of the field K(x) of rational functions.


In this setting, a valuation ring of the function field F is a proper subring 
K ⊊𝒪⊊ F that is maximal for inclusion. Such rings are
discrete valuation rings, i.e. they are local rings whose unique maximal
ideal is principal. Following <cit.> and other authors we call places of F the maximal ideals of the discrete valuation rings of F. The induced discrete valuation of a place P is denoted
v_P : ^×→ℤ.
In most of the article, with the notable exception of Section <ref>, the field 
 will be supposed to be algebraically closed, so that we have Ø / P ≃, meaning places have degree 1.
The image of a function f by the evaluation map at P: 𝒪→𝒪 /P, will be denoted by f(P).



A divisor of F is a formal sum n_1 P_1 + ⋯ + n_s P_s
where the n_i's are integers and the P_i's are places of F.
The degree of D = n_1 P_1 + ⋯ n_s P_s is defined as ∑_i n_i when the places P_i are of degree 1.
A divisor D = n_1 P_1 + ⋯ n_s P_s is said to be positive if n_i ≥ 0 for all i. 
We write D ≥ D' to mean that D-D' is positive. Since 
a function f ∈ F^× has non-zero valuation at only a finite number of places,
we can associate to f its principal divisor

    (f)_F ∑_P v_P(f) P,

where the sum is taken over all the places of F. The positive and negative part 
of the above divisor are denoted (f)_F,0 and (f)_F,∞ which means that
(f)_F = (f)_F,0 - (f)_F,∞. To lighten notation, when there is no ambiguity on the ambient field, the subscript F is omitted and we write (f), (f)_0, (f)_∞.
Principal divisors are known
to have degree 0.
Finally, given a divisor D, the Riemann–Roch space associated to D
is defined as

    Ł_F (D) {f ∈ F^× |  (f)+D ≥ 0}∪{0}.

This is a finite dimensional -vector space. We will again omit the subscript
F and write Ł (D) when the ambient field is clear from context.
Finally, to any function field is associated
a non-negative invariant g called its genus and the Riemann–Roch Theorem asserts that

    _Ł(D) ≥ D + 1 - g

with equality when D ≥ 2g-2.




Products of spaces
As mentioned in the introduction, when S,T are K-vector subspaces of an
extension field F of K, we denote by ST the K-vector space generated by
products st of elements of S with elements of T. Note that this is simply the
set of sums of products of elements of S and T. In the particular case when
T=M is itself a subfield of F, MS is simply the M-subspace generated by
S. Also, if S and T are not only K-subspaces of F but are also
M-subspaces, for some intermediate field K⊂ M⊂ F, then we have
M(SF)=(MS)(MT) and we denote this space simply by MST. More generally, the product ST of two K–vector spaces can be defined in any –algebra. In particular, in Section <ref>, products of spaces in the product ring ^n are involved.

We assume the setting of Theorem <ref>. Precisely, let K be a fixed
perfect field and F/K be an extension field such that no element of
F∖ K is algebraic over K. Let S be a finite-dimensional
K-subspace of F, such that K⊂ S and F=K(S), i.e. S generates F
as an extension of K. We denote henceforth the dimension _KS  of S by
k. Finally, we suppose that S has combinatorial genus 0≤γ≤ k-3,
where the combinatorial genus γ is defined to be the integer such that

    S^2=2k-1+γ.



The fields F and K in Theorem <ref>

The transcendence degree of F/K.

As already mentioned in the introduction, given the above assumptions on
K,S,F, it is already proved in <cit.> that the extension F/K
has transcendence degree 1.

Hence, one can assume from now on that F is a one variable function field over K.

The base field K can be assumed to be algebraically closed.
We will prove Theorem <ref> in the special case when K is an
algebraically closed field. 
We will address the issue of the generalisation from an algebraically closed field K
to all perfect fields K in Section <ref>.

From now and till the end of Section <ref>, K is assumed to
be algebraically closed.

Choosing the function x∈ S

Translating S
One easily checks that, since K⊂ S, we have F=K(S)=K(u^-1S) for any
u ∈ S∖{0}, and that proving Theorem <ref> for S is
equivalent to proving it for u^-1S. It will be convenient for us to thus
switch to some multiplicative translate of S.

Specifically, start by fixing an arbitrary place of F that we shall denote by Q_∞. 
The reason for the subscript ∞ will become apparent later
(see Remark <ref>).
Consider a filtered basis f_0, …, f_k-1 of S, 
i.e. a basis of S such that v_Q_∞(f_0) > v_Q_∞(f_1) > ⋯ >
v_Q_∞(f_k-1), where v_Q_∞ is the discrete valuation at Q_∞ (such a basis is easily constructed). 
Replacing if need be S by f_0^-1S
and setting e_i  f_0^-1f_i for any 0 ≤ i <k, we get a filtered basis
e_0, …, e_k-1 with e_0 = 1 and

    0 = v_Q_∞(e_0) > v_Q_∞(e_1) > ⋯ > v_Q_∞(e_k-1).

We hereafter suppose that S admits a basis e_0=1,…,e_k-1 satisfying
(<ref>).

The poles of x and the divisor D.
Note that for almost any place Q of F, we have v_Q(e_i) ≥ 0
for all 0 ≤ i < k. Consequently, the pole locus of S is finite,
i.e.  there is only a finite set of places at which some element
of S∖{0} has a negative valuation. Therefore, there exists
a divisor D such that S ⊂Ł(D) and
S ⊄Ł (D_0) for any D_0 < D. Since 1 ∈ S, the divisor D is positive.

We can think of Ł(D) as the smallest Riemann-Roch space which contains S.


  There exists x ∈ S such that its pole divisor (x)_∞ satisfies
  
    (x)_∞= D.




  If we do not have (x)_∞= D, then there is a function
  y∈ S that has v_Q(y)<v_Q(x) for some place Q in the support
  of D. Since  is infinite, we may add a -multiple
  λ y of y to x to obtain a function x+λ y such
  that v_Q(x+λ y)=v_Q(y) and v_Q'(x+λ y)≤ v_Q'(x) for
  every place Q'≠ Q and supporting D. We then replace x by x+λ y: a finite number of
  iterations of this process leads to an x ∈ S satisfying the
  required property (<ref>).



Note that the previously introduced place Q_∞ of F
is a pole of x and hence this place lies above the place at infinity of K(x).


Choosing x to be separating
In Section <ref> we will need the extension F/ (x) to be separable. 
For this reason we potentially modify the choice of x
while keeping Property (<ref>).


  If F/ (x) is inseparable, there exists x' ∈ S such that F/(x')
  is separable and (x')_∞ = (x)_∞.



  Denote by p the characteristic of . Suppose p> 0 (otherwise
  any extension is separable).  From <cit.>,
  F/ (x) is separable if and only if x ∉ F^p. So if x ∉ F^p ti suffices to choose x'=x. Suppose now that x ∈ F^p. By assumption F = K(S), and hence S cannot be
  contained in F^p. Consequently, there exists y∈ S
  such that y ∉ F^p and for any λ∈,
  x + λ y ∉ F^p.  Since y ∈Ł(D), we have
  (y)_∞≤ (x)_∞ and one observes that for
  almost any λ∈, 
  (x + λ y)_∞ = (x)_∞.  Setting x' = x+λ y
  for such a λ yields the result.


Replacing x by x' thus constructed if need be, we may suppose that 
(x)_∞ = D and that F/K(x) is separable. Furthermore, since
x has been constructed so as to minimise, among functions of S, its valuation at any place,
we have that the Q_∞–filtered basis e_0,… ,e_k-1 satisfies v_Q_∞(e_k-1)=v_Q_∞(x). We may therefore choose to have e_k-1=x without
losing the Q_∞–filtered structure of the e_i.

The zeros of x
    In the sequel we will need one last property of x, namely that all its
    zeros in F be simple. In other words, we want the divisor (x)_0 to be a sum of places all occurring with multiplicity 1. Note that this divisor is nothing but the decomposition in F of the
    single place O of (x) at which x vanishes (the
place O will play a significant role in Section <ref>). Now,
non-simple zeros of x appear in F if and only if the
place O of K(x) ramifies in F. Note that the places of K(x) are the place at infinity and
    the unique zero of (x-a) when a ranges over K.
    Since in a given extension there are only finitely many ramified places, replacing if need be
    x by x-a for some a, one can suppose that the zero of x in (x) is unramified
    in F. Finally, note that this last change of variable has no influence on the pole locus of x
    and does not change the field (x). Hence it can be applied without harming the previous properties of x and its pole locus.



Summarising: from now on, we suppose that

    e_0 = 1,   e_k-1 = x,
         (x)_∞ = D, 
       F/(x) is separable,
       All the zeros of x in F are simple.





Finally, we recall that:

    (x)_∞ =  D= [F:(x)].


Applying Kneser's Theorem for extension fields
Let us recall Kneser's Theorem for extension fields <cit.>.


Let E⊂ M be fields and let A,B be finite-dimensional E-subspaces of
M such that A≠{0},B≠{0}. Then

    _EAB ≥_EA+_EB-_ESt(AB)

where St(AB)={x∈ M, xAB⊂ AB} is the stabiliser of AB in M.

Note that since AB is finite-dimensional over E, so is St(AB), which
therefore must be a field.

We shall apply Theorem <ref> to E=K(x), M=F, and A=B=K(x)S.
Let L=St(K(x)S^2) be the stabiliser of K(x)S^2. We first show that L is
non-trivial, i.e. L≠ K(x).

L is a non-trivial extension of K(x)

Consider the subspace S+xS of S^2. We have that S ∩ xS=Kx. Indeed, an element
f ∈ S ∩ xS is an element of xS and hence satisfies
(f)_∞≥ (x)_∞. On the other hand, as an element of
S it should satisfy (f)_∞≤ (x)_∞.  Therefore,
(f)_∞ = (x)_∞ and then (x^-1f) ≥ 0, i.e. x^-1f has no
poles, meaning  x^-1f ∈ and hence f ∈ Kx.
Consequently,
  
    _(S+xS) = 2k-1.

  Since _KS^2=2k-1+γ, there exists some –subspace T of F satisfying
  
    S^2=(S+xS)+T  and  _ T = γ.

  Now, consider the spaces (x)S and
  (x)S^2. Since all Q_∞-adic valuations of functions in (x)
  are multiples of v_Q_∞(x), we have that
  
    1,e_1,… ,e_k-2

  must be linearly independent over (x), hence
  
    _(x) (x)S=k-1.

  We also have (x)(S+xS)= (x)S, which, together with
  (<ref>) entails
  
    _(x) (x)S^2≤_(x) (x) S+_(x) (x) T≤ k-1+γ.

From (<ref>) and (<ref>) we therefore have

    _ (x) (x)S^2 ≤ k-1+γ   = _(x)(x)S + γ
        < 2 _ (x) (x)S - 1.

since γ≤ k-3. Theorem <ref> therefore implies that
L K(x).





The hypothesis L=F
It could happen that L=F, i.e. that K(x)S^2 is stabilised by F, which
simply means that K(x)S^2=F. We now proceed to show that if this is the case, then the conclusion of Theorem <ref> holds. 
This is captured by the following proposition.

If K(x)S^2=F then the genus g of F satisfies g≤γ and 
 _Ł (D) - _KS ≤γ - g, i.e.  the space S is of
codimension at most
γ-g inside a Riemann-Roch space.


From (<ref>) and (<ref>),
the hypothesis L=F gives 

    D ≤ k-1+γ.

Let us now prove that the genus g of F
satisfies g≤γ. By Clifford's Theorem <cit.>, if D≤ 2g-2, then
we have Ł(D)≤ 1+1/2 D. Since S⊂Ł(D) we have
k≤Ł(D) so that

    2k-2≤ D,

which together with (<ref>) gives k≤γ +1 in
contradiction with the hypothesis k≥γ +3. Therefore,
D≤ 2g-2 cannot hold, and by the Riemann-Roch Theorem D≥ 2g-1 implies Ł(D)= D+1-g. Hence,

    k   ≤Ł(D) =  D+1-g
       ≤ k-1+γ +1-g,

where the last inequality is a consequence of (<ref>).
From this, we obtain

    g≤γ.

Furthermore, Ł(D)= D+1-g and (<ref>) also gives

    _Ł (D) - _KS ≤γ - g.



We have therefore shown that if L=F all conclusions of Theorem <ref> hold. 
It
remains to study the situation when (x)  L  F.
In the following sections we proceed to show that such a situation cannot
happen.

Choosing a basis of LS
We conclude this section by fixing a basis of LS that will play a role in the two subsequent sections.
Let us set ℓ [L: (x)] and κ = _LLS, so that we have
_K(x)LS=κℓ. 



We have (2κ-1)ℓ≤ 2k-4.


Since L is the
stabiliser of LS^2=K(x)S^2, Kneser's Theorem <ref> gives us 
that _K(x)LS^2≥ 2_K(x)LS - _K(x)L=2κℓ-ℓ.
Together with (<ref>) and γ≤ k-3 this gives us the lemma.



 Let us focus on the
place Q_∞ relative to which we have chosen a filtered basis of S. We have v_Q_∞(x)=-N for a certain positive integer N,
and the set of Q_∞–adic valuations of the functions of K(x) is therefore N. The set
of Q_∞–adic valuations of the functions of L is therefore a subgroup of  which
contains N, so it is the subgroup d for some d|N. Since elements of
L of valuations 0,-d,-2d,… ,-(N/d-1)d must be linearly independent over
K(x), we have N/d≤ℓ.
Now in the interval [-N,0] we have N/d+1 integers that are multiples of d,
and exactly N/d representatives of every non-zero class of integers modulo
d. Therefore, the number of integers of [-N,0] that fall into the union of
some κ-1 classes modulo d is bounded from above by (κ-1)ℓ +1.
But Lemma <ref> implies that (κ-1)ℓ +1<k.
Since the set of valuations of the functions of S must have cardinality equal
to k=_KS, it has to include at least κ distinct classes modulo d. Let us choose
κ elements of S, s_1=1,s_2,… ,s_κ that have distinct
valuations v_Q_∞(s_i) modulo d: these elements are necessarily linearly
independent over L. Summarising for future reference:



There exist κ elements s_1=1,s_2,…,s_κ of S that have
valuations v_Q_∞(s_i) at Q_∞ that are distinct modulo d, where
v_Q_∞(L)=d, and make up therefore a basis of LS.





 

§ THE DIMENSION OF S ∩ L


Let τ be the integer defined as:

    τ_(x) LS - _(x)(x)S.

Our objective in this section is to prove the following lower bound on the dimension of S∩ L:


    We have
    
    _K(S∩ L)≥ℓ + 1-τ  or equivalently  _ S∩ L ≥_ S - (κ - 1)ℓ.


Whereas the previous section was mainly focused on the poles of x and the space Ł(D) for D=(x)_∞, in the present section the zeros of x will play a dominant role. Since we have taken care to ensure that every pole of every function of S is also a pole of x, it must be that every zero of x is not a pole of any function of S. It makes sense, therefore, to evaluate the functions of S on the zeros of x and study the K-vector space generated by these evaluations.
In the first part of this section we will prove that the subspace S_0 of those functions of S that evaluate to blockwise constant vectors (to be defined formally below) satisfies the lower bound of Proposition <ref>: this will be given by Proposition <ref>. The second part of the section will then be devoted to proving that this space S_0 must coincide with the space S∩ L, which will yield Proposition <ref>.

The dimension of the space S_0

Decomposing places in extensions
Given a separable extension M' of a function field M over K, and given a place P' of M', there is a unique place P of M defined as P  P' ∩ M, called the
restriction of P' to M. 
According to the representation on Figure <ref>, we follow standard practice and
say that P' is above P and that the latter is below
the former.

In the other direction, starting from a place P of M, there exist finitely many places P_1', …, P_s' of M whose restriction to M equals P. Given an element
t ∈ M of valuation 1 at P, then for any P' above P the ramification index of P' above P is defined as

    e(P'|P)  v_P'(t)

and is independent of the choice of t. We recall the fundamental identity (for K algebraically closed)

    [M' : M] = ∑_P'  above P e(P'|P).



The place O and its decomposition in L and F
According to Section <ref>, the function x has only simple zeros in F or equivalently, the place O  (x)_K(x),0 of (x) does not ramify in F.
Consequently, O is also
unramified in L and decomposes into places
_1, …, _ℓ of L. Next, these places of L all split totally into places Q_11, …, Q_1m, ⋯, Q_ℓ 1, …, Q_ℓ m of F. 
  Denote by  the local ring at O of (x),
  that is to say:
  
    {f(x)/g(x)∈(x)  |  g(0) ≠ 0
        }.

  Then, denote by  the integral closure of  in L and by  the
  integral closure of  in F. The rings  and  are
  semi-local (i.e., have a finite number of maximal
  ideals) and, from <cit.>, they satisfy
  
    = ⋂_1 ≤ i ≤ℓØ__i  and   = ⋂_1 ≤ i ≤ℓ
    1 ≤ j ≤ mØ__ij.

 Moreover, we have
  
    / x ≃,      / x ≃^ℓ  and   / x ≃^n

  where n [F:K(x)] which is equal, from (<ref>), to (x)_F,∞=(x)_F,0.
  Finally,  is a discrete valuation ring, and, from
  <cit.> the semi–local rings  and  are
  principal ideal domains.
  
The situation is summarised in 
Figure <ref>.



The space of evaluations
Note that when regarded as belonging to a subspace of / x≃^n, the
elements of /x are constant vectors and those of /x are
blockwise constant vectors. That is to say, vectors indexed by the
places _ij whose entries are equal on places _i_0 j for
j ∈{1, …, m} and a fixed i_0. They are the vectors of evaluations
of functions of  at places _ij and hence take the same value
at places lying above a given place of L.

We now focus on the functions of S that have blockwise constant vectors in the sense just introduced.
Recall that the integer τ is defined by (<ref>).



  Let
    S_0 {f ∈ S  | ∀ i ∈{1, …, ℓ},    f(_i1) = ⋯ = f(_im)}.
  In words, let S_0 be the subspace of functions of S whose evaluation vectors
are blockwise constant. The dimension of S_0 then satisfies:
  
    _ S_0 ≥ℓ + 1 - τ.




We will prove Proposition <ref> in several steps. Observe first that S_0 contains both 1 and x
and hence has dimension at least 2. Next, we will focus on
the evaluation map itself

    π : ^nf(f(_11), …, f(_ℓ m)),

which, by the Chinese remainder theorem, is equivalent to the canonical map

    ⟶/x.

In the sequel, for an element a or a subspace A of , we
denote respectively by a̅ and A their images by the map π of
(<ref>), or equivalently by the map (<ref>).

Our objective is to evaluate the dimension of  from which the
dimension of S_0 can be easily be deduced as explained by the two
following statements. Notice that Ł(D)⊂, so that
π is well-defined on Ł(D).


  The kernel of the map π_|Ł(D) : Ł(D) ⟶^n is the
  space Ł (D - ∑_i,j Q_ij), which equals the
  one–dimensional space spanned by x.



  Let f ∈Ł(D)∖{0} such that π(f) = 0. By (<ref>)
  f vanishes at all the places Q_ij and hence (f) ≥∑_i,j Q_ij -D.
  Recall that D was defined as D=(x)_∞ and since the zeros of x are the Q_ij, we have

    (x) = (x)_0-(x)_∞ = ∑_i,jQ_ij - D.

  Thus, (f) ≥ (x) and, since principal divisors have
  degree 0, we deduce that (f) = (x) or equivalently (f/x) = 0 and hence that f and x
  are –collinear.



  The spaces  and  have respective –dimensions
  _ S_0 -1 and _ S - 1.



  Both spaces are contained in Ł(D) and contain x.


From now on, our objective will be to derive a lower bound for the dimension of
. For this purpose, we will study the dimensions of  and .


  _ = ℓ



  First observe that  is isomorphic to
  / (∩ x ) =  / x. From <cit.>,
   is an –free module of rank ℓ. Then, by Nakayama's
  Lemma <cit.>, / x is a vector space over / x≃
  of dimension ℓ.

The previous lemma can be made more explicit: indeed, since / x, when regarded as a
subspace of ^n, is made up of blockwise constant vectors, it follows from _K = ℓ that 
is exactly the subspace of blockwise constant vectors of ^n, meaning that it
is spanned by the vectors
  
    v̅_1     =     (1, …, 1, 0, …, 0, ⋯, 0, …, 0);
    v̅_2     =     (0, …, 0, 1, …, 1, ⋯, 0, …, 0);
       ⋮   
    v̅_ℓ    =     (0, …, 0, 0, …, 0, ⋯, 1, …, 1).



  Now that we have the dimension of , observing that  contains , we
  will therefore obtain the dimension of  if we compute the codimension of  in .
  For this, we proceed as follows. We observe that since both S and
   contain 1, then both  and  are contained in
  =, where the product  is taken   in the algebra /x or equivalently in ^n equipped with the component-wise product.


  
    The module S is a free –module whose rank satisfies
    
    Rk_ S ≤_(x)LS = k-1+τ,

    where τ is defined by (<ref>). 

  
    Since S is finite dimensional and  is free and finitely
    generated over  (<cit.>), we deduce that
    S is a finitely generated module over  which is a
    principal ideal domain. Hence, by the structure theorem of
    finitely generated modules over a principal ideal domain, S
    is the direct sum of a free -module and a torsion module.
    However, we have that S ⊆ F and that F, being a field, has no zero
    divisors and is therefore torsion–free as an –module. Therefore, S is torsion–free also and hence is a free finitely generated –module.

    Next, let r be the –rank of the module S and let
    t_1, …, t_r be an –basis of S, we claim that
    t_1, …, t_r are (x)–independent. Indeed, suppose that
    
    f_1 t_1 + ⋯ + f_r t_r = 0

    for some f_1, …, f_r ∈ (x). Then, since  is a
    discrete valuation ring of (x) with local parameter x, for
    some integer j, the functions x^j f_1, …, x^j f_r all lie
    in . Therefore, multiplying (<ref>) by a suitable x^j we get an -linear relation between the t_i's which
    entails that the x^j f_i's are all zero and hence so are the
    f_i's. Thus, the t_i's are (x)–independent in LS and hence
    r ≤_ (x) LS.
  

  
    Actually, pushing further the arguments of the above proof, one can
    easily obtain that S has –rank exactly k-1+τ
    but we do not require this in the sequel.
  

  We now have all we need to prove Proposition <ref>.

  
    Since both  and  contain 1̅, we deduce that
     contains both  and . Since  is exactly the subspace of blockwise vectors of K^n, we have from the definition of S_0 that
    
    = ∩.

    Let
    us estimate the dimension of =.
    The evaluation map
    
    S ⟶ / x

    has x  S in its kernel, inducing a map
    
    S/ x S ⟶ / x

    whose image is nothing but .
    Thus, by the rank nullity theorem,
    
    _≤_ S / x  S

    and, from Nakayama's Lemma <cit.> the dimension of
    the right–hand side is the –rank of S. Hence, from
    Proposition <ref>, we deduce that
    
    _≤ k-1+ τ.

    From Corollary <ref>, _ = k-1.
    Thus, (<ref>) entails that  has codimension
    at most τ in .
    From Lemma <ref>, _ = ℓ
    and hence = ∩ has dimension
    at least ℓ - τ, which, again from Corollary <ref>, yields
    
    _ S_0 ≥ℓ + 1 - τ.

  

The space S_0 equals S ∩ L 
Since the evaluation of elements of S on the places Q_ij is well-defined,
we have that S∩ L⊂, and since we have seen that functions of  have
blockwise constant evaluations, we have that S∩ L⊂ S_0.
Our present objective is now to prove that the converse statement holds:

  S_0  = S ∩ L.

Together with Proposition <ref>, Proposition <ref>
will therefore yield Proposition <ref>.










Let us briefly recall the setting described in Section <ref>. There is a place  above the place
at infinity of (x) for which S has a filtered basis with
decreasing valuations from 0 to -N for some N>0.  Denote by
 the restriction of  to L. The elements of L have a
-adic valuation in d for some integer d or
equivalently, the ramification index e( | ) equals d.

To prove Proposition <ref>, we suppose the existence of u ∈ S_0 ∖ S∩ L and work towards a contradiction.
We will use Galois theory and since F/L has no reason
to be a Galois extension, we introduce  to be the Galois closure of
F/L, that is to say the smallest Galois extension of L containing F.
Since F/(x) is separable, then so is F/L and hence such a
closure exists and is a finite extension of F. We denote by G
the Galois group of the extension F'/L.

Let us observe some first properties of the Galois action of  on
the objects we are dealing with. Recall first a well–known fact
 (see for instance <cit.>). Given f ∈,  a
place of  and σ∈, then

    v_σ()(σ(f)) = v_(f).

Finally, we denote by  the conorm of D in , i.e.,
the pole locus of x when regarded as an element of . In
addition, let  be a place of  above , which is a place
in the support of .


  The divisor  is –invariant and Ł() is
  a linear representation of .



  Since  is the pole locus of x and x ∈ L,
  it is –invariant and hence, so is its pole locus .
  Now, if for some
  f ∈ we have (f) ≥ -, then,
  from (<ref>), we deduce that for any
  σ∈, we have
  (σ (f)) ≥ -σ () = -. Thus, σ leaves
  Ł () globally invariant and the action is linear since σ
  is a field automorphism of .


From now on let u be a fixed, hypothetical, element u ∈ S_0 ∖ S ∩ L.


  There
  exists an additive character χ_u : →,
  satisfying
  
    ∀σ∈,  σ(u) = u + χ_u(σ)x.




  Let σ∈. By definition of S_0, for any
  i ∈{1, …, ℓ}, u has constant evaluation at the places Q_ij that lie above P_i in F.
  Since u∈ F, we also have that u, viewed as a function of F', has constant evaluation at all the places of F' that lie above any given Q_ij. Therefore, u has constant evaluation at all the places of F' that lie above a given place P_i of L.
Since σ acts by permutation
  on the places of F' above _i, the evaluations of σ(u) at the
  places above _i are exactly the same as the evaluations of u.
  Equivalently, σ(u) - u vanishes at any place of F' above the
  _i's. In other words, σ(u)- u vanishes at any zero of x.

  Moreover, since u ∈Ł_F(D) and hence u ∈Ł_F'() when
  regarded as an element of , then from
  Lemma <ref>, σ (u) ∈Ł() and hence
  we have σ(u) - u∈Ł() also. Since  is nothing but the pole locus of x
  when regarded as an element of , we deduce that
  
    (σ (u) - u)_≥ (x)_

  and since principal divisors have degree zero, we deduce that the
  above inequality is an equality. Thus σ (u) - u = λ x
  for some λ∈ which we now denote by χ_u(σ).

  There remains to prove that the map
  σχ_u(σ) is an additive character.
  Let μ∈,
  
    μ∘σ (x)     = μ (u + χ_u(σ)x) 
       = μ(u) + μ(χ_u(σ)x) 
        = u + χ_u(μ)x + μ (χ_u(σ) x)

  Now, since χ_u(σ)x ∈ L, it is fixed by any element of 
  and hence
  
    μ∘σ (x) = u + (χ_u(μ)+χ_u(σ)) x

  and hence χ_u(μ∘σ) = χ_u(μ) + χ_u(σ)
  which means that χ_u is a group morphism.



  Note that if K is of characteristic zero, the proof of Proposition <ref> stops here since
  the additive character χ_u would be a morphism from a finite group
  into a torsion-free group, hence must be trivial, implying u∈ L.



Actually in characteristic zero, Proposition <ref> can be proved without involving Galois action as follows. Given f ∈ S_0, consider t = 1/m_F/L(f). One can prove using
<cit.> that t, when regarded as an element of F is in Ł(D) and that
for any i,j, f(Q_ij) = t(Q_ij). Therefore, from Lemma <ref>,
f = t + λ x for some λ∈ K and since both t and x are in L we deduce
that f ∈ L. Details are left to the reader.



  The extension L(u)/L is Galois, with Galois group / χ_u.



For σ∈ G, by definition of χ_u, we have that σ(u)=u if and only if σ∈χ_u.
Therefore, L(u) is exactly the subfield of  fixed by χ_u.
Moreover,  χ_u is a normal subgroup of , and by Galois
  correspondence its invariant subfield L(u) is Galois over
  L with Galois group / χ_u.


Now, let us consider the places at infinity. Recall that we chose a
pole  of x in F at which x has valuation -N. We also chose a
place  above . Next,  is the restriction of 
(or of ) to L. Finally, we denote by  the restriction of  
to L(u). The situation is summarised in Figure <ref>.


  Recall that elements of L have –adic valuation in d
  which means that the ramification index e( | ) equals d
  as shown on the figure.

  Recall also that u ∈Ł(D), which entails that
  v_(u) ≥ v_(x). When regarded as elements of
  L(u), we deduce that v_(u) ≥ v_(x).
  Note that, 
if the inequality is not strict, then for some choice of δ∈, we have v_(u+δ x) > v_(x).
  Moreover, when replacing u by u' = u + δ x, the character χ_u
  is unchanged. Indeed, for σ∈,
  
    σ (u')     = σ (u + δ x) = σ (u) + σ (δ x_∈ L) = σ (u) + δ x 
        = u + δ x + χ_u(σ)x
        = u' + χ_u(σ) x.

  In short, χ_u = χ_u'. Thus, replacing u by u' if need be, from now on one can assume that
  
    v_(u) > v_(x).


  
    The place  is unramified over .
  

  
    From <cit.>, the decomposition group
    
    D_|{σ∈Gal(L(u)/L)  | σ() = }
 is non-trivial if and only if the place is ramified.
    Let σ be an element of the decomposition group.
    Then, from (<ref>),
    
    v_ (u)     = v_σ ()(σ (u)) 
        = v_(σ (u)) 
       v_ (u)      = v_ (u + χ_u(σ)x),

    the second equality being due to the definition of the decomposition group.
    If χ_u(σ) ≠ 0, then, from (<ref>),
    the right–hand side of (<ref>) is v_(χ_u(σ) x), but then equality with the left-hand side
    immediately contradicts (<ref>) itself.

    Thus, χ_u (σ) = 0.  But from
    Corollary <ref>, the Galois group of L(u)/L is
    / χ_u. Hence, for any 
    σ∈Gal(L(u)/L) ∖{id}, χ_u(σ) ≠ 0.  Thus,
    σ = id and the decomposition group is trivial. Hence
    the result.
  

  Consequently, since
  
    e( | ) = e( | ) e( | ) = d,

  Lemma <ref> yields
  
    e( | ) = d.

  
  
    The –adic valuations of elements of L(u) are in d.
  

  Now we have all we need to finally prove Proposition <ref>.

  
    Let u ∈ S_0 ∖ S ∩ L. Using the L–basis of LS guaranteed by Lemma <ref>,
    and since u ∈ S, there exist ℓ_1, …, ℓ_κ∈ L such that
    
    u     = ℓ_1 + ℓ_2 s_2 + ⋯ + ℓ_κ s_κ
     
          u - ℓ_1     =  ℓ_2 s_2 + ⋯ + ℓ_κ s_κ.

    The left–hand side is in L(u) and, from
    Corollary <ref>, has its –adic valuation in
    d.  The terms ℓ_i s_i on the right–hand side have distinct valuations since, from Lemma <ref>, they must be distinct
    modulo d. Thus, the
    –adic valuation of the right–hand side is that of one of
    its terms, but all of them have a valuation non divisible by d.
    Thus, the only possibility would be that both sides are zero, yielding u = ℓ_1 ∈ L. A contradiction.
  


 

§ FROM PROPOSITION <REF> TO A PROOF OF THEOREM <REF>:
COMBINATORIAL ARGUMENTS


Laying the groundwork

Notice that the hypothesis of Theorem <ref>, namely _KS^2≤
2k-1+γ and γ≤ k-3, where k=_KS, implies

    _KS^2≤ 3k-4

which we will refer to as the Freiman hypothesis (hence the name
“3k-4 Theorem”). 

Recall that κ denotes κ=_LLS so that we have
_K(x)LS=κℓ, and recall from definition (<ref>)

    τ=_K(x)LS-_K(x)K(x)S=κℓ -k+1.


From Theorem <ref> (Kneser for extension fields) we have

    _LLS^2≥ 2κ -1

since the stabiliser of LS^2 is L, by definition of L.
If κ=1 we have LS^2=LS=L and, since S generates F over , we have L=F.
That situation was dealt with in
Section <ref>, so we are now in the setting κ >1. Our goal is to
derive a contradiction with the Freiman hypothesis (<ref>), thus
proving that we can only have κ=1 which will prove
Theorem <ref>. 

The following lemma will be useful:


Let W be a K-subspace of S^2 such that _LLW=w≤ 2κ -1. Then

    _K S^2≥_KW+(2κ-1-w)ℓ.



Let W' be a complementary subspace of W in S^2. Since (x)S^2=LS^2, from (<ref>)
we have _K(x)K(x)S^2≥ (2κ-1)ℓ. Since _LLW=w, we have _K(x)LW=wℓ and _K(x)K(x)W≤ wℓ. Hence, K(x)S^2=K(x)W+K(x)W' implies _K(x)K(x)W'≥ (2κ-1-w)ℓ. Finally,
S^2=W⊕ W' and _KW'≥_K(x)K(x)W' imply therefore the result.

Applying Lemma <ref> to W=S^2∩ LS gives therefore

    _KS^2≥_K(S^2∩ LS) + (κ-1)ℓ.

We have (S^2∩ LS)⊃ (S+xS), and recall from
Section <ref> that _K(S+xS)=2k-1. Inequality
(<ref>) therefore gives us

    _KS^2≥ 2k-1 + (k+τ-1) - ℓ =3k-2 -(ℓ -τ).

This does not contradict the Freiman hypothesis (<ref>), but it is
not too far from it: to obtain a contradiction, what we need to do is to improve
the lower bound (<ref>) by ℓ-τ-1.

We will improve upon (<ref>) 
in two ways. We will use valuation arguments to obtain that (S^2∩ LS) is
potentially strictly larger than S+xS, yielding a larger term than 2k-1 in
(<ref>). We will also need to improve upon the (κ-1)ℓ
estimate in (<ref>). For this we will use the Cauchy-Davenport inequality:


If U and V are K-vector subspaces of F of finite dimension, then 

    _KUV≥_KU+_KV-1.



This is a direct consequence of Theorem <ref>, since F does not
contain finite-dimensional extensions of K.

The strategy will be to look for K-subspaces U,V of S for which
Lemma <ref> will yield a subspace UV of S^2 such that  
_KUV > ℓ_LLUV. Care will be needed to ensure that these “extra
K-dimensions” add up to a sufficiently large improvement of the lower bound
(<ref>).

Proposition <ref> allows us to forget how S_0 was originally defined in Section <ref>, and we may now
use the notation S_0 simply to refer to S∩ L.


Finally, recall from Lemma <ref> that there exist κ elements s_1=1,s_2,…,s_κ of S
whose valuations at Q_∞ lie in distinct classes modulo d. We may suppose
additionally that each v_Q_∞(s_i) is maximal within
its class modulo d. Let these elements s_i of S remain fixed
throughout Section <ref>. 

The decomposition of LS and the induced subspaces of S

The following technical lemma gives us the existence of a special decomposition of the
space LS that will play an important role in bounding from below the dimension
of S^2.
We use bold uppercase letters to denote
vector spaces over L, while K-subspaces of S are denoted with
regular uppercase letters.




There exists an element,
s ∈{s_2, s_3, …, s_κ},
and three L-subspaces 𝐀, 𝐁, 𝐂
of respective L-dimensions a≥ 1,b≥ 1 and c, such that L ⊂𝐀 
and

    LS = 𝐀⊕𝐁⊕𝐂,

that satisfy the following properties:


  *  𝐀s ⊂ LS;

  *  LS ∩𝐁s = {0};

  *  The subspace A = S ∩𝐀
has a dimension _K A satisfying 
    _K A ≤_K S_0 + (a-1)ℓ,
 

  *  The subspace A^+ = S ∩ (𝐀 + 𝐁)
has a dimension satisfying 
    _K A^+≥_K S - cℓ.
 




  We will iteratively construct a sequence of triples
  𝐀_i, 𝐁_i, 𝐂_i satisfying all conditions
  of the lemma except possibly condition <ref>, and show that we must
  eventually arrive at a triple satisfying all conditions, including condition <ref>. 
  The associated element s changes at each iteration,
  though we will keep denoting it by s to lighten notation: this should not generate undue confusion. 
  The L–dimensions of the
  spaces 𝐀_i, 𝐁_i, 𝐂_i will be denoted as
  a_i, b_i, c_i respectively, and the associated subspaces of S
  will be denoted as A_i and A_i^+.

We start by choosing any s ∈{s_2,… s_κ}. 
We then define the L-subspace 𝐀_0 of LS consisting of those 𝐲∈ LS such that
𝐲s ∈ LS. Clearly L⊂𝐀_0; 
note also that LSs ≠ LS, otherwise
LS would be stabilised by s, and L would not be the
stabiliser of LS^2, since s∉L. Therefore, 𝐀_0≠ LS.
For 𝐁_0 we take any subspace of LS such that 
LS = 𝐀_0 ⊕𝐁_0, and we set 𝐂_0 = {0}, so
that we have LS=𝐀_0 ⊕𝐁_0 ⊕𝐂_0, with
a_0≥ 1 and b_0≥ 1.
Note that conditions <ref> and <ref> of the lemma are fulfilled by
construction. Condition <ref> is trivially satisfied since A_0^+ = S.


There remains the issue of condition <ref>: it is either satisfied, and we are done: or it is not, in which case we will look for another triple (𝐀_1,𝐁_1,𝐂_1)
(together with an element s)
satisfying all conditions of the Lemma except possibly the third one. More generally, given 
a triple (𝐀_i,𝐁_i,𝐂_i) satisfying all the requirements except condition <ref>,
we now construct another triple (𝐀_i+1,𝐁_i+1,𝐂_i+1) satisfying all requirements except possibly condition <ref>, and such that 𝐀_i+1⊊𝐀_i.

First observe that we have 

    a_i = _L 𝐀_i≥ 2,

otherwise we would have 𝐀_i = L and
A_i = S_0, satisfying condition <ref>.
Next, we change the current value of s to a value s ∈{s_2, … , s_κ} such
that 𝐀_i s ⊄LS. Such an s exists because
otherwise LS 𝐀_i ⊂ LS and LS would be stabilised by
𝐀_i, implying that the stabiliser of LS^2 would be not L but a
field containing 𝐀_i. We now set
𝐀_i+1 = {𝐚∈𝐀_i, 𝐚s ∈ LS}⊊𝐀_i and take 𝐁_i+1 to be any
complementary subspace of 𝐀_i+1 in 𝐀_i. Finally we set
𝐂_i+1 = 𝐁_i ⊕𝐂_i, so that 
LS=𝐀_i+1⊕𝐁_i+1⊕𝐂_i+1.

Note that L⊂𝐀_i+1⊊𝐀_i, and that b_i+1≥ 1.
By definition of A_i+1^+, we have A_i+1^+ = A_i: since A_i does not
satisfy condition <ref>, we have

    _K A_i+1^+ > _K S_0 + (a_i - 1)ℓ.

Applying Proposition <ref>, we therefore get

    _K A_i+1^+ > _K S - (κ - a_i)ℓ.

But κ - a_i = b_i + c_i = c_i+1: thus, the new triple
(𝐀_i+1, 𝐁_i+1, 𝐂_i+1) satisfies
condition <ref>. Conditions <ref> and <ref> are naturally satisfied by construction,
therefore we again have a triple satisfying all conditions of the Lemma except possibly for condition <ref>.

As already observed, condition <ref> is
trivially satisfied when 𝐀_i = L. We have therefore constructed a sequence of triples
(𝐀_i,𝐁_i,𝐂_i) that, since the 𝐀_i are strictly decreasing,
must eventually lead to L-spaces 𝐀, 𝐁, 𝐂 and an
element s satisfying all conditions of the lemma.


From now on we let 𝐀, 𝐁, 𝐂 and s ∈ S be fixed
as guaranteed by Lemma <ref>. 

Studying S_0S and S_0s∩ S
Let us write S_0 = S_0' ⊕ S_0”
where S_0' is the subspace of S_0 consisting of the elements y such
that ys ∈ (S + xS), i.e. S_0' = S_0 ∩ s^-1(S + xS), and
S_0” is any complementary subspace of S_0' in S_0. 
Since 1,x∈ S_0 we have S_0S⊃ (S+xS), and we also have S_0S⊃ S_0”s. By definition of S_0” we have S_0”s∩(S+xS)={0}, therefore

    _K S_0S ≥_K(S + xS) + _K S_0”.

Note that 1, x ∈ S_0'. Let T_0 be the subspace of elements of S_0' whose valuation at Q_∞ is larger than -N=v_∞(x). We clearly have S_0'=T_0⊕ Kx, so that _K T_0=_K S_0'-1.



We have T_0s⊂ S.



We have T_0⊂ S_0' and by definition of S_0' we have have therefore T_0s⊂ S+xS.
Suppose there exists λ∈ T_0 such that λ s∉S. 
Then, since λ s∈ S+xS, there must be some y∈ S such that v_Q_∞(λ s)=v_Q_∞(yx), i.e.

    v_Q_∞(λ)+ v_Q_∞(s)=v_Q_∞(y)+v_Q_∞(x).

Since x,λ∈ L, their valuations at Q_∞ are in d, so we have
v_Q_∞(y) = v_Q_∞(s)  d. Now recall from Lemma <ref>
that s is chosen to be one of the elements s_1,… ,s_κ and therefore 
(from the end of Section <ref>)
v_Q_∞(s) is maximal within its class modulo d. In other words we have v_Q_∞(y) ≤ v_Q_∞(s).
But the definition of T_0 implies v_Q_∞(x) < v_Q_∞(λ), so we have a contradiction with (<ref>).


Since A^+⊂ S, Lemma <ref> implies T_0sA^+⊂ S^2, and we have
S^2 ⊃ S_0S + T_0sA^+. 
We will study the sum S_0S + T_0sA^+ and its dimension, aiming to derive a contradiction
with the Freiman hypothesis (<ref>).  More precisely, since A^+⊂𝐀+𝐁, by condition 1 of Lemma <ref> we have A^+s⊂ LS+𝐁s: and since T_0⊂ L and LS+𝐁s is L-linear,
we have

    T_0sA^+ ⊂ LS + 𝐁s,

and finally, since S_0⊂ L, we also have S_0S + T_0sA^+ ⊂ LS + 𝐁s.
Therefore,

    _L L(S_0S + T_0sA^+) ≤κ + b

(where b = _L 𝐁). 
Applying Lemma <ref>, we therefore get:

    _K S^2 ≥_K(S_0S + T_0sA^+) + (κ - 1 - b)ℓ.

We will find a contradiction with the Freiman hypothesis as soon as we
show that

    _K (S_0S + T_0sA^+) ≥ 3_K S - 3 - (κ - 1 - b)ℓ,

which is now our objective. The dimension of S_0S + T_0sA^+ will be bounded from below by the sum of two terms: 
the dimension of S_0S given by (<ref>) and the contribution of elements of T_0sA^+ that
“fall outside of LS”, in other words, those that have a nonzero
component in 𝐁s.

The lower bound on S_0S+T_0sA^+ that contradicts the Freiman hypothesis

Consider the projection map

    Φ : LS⊕𝐁s𝐁sσ+𝐛s𝐛s

where σ and 𝐛 are arbitrary elements of LS
and 𝐁 respectively. The following lemma is a simple remark, but is a key to the argument to follow.


For any finite-dimensional K-subspace V of LS+𝐁s and any finite-dimensional K-subspace U of
L, we have Φ(UV)=UΦ(V).


This is a simple application of the L-linearity of the map Φ.





All dimensions below are K-dimensions. Let us write

    (S_0S+T_0sA^+)    ≥ S_0S+  T_0sA^+ -(S_0S∩ T_0sA^+)
       ≥ S_0S+  T_0sA^+ - (LS∩ T_0sA^+)

since S_0S⊂ LS. From (<ref>) and the rank-nullity theorem we have 

    T_0sA^+=Φ(T_0sA^+) + (Φ∩ T_0sA^+).

Therefore, since LS=Φ,

    (S_0S+T_0sA^+) ≥ S_0S + Φ(T_0sA^+).

Applying Lemma <ref> we therefore have, since T_0⊂ L,

    (S_0S+T_0sA^+) ≥ S_0S +  T_0Φ(sA^+).

Since A^+s⊂ LS+𝐁s, applying again the rank-nullity theorem,

    Φ(A^+s)   = A^+s - (Φ∩ A^+s)
       = A^+ -  (LS∩ A^+s).

But from the definition of A^+=S∩(𝐀+𝐁) and conditions 1 and 2 of Lemma <ref>,
we have LS∩ A^+s=As. Therefore,

    Φ(A^+s) =  A^+ -  A.

Lemma <ref> now gives us the “Cauchy-Davenport inequality”:

    T_0Φ(sA^+)    ≥ T_0 + Φ(A^+s) - 1
       ≥ T_0 +  A^+- A -1

by the above computation of Φ(A^+s).
Substituting into inequality (<ref>) we therefore have:

    (S_0S + T_0sA^+) ≥ S_0S +  T_0 +  A^+ -  A - 1.

Applying (<ref>) we now obtain:

    (S_0S + T_0sA^+)    ≥(S + xS) +  S_0” +  T_0 +  A^+ -  A - 1 
       = (S + xS) +  S_0” +  S_0' +  A^+ -  A - 2 
       = (S + xS) +  S_0 +  A^+ -  A - 2

where we have used S_0=S_0'⊕ S_0” and T_0= S_0'-1.
Applying conditions <ref> and <ref> from Lemma <ref>, we obtain:

    (S_0S + T_0sA^+)    ≥(S + xS) - 2 +  S - cℓ - (a - 1)ℓ
       ≥ 2 S - 1 - 2 +  S - (κ - 1 - b)ℓ
       = 3 S - 3 - (κ - 1 - b)ℓ

where we have used _LLS=κ=a+b+c.
Finally, substituting into (<ref>) gives the desired contradiction
with the Freiman hypothesis (<ref>). This concludes the proof of Theorem <ref> for algebraically closed fields K.

 

§ WHEN K IS NOT ALGEBRAICALLY CLOSED


Once Theorem <ref> is proved in the special case of an algebraically closed ground field K, there remains to extend it to the more general case a perfect ground field K. Suppose that K is perfect and algebraically closed
in F. The proof to follow is in the same spirit as that of <cit.>.
Denote by  the algebraic closure of K. Since K is algebraically closed in F,
the compositum F'  F is a function field over . Moreover, from <cit.>,
_ T = _K T for any K-subspace T, so that we have

    _ S = _K S

and also 

    _ S^2 = 2 _ S - 1 + γ.

Theorem <ref> for algebraically closed fields therefore applies, so that
F' has genus g ≤γ and the least divisor D' of F' such that 
S ⊂Ł_F' (D') satisfies

    _Ł_F' (D') - _ S ≤γ - g.







From <cit.>, F' has the same genus as F which yields the upper
bound on the genus. Let D be the least divisor of F such that S ⊂Ł_F (D), there remains to evaluate the codimension of S in Ł_F (D). 
Following the lines of <cit.>,
one proves easily that D' is nothing but the conorm (also called the pullback)
of D in F' (see <cit.>) and, from <cit.>,

    _Ł_F'(D') = _KŁ_F (D).

Therefore, (<ref>) and (<ref>) together with (<ref>)
permits to conclude.





Note that the assumption “K is perfect” can be relaxed to “any finite extension of K is primitive”. This is indeed the only requirement for the above argument to hold. Therefore, Theorem <ref> remains true, for example, when K is an algebraic extension of the rational function field 𝔽_q(T).


 


§ ACKNOWLEDGEMENTS

The authors express their deep gratitude to Christine Bachoc and Hugues Randriambololona for inspiring discussions. This work was supported by
the French Agence Nationale de la Recherche project ANR-21-CE39-0009 Barracuda and by Horizon-Europe MSCA-DN project Encode.
  
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Revisiting Kneser's theorem for field extensions.
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 Paper Title: A strengthening of McConnel's theorem on permutations over finite fields.




§ ABSTRACT 
Let p be a prime, q=p^n, and D ⊂_q^*. A celebrated result of McConnel states that if D is a proper subgroup of _q^*, and f:_q →_q is a function such that (f(x)-f(y))/(x-y) ∈ D whenever x ≠ y, then f(x) necessarily has the form ax^p^j+b. In this notes, we give a sufficient condition on D to obtain the same conclusion on f. In particular, we show that McConnel's theorem extends if D has small doubling.


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§ INTRODUCTION

Throughout this paper, let p be a prime. Let q=p^n and _q be the finite field with q elements with _q^*=_q ∖{0}. 

A celebrated result of McConnel <cit.> states that if D is a proper subgroup of _q^*, and f:_q →_q is a function such that 

    f(x)-f(y)/x-y∈ D,

then there are a,b ∈_q and an integer 0 ≤ j ≤ n-1, such that f(x)=a x^p^j+b for all x ∈_q. This result was first proved by Carlitz <cit.> when q is odd and D consists of squares in _q^*, that is, D is the subgroup of index 2. Carlitz's theorem and McConnel's theorem have various connections with finite geometry, graph theory, and group theory; we refer to a nice survey by Jones <cit.>. In particular, they have many applications in finite geometry; see for example <cit.>. We also refer to variations of McConnel's theorem in <cit.> via tools from group theory. 



One may wonder if the assumption that D is a multiplicative subgroup plays an important role in McConnel's theorem and if it is possible to weaken this assumption. Inspired by this natural question, in this paper, we find a sufficient condition on D so that if equation (<ref>) holds for a function f:_q →_q, then f necessarily has the form f(x)=a x^p^j+b. In particular, our main result (Theorem <ref>) strengthens McConnel's theorem. 

Before stating our results, we introduce some motivations and backgrounds from the theory of directions and their applications.  Let AG(2,q) denote the affine Galois plane over the finite field _q. Let U be a subset of points in AG(2,q); we use Cartesian coordinates in AG(2,q) so that U={(x_i,y_i):1 ≤ i ≤ |U|}.
The set of directions determined by U ⊂ AG(2, q) is 

    𝒟_U={y_j-y_i/x_j-x_i 1≤ i <j ≤ |U| }⊂_q ∪{∞},

 where ∞ is the vertical direction. If f:_q →_q is a function, we can naturally consider its graph U(f)={(x,f(x)): x ∈_q} and the set of directions determined by f is 𝒟_f:=𝒟_U(f). Indeed, 𝒟_f precisely computes the set of slopes of tangent lines joining two points on U(f). Using this terminology, equation (<ref>) is equivalent to 𝒟_f ⊂ D. 
 
 
 The following well-known result is due to Blokhuis, Ball, Brouwer, Storme, and Szőnyi <cit.>.


Let p be a prime and let q=p^n. Let f:_q →_q be a function such that f(0)=0. If |𝒟_f|≤q+1/2, then f is a linearized polynomial, that is, there are a_0,a_1, …, a_n-1∈_q, such that 

    f(x)=∑_j=0^n-1α_j x^p^j,   ∀ x ∈_q.



In particular, when q=p is a prime, Theorem <ref> implies McConnel's theorem; this was first observed by Lovász and Schrijver <cit.>. We remark that when q=p is a prime, Müller <cit.> showed a stronger result, namely, if D is a non-empty proper subset of _p^* such that f(x)-f(y)∈ D whenever x-y ∈ D, then there are a,b ∈_p such that f(x)=ax+b for all x ∈_p. For a general prime power q, Theorem <ref> does not imply McConnel's theorem directly. However, using the idea of directions, Muzychuk <cit.> provided a self-contained proof of McConnel's theorem. 


While Theorem <ref> is already quite powerful, in many applications, if some extra information on 𝒟_f is given, it is desirable to obtain a stronger conclusion, namely, f(x) has of the form ax^p^j+b for some j, or even f(x) must have the form ax+b. As an illustration, we mention two recent works in applying a stronger version of Theorem <ref> to prove analogues of the Erdős-Ko-Rado (EKR) theorem in the finite field setting <cit.>. Asgarli and Yip <cit.> proved the EKR theorem for a family of pseudo-Paley graphs of square order, and their main result roughly states that if 𝒟_f arises from a Cayley graph with “nice multiplicative properties" on its connection set, then f(x) has the form ax+b; see <cit.> for more discussions. As another example, very recently, Aguglia, Csajbók, and Weiner proved several EKR theorems for polynomials over finite fields <cit.>. Again, one key ingredient in their proof is a strengthening of Theorem <ref>. In <cit.>, they showed that if 𝒟_f is a proper _p-subspace of _q, then f(x) has the form ax+b; in <cit.> (see also <cit.>), they showed that if 𝒟_f∖{0} is a subset of a coset of K, where K is the subgroup of _q^* with index 2, then f(x) has the form ax^p^j+b for some j.


Inspired by the connection above between the theory of directions and McConnel's theorem, we establish the following result. 


Let p be a prime. Let q=p^n and D ⊂_q^*. Suppose f:_q →_q is a function such that 

    f(x)-f(y)/x-y∈ D

whenever x,y ∈_q with x ≠ y. If 

    |DD^-1D^-1∪ D^-1|≤q+1/2,

then there are a,b ∈_q and an integer 0 ≤ j ≤ n-1, such that f(x)=a x^p^j+b for all x ∈_q. In particular, if c>0 is a real number such that |DD|≤ c|D|, and (c^3+1)|D|≤q+1/2, then the same conclusion holds.


If D is a proper subgroup of _q^*, then clearly DD^-1D^-1∪ D^-1=D and thus Theorem <ref> recovers McConnel's theorem. Indeed, in this case, the doubling constant of D is simply |DD|/|D|=1. Our result shows that if the doubling constant |DD|/|D| of D is small (for example, if D=K ∪ E, where K is a subgroup of _q^*, and |E| is small compared to K), and |D| is not too large, then the analgoue of McConnel's theorem still holds. There are many ways to construct a set D ⊂_q^* with small doubling. For example, we can set D=K ∪ E, where K is a subgroup of _q^*, and E is an arbitrary subset of _q^* such that |E| is small. Alternatively, D can be taken to be the union of some cosets of a fixed subgroup K of _q^*.

For a linearized polynomial f(x), note that 𝒟_f=Im(f(x)/x). We refer to <cit.> and references therein on the study of Im(f(x)/x) for linearized polynomials f. In particular, it is an open question to determine all the possible sizes of Im(f(x)/x) among linearized polynomials f <cit.>. Theorem <ref> implies the following corollary, which partially addresses this question. It states that if D is such an image set and D satisfies inequality (<ref>), then D is necessarily a coset of a subgroup of _q^* with a restricted index.


Let p be a prime and let q=p^n. If D ⊂_q^* and |DD^-1D^-1∪ D^-1|≤q+1/2, then D=𝒟_f for some function f: _q →_q with f(0)=0 if and only if D=aK, where a ∈_q^* and K is a subgroup of _q^* with index p^r-1, where r is a divisor of n.




Notations. We follow standard notations for arithmetic operations among sets.  Given two sets A and B, we write AB={ab: a ∈ A, b ∈ B}, A^-1={a^-1: a ∈ A}.



§ PROOFS


We start by proving Theorem <ref>. Our proof is inspired by several arguments used in <cit.>.


Without loss of generality, by replacing the function f(x) with f(x)-f(0), we may assume that f(0)=0. Since |𝒟_f|≤q+1/2, Theorem <ref> implies that f is linearized. Let g(x)=1/f(x^-1) for x ∈_q ∖{0} and set g(0)=0. We claim that g is also linearized.

Let x,y ∈_q^* with x ≠ y. Then 

    g(x)-g(y)/x-y=1/f(x^-1)-1/f(y^-1)/x-y=f(y^-1)-f(x^-1)/(x-y)f(x^-1)f(y^-1)=f(y^-1-x^-1)/(x-y)f(x^-1)f(y^-1)

since f is linearized. It follows that

    g(x)-g(y)/x-y=f(x-y/xy)/(x-y)f(x^-1)f(y^-1)=f(x-y/xy)/x-y/xy·x^-1/f(x^-1)·y^-1/f(y^-1)∈ DD^-1D^-1.

On the other hand, if x ∈_q^*, then 

    g(x)-g(0)/x-0=g(x)/x=1/xf(x^-1)=x^-1/f(x^-1)∈ D^-1.

We conclude that 

    𝒟_g ⊂ DD^-1D^-1∪ D^-1.

Then inequality (<ref>) implies that |𝒟_g|≤q+1/2, and Theorem <ref> implies that 
g is also linearized. 

Since f and g are both linearized, we can find α_0, β_0, …, α_n-1, β_n-1∈_q, such that

    f(x)=∑_j=0^n-1α_j x^p^j,   and   g(x)=∑_j=0^n-1β_j x^p^j,   ∀ x ∈_q.

By the definition of g, we have f(x^-1)g(x)=1 for each x ∈_q^*, and thus (x^p^n-1f(x^-1))(g(x)/x)=x^p^n-1-1 for all x ∈_q^*. Equivalently, 

    h(x):=(∑_j=0^n-1α_j x^p^n-1-p^j) (∑_j=0^n-1β_j x^p^j-1)=x^p^n-1-1

holds for all x ∈_q^*. Note that h(x)-x^p^n-1 is a polynomial with degree at most 2p^n-1-1≤ p^n-1=q-1, and h(x)-x^p^n-1 vanishes on _q^*. It follows that there is a constant C ∈_q, such that h(x)-x^p^n-1-1=C(x^q-1-1) as polynomials. By setting x=0, we get C=0. Therefore, h(x)=x^p^n-1-1 as polynomials. In particular, g(x) is a factor of x^p^n-1. Thus, there are γ∈_q^* and 0 ≤ j ≤ n-1 such that g(x)=γ x^p^j for all x ∈_q, and it follows that f(x)=γ^-1 x^p^j for all x ∈_q, as required.

Finally, assume that |DD|≤ c|D|, and (c^3+1)|D|≤q+1/2. Then Plünnecke–Ruzsa inequality (see for example <cit.>) implies that 

    |DD^-1D^-1|≤ c^3|D|,

It follows that

    |DD^-1D^-1∪ D|≤ (c^3+1)|D| ≤q+1/2,

and the same conclusion holds.


Now we use Theorem <ref> to deduce Corollary <ref>.

Assume that f:_q →_q is a function such that f(0)=0 and 𝒟_f=D. Then Theorem <ref> implies that there exist a ∈_q^* and an integer 0 ≤ j ≤ n-1, such that f(x)=ax^p^j for all x ∈_q. Therefore, D=𝒟_f={ax^p^j-1: x ∈_q^*}. Note that (p^j-1, q-1)=(p^j-1, p^n-1)=p^(j,n)-1. It follows that D={ax^p^(j,n)-1: x ∈_q^*}=aK, where K is the subgroup of _q^* with index p^(j,n)-1.     

Conversely, let r be a divisor of n and a ∈_q^*. Let f(x)=ax^p^r for all x ∈_q. Then we have 𝒟_f={ax^p^r-1: x ∈_q^*}=aK, where K is the the subgroup of _q^* with index p^r-1.     


abbrv




 Paper Title: A note on transcendence of special values of functions related to modularity.



Transcendence of special values of modular functions]A note on transcendence of special values 
 of functions related to modularity

T. Bhowmik]Tapas Bhowmik

S. Pathak]Siddhi Pathak

Department of Mathematics, University of South Carolina, 1523 Greene Street, LeConte College, Columbia, South Carolina, USA 29208.

Chennai Mathematical Institute, H-1 SIPCOT IT Park, Siruseri, Kelambakkam, Tamil Nadu, India 603103.

tbhowmik@email.sc.edu

siddhi@cmi.ac.in

[2010]11J91, 11F03, 11F11



Research of the second author was partially supported by an INSPIRE Faculty fellowship.



§ ABSTRACT 
In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This approach unifies many of the known results, and leads to generalizations of the theorems of Schneider, Nesterenko and others. 


[
    [
    
=====





§ INTRODUCTION



The study of the arithmetic nature of values of special transcendental functions at algebraic arguments has been a well-established theme in number theory. Continuing in this spirit, this note focuses on the transcendental nature and algebraic independence of values of functions arising in the modular world, such as modular functions, modular forms and quasi-modular forms (see Section 2 for definition of the functions appearing below). The genesis of this study can be traced back to a 1937 theorem of Schneider <cit.>, namely,

    If τ∈ℍ is algebraic but not imaginary quadratic, then j(τ) is transcendental.

It is known from the theory of complex multiplication that if τ∈ℍ generates an imaginary quadratic field (τ is a CM point), then j(τ) is an algebraic number. Therefore, Schneider's theorem translates to the statement: if τ is algebraic, then j(τ) is algebraic if and only if τ is CM. A further conjecture by Mahler <cit.> in this regard, proved by Barré-Sirieix, Diaz, Gramain and Philibert <cit.> states that

For any τ∈ℍ, at least one of the two numbers e^2π iτ and j(τ) is transcendental.

This can be derived as a consequence of a remarkable theorem of Nesterenko <cit.> with several applications.
 
If τ∈ℍ, then at least three of the numbers

    e^2 π i τ,   E_2(τ),   E_4(τ),   E_6(τ)

are algebraically independent over ℚ.


Although the above theorems are about specific functions of “level 1", it is the aim of this note to highlight that they are sufficient to deduce the corresponding results for functions associated with arbitrary congruence subgroups. The authors believe that this fact may be known to experts, but is not well-documented. Often, the congruence subgroup in question is restricted to be the group Γ_0(N). The results in this paper apply to functions satisfying appropriate modularity properties with respect to Γ(N), and hence, arbitrary congruence subgroups.


A detailed investigation of algebraic independence of values of modular forms and quasi-modular forms was carried out by S. Gun, M. R. Murty and P. Rath <cit.> in 2011. Their results on values of modular forms were further elaborated upon for higher level in <cit.> by A. Hamieh and M. R. Murty[A small correction in their statement of Theorem 1.2 is required. The conclusion should read as (π/ω_τ)^k   L_q(k,χ) is algebraic, without the L(1-k,χ) term.]. These theorems will follow from our discussion later. In the context of quasimodular forms, it was proven independently by Gun, Murty and Rath <cit.> and C. Y. Chang <cit.> that

    If f is a quasi-modular form of non-zero weight for SL_2(ℤ) with algebraic Fourier coefficients and τ∈ℍ is such that j(τ)∈ℚ, then either f(τ) = 0 or f(τ) is transcendental.  

The above statement is also proved for quasimodular forms with respect to Γ_0(N) in <cit.>.



Another instance of investigation is a recent paper of D. Jeon, S.-Y. Kang and C. H. Kim <cit.>, where they prove the following. Let N ∈ℕ, 𝔤:=𝔤_0(N) = genus of X_0(N), the modular curve obtained as the quotient of the extended upper half plane by Γ_0(N). Suppose that 𝔤 > 0 and m ≥𝔤+1 is an integer. Let 𝔣_N,m denote the unique modular function with respect to Γ_0(N) constructed in <cit.> such that 𝔣_N,m is holomorphic on the upper half plane and

    𝔣_N,m(q) = 1/q^m + ∑_l=1^𝔤 a_N(m,-l) 1/q^l + O(q),

with the coefficients of powers of q being algebraic numbers. Then, they show the following.

   Let f be a non-zero meromorphic modular form with respect to Γ_0(N) with algebraic Fourier coefficients. If τ is either a zero or a pole of f, then 𝔣_N,m(τ) is algebraic for all m. 

As a corollary, they deduce that any zero or pole of f should be either CM or transcendental.


In this note, we first give an exposition of the algebraic structure of modular functions of higher level, following Shimura <cit.>. Building upon this and using Schneider's theorem, we prove

Let g be a non-constant modular function with respect to a congruence subgroup Γ of level N, with algebraic Fourier coefficients at i∞. 

    
  (a) If τ∈ℍ is either a zero or a pole of g, then j(τ) is algebraic, and hence, τ is either CM or transcendental.
    
  (b) If τ∈ℍ is such that τ is not a pole of g, then j(τ)∈ℚ⇔ g(τ)∈ℚ. Thus, at least one of g(τ) and e^2π iτ is transcendental for any τ∈ℍ.




In the context of meromorphic modular forms, we establish a generalization of Theorem <ref> to arbitrary modular forms and arbitrary modular functions. 

Let f be a non-constant meromorphic modular form of weight k ∈ℤ with respect to a congruence subgroup of level N and g be a non-constant modular function with respect to a congruence subgroup of level M. Suppose that both f and g have algebraic Fourier coefficients at i∞. Let τ∈ℍ be a zero or a pole of f. Then either g(z) has a pole at z=τ or g(τ) is algebraic.



Furthermore, we generalize the theorems in <cit.> and <cit.> to the setting of meromorphic modular forms. In particular, we show the following. 

Let f be a non-constant meromorphic modular form with respect to a congruence subgroup Γ of level N. Suppose that f has algebraic Fourier coefficients at i∞. Suppose that τ is not a pole of f. 
  
      
  (a) If τ∈ℍ is such that e^2π iτ is algebraic, then f(τ) is transcendental.
      
  (b) If τ∈ℍ is such that j(τ)∈ℚ, then there exists a transcendental number ω_τ which depends only on τ and is ℚ-linearly independent with π, such that (πω_τ)^k f(τ)∈ℚ. Therefore, f(τ) is either zero or transcendental.
    


With regard to quasi-modular forms, we extend the previously known results to meromorphic quasi-modular forms and prove

    Let f be a non-constant meromorphic quasi-modular form with depth p ≥ 1, with respect to a congruence subgroup Γ of level N. Suppose that f has algebraic Fourier coefficients at i∞ and that τ is not a pole of f. 
  
      
  (a) If τ∈ℍ is such that e^2π iτ is algebraic, then f(τ) is transcendental.
      
  (b) Let 
      
    f = ∑_r=0^p f_r  E_2^r      with      f_r∈ M^m_k-2r, N, ℚ.

      Here M^m_j,N,ℚ denotes the space of meromorphic modular forms of weight j, level N and algebraic Fourier coefficients at i∞. If τ∈ℍ is such that j(τ)∈ℚ, then 
      
    f(τ) = 0          f_r(τ) = 0   for all   0 ≤ r ≤ p.

      Moreover, if f(τ) ≠ 0, then f(τ) is transcendental.
    

This generalizes Theorem <ref> as well as <cit.>.  


Let 𝒵_j, ℚ:= {τ∈ℍ :   j(τ) ∈ℚ}. From the above results, we deduce the following interesting corollary.

Let 𝒵_mdfn be the set of zeros and poles of modular functions of arbitrary level with algebraic Fourier coefficients, 𝒵_mdfrm be the set of zeros and poles of meromorphic modular forms of arbitrary level with algebraic Fourier coefficients and 𝒵_quasi-mdf be the set of zeros and poles of meromorphic quasi-modular forms of arbitrary level with algebraic Fourier coefficients. Then

    𝒵_quasi-mdf⊆𝒵_mdfrm⊆𝒵_mdfn⊆𝒵_j, ℚ.

In particular, zeros and poles of quasi-modular forms, modular forms and modular functions are either CM or transcendental.



Finally, we have the following generalization of Theorem <ref>. 

Let f be a non-constant meromorphic modular form of weight k ∈ℤ with respect to a congruence subgroup of level N, g be a non-constant modular function with respect to a congruence subgroup of level M and f be a non-constant meromorphic quasi-modular function of depth at least 1, with respect to a subgroup of level N. Suppose that f, g and f have algebraic Fourier coefficients at i∞. If τ∈ℍ is such that f(τ)≠ 0, f(τ)≠ 0 and τ is not a pole of f, g and f, then 
    
    trdeg_ ℚ  ℚ( e^2π iτ,  f(τ),  g(τ), f(τ))≥ 3.


This theorem is in the same spirit as <cit.>, where W. Wang considers the algebraic independence of the values of three algebraically independent quasi-modular forms. However, Theorem <ref> allows one to also compare values of modular functions with those of quasi-modular forms. It can be shown that a modular function, a modular form of non-zero weight and a quasi-modular form of positive depth are algebraically independent. We include a proof of this assertion in Theorem <ref> for completeness. 


We also remark that in Theorem <ref> and Theorem <ref>, one can replace a meromorphic modular form by a half-integer weight modular form with algebraic Fourier coefficients since the square of a half-integer weight modular form is an integer weight modular form.





§ PRELIMINARIES



The aim of this section is to study the algebraic structure of the field of modular functions and to record the required results from transcendental number theory. For the sake of completeness and clarity of exposition, a brief account of the proofs is included, and appropriate references are given.




 §.§ Modular and quasi-modular forms

For each N∈ℕ, 
    Γ(N):={( 
    [ a b; c d ])∈ SL_2(ℤ):( 
    [ a b; c d ])≡( 
    [ 1 0; 0 1 ]) (modN) },
 
with Γ(1)=SL_2(ℤ). A subgroup Γ of SL_2(ℤ) is said to be congruence subgroup if there exists N∈ℕ such that Γ(N)⊆Γ. The smallest such N is said to be the level of Γ.


A holomorphic modular form of integer weight k ≥ 0 with respect to a congruence subgroup Γ is a holomorphic function on the upper half-plane ℍ which satisfies 

    
  (i) f |_k γ = f for all γ∈Γ
    
  (ii) f |_k α is holomorphic at i∞ for all α∈ SL_2(ℤ).

The function f is said to be a weakly holomorphic modular form if f |_k α is allowed to have poles at i∞, that is, f is meromorphic at the cusps. More generally, f is called a meromorphic modular form if it is meromorphic on ℍ and also at cusps. 


We will say that f has algebraic Fourier coefficients if the Fourier coefficients of f |_k α at i ∞ are algebraic numbers. Denote the space of all holomorphic, weakly holomorphic and meromorphic modular forms, with algebraic Fourier coefficients by M_k,ℚ(Γ), M_k,ℚ^w(Γ) and M_k,ℚ^m(Γ) respectively. Clearly, M_k,ℚ(Γ)⊂ M_k,ℚ^w(Γ)⊂ M_k,ℚ^m(Γ). 


For even integer k≥ 2, define the normalized Eisenstein series of weight k for S L_2(ℤ) by

    E_k(τ) = 1-2 k/B_k∑_n=1^∞σ_k-1(n) q^n,    where q=e^2π iτ and σ_s(n)=∑_d|n
     d>0d^s.
 
Here, B_k is the k-th Bernoulli number. Define

    Δ(τ):=E_4(τ)^3-E_6(τ)^2/1728.

For k≥ 4, the function E_k∈ M_k,ℚ(S L_2(ℤ)) and Δ∈ M_12,ℚ(S L_2(ℤ)). But E_2(τ) is not a modular form (see <cit.>, Chapter 5), as

    E_2 ( -1/τ) = τ^2 E_2(τ) + 6/iπτ.

This motivates the definition of a quasi-modular form, of which there are several equivalent formulations. We use the following characterization, which was established for holomorphic quasi-modular forms in <cit.> and meromorphic quasi-modular forms in <cit.>.

Every meromorphic quasi-modular form for a congruence subgroup Γ is a polynomial in E_2 with modular coefficients. More precisely, if f is a meromorphic quasi-modular form of weight k and depth p with respect to Γ, then f can be uniquely written as f = ∑_r=0^p  f_r  E_2^r, where f_r is a meromorphic modular form with respect to Γ of weight k-2r for all 0≤ r≤ p and f_p≠ 0. 

A quasi-modular form f is said to have algebraic Fourier coefficients if all the modular coefficients in the above expression of f have algebraic Fourier coefficients. 




 §.§ The Weierstrass ℘-function

 Let L=ω_1 ℤ ⊕ ω_2 ℤ be a two-dimensional lattice in ℂ with ω_1/ω_2 ∈ℍ. The Weierstrass ℘-function associated to L, given by

    ℘ (z)=1/z^2+∑_ω∈ L
    ω≠ 0(1/(z-ω)^2-1/ω^2),   for z∈ℂ∖ L,

defines a meromorphic function on ℂ. It satisfies the differential equation

    ℘'(z)^2=4℘(z)^3-g_2(L)℘(z)-g_3(L),

where 

    g_2(L)=60 ∑_ω∈ L
    ω≠ 01/ω^4    and     g_3(L)=140 ∑_ω∈ L
    ω≠ 01/ω^6.

For the lattice L_τ=τ ℤ⊕ℤ with τ∈ℍ, 

    g_2(L_τ)=4π^4/3E_4(τ)     and     g_3(L_τ)=8π^6/27E_6(τ).

Define the discriminant of a lattice, 
    Δ_0(L):=g_2(L)^3-27g_3(L)^2≠ 0,  for any two dimensional lattice L.
 In particular, we have Δ_0(L_τ)=(2π)^12 Δ(τ) for all τ∈ℍ.


The Weierstrass zeta-function associated to L=ω_1 ℤ⊕ω_2 ℤ is defined by

    ζ_L(z)=1/z+∑_ω∈ L
    ω≠ 0(1/z-ω+1/ω+z/ω^2)  for  z∈ℂ∖ L.

Note that ζ_L'(z)=-℘_L(z) is a periodic function with each point of L as a period. Hence, the functions 
    ζ_L(z+ω_1)-ζ_L(z)   and  ζ_L(z+ω_2)-ζ_L(z)
 are constants. These constants are denoted by η_1(L) and η_2(L) respectively and are called quasi-periods. Moreover, for ω_1 = τ and ω_2 = 1, it is known that 

    η_2(L_τ)=G_2(τ)=π^2/3  E_2(τ)  
      for all τ∈ℍ.

The reader is referred to (<cit.>, chapter 18.3) for proof of these results.


The Weierstrass ℘-function satisfies the following addition formula (<cit.>, chapter 11).
  
 For z_1, z_2∈ℂ such that z_1± z_2∉ L we have

    ℘(z_1+z_2)=-℘(z_1)-℘(z_2)+1/4(℘^'(z_1)-℘^'(z_2)/℘(z_1)-℘(z_2))^2 .


Using the addition formula of the Weierstrass ℘-function, Schneider <cit.> proved that 

    Let L=ω_1 ℤ⊕ω_2 ℤ be a lattice such that both g_2(L), g_3(L) are algebraic. If α is an algebraic number with α∉ L, then ℘(α) is transcendental.

The addition formula also implies the following important proposition (see <cit.>). Two generators ω_1 and ω_2 of a lattice L are said to be primitive if both have minimal absolute value among all generators of L.

Let L=ω_1 ℤ⊕ω_2 ℤ be such that both g_2(L) and g_3(L) are algebraic. Assume that ω_1 and ω_2 are primitive generators of L. Then, for any natural number n>1, the numbers ℘(ω_1/n) and ℘(ω_2/n) are algebraic. Moreover, any non-zero period of L is necessarily transcendental.





 §.§ Modular functions


A meromorphic function g on ℍ is said to be a modular function if it satisfies

    g(a τ+b/c τ+d)=g(τ)  for all γ=([ a b; c d ])∈Γ,

that is, g |_0 γ = g, and is also meromorphic at all the cusps. In particular, we call a modular function on the congruence subgroup Γ(N) to a be modular function of level N. Note that if g is a modular function with respect to Γ, which is of level N, then g is a modular function of level N.


An example of modular function of level one is given by 

    j(τ):=E_4^3(τ)/Δ(τ) for all τ∈ℍ,

which has the following Fourier expansion at i∞

    j(τ)=1/q+744+196884q+⋯,

where q=e^2π iτ. 



More specifically, for any modular function (or form) f, define 

    ℚ(f) := ℚ( {Fourier coefficients of f at all Γ-in-equivalent cusps})

The j-function is a canonical example of a modular function of level one, and governs properties of modular functions of higher levels as well. This is made precise in the following series of propositions.
 
Let g be a non-constant modular function of level one. Let ℱ denote the standard fundamental domain for the action of S L_2(ℤ) on ℍ. Suppose that the poles of g in ℱ are τ_1,τ_2,⋯,τ_m. Then g(τ) is a rational function in j(τ) with coefficients in the field ℚ(g) ( j(τ_1),⋯,j(τ_m) ).


By the compactness of ℱ∪ i∞, we know that g has only finitely many poles in ℱ. Consider the function 

    h(τ):=(∏_j=1^m (j(τ)-j(τ_j))^-ord_τ_j(g))g(τ),

where ord_τ_j(g) = - order of the pole of g at τ_j.  Then h is holomorphic on ℍ.


Suppose h has pole of order M ≥ 1 at i∞. Then the Fourier expansion of
h(τ) at i∞ has the form

    h(τ)=∑_n=-M^∞ c_n q^n,   where  c_-M≠ 0.

Note that the modular function h(τ)-c_-M j(τ)^M is holomorphic on ℍ and its Fourier expansion starts with at most a polar term of order M-1. Iterating this process, we can subtract a polynomial in j(τ) to get a holomorphic modular function that vanishes at i∞, and hence is identically zero. Thus, h(τ) is a polynomial in j(τ) over ℚ(g), and so g(τ) is a rational function of j(τ) over ℚ(g)(j(τ_1),⋯,j(τ_m)).


One can also conclude the following important fact from the above proof.

If g is a modular function of level one which is holomorphic on ℍ with a pole of order M at i∞, then g(τ) is a polynomial in j(τ) of degree M with coefficients in ℚ(g).


The j-function is sufficient to `generate' all higher level modular functions as well. This is proved below.
 
Let g be a modular function with respect to a congruence subgroup Γ and let τ_1, τ_2, ⋯,  τ_m be the poles of g in ℱ. Set

    K:=ℚ(g)(j(τ_1),   j(τ_2), ⋯,   j(τ_m) ).

Then there exists a monic polynomial P_g(X)∈ K(j)[X] such that P_g(g)=0. 


Let [S L_2(ℤ):Γ]=r and {γ_1=I,γ_2,⋯,γ_r} be a complete set of right coset representatives so that 
    S L_2(ℤ)=_i=1^rΓγ_i.

For all 1≤ i ≤ r, define the functions 
    g_i(τ):=g(γ_iτ)   for τ∈ℍ.
 Each g_i is independent of the choice of coset representatives as g is Γ-invariant. Moreover, each g_i is a modular function with respect to Γ and the Fourier expansion of g_i at i ∞ is precisely the expansion of g at the cusp γ_i(i∞). For any γ∈ S L_2(ℤ), we have g_i(γτ)=g(γ_iγτ)=g_j(τ) for some j with 1 ≤ j ≤ r such that γ_iγ∈Γγ_j, i.e., the set {g_1,g_2,...,g_r} gets permuted under the action of S L_2(ℤ). This observation implies that any elementary symmetric polynomial in g_1,g_2,⋯,g_r is a modular function of level one, and is in K(j) by Theorem <ref>. Note that the polynomial 
    P(X)=∏_i=1^r(X-g_i)
is satisfied by g as g=g_1 and has coefficients that are elementary symmetric polynomials in g_1,g_2,⋯,g_r. This proves the theorem.





Let g be a modular function with respect to a congruence subgroup Γ which is holomorphic on ℍ and K = ℚ(g). Then the monic polynomial P_g(X) satisfied by g has coefficients in K[j][X].


The coefficients of P_g(X) constructed in the proof above are modular functions of level one, holomorphic on ℍ with Fourier coefficients in K. The result now follows from Corollary <ref>. 



An important point to note here is that if g is a modular function with respect Γ that has algebraic Fourier coefficients, Theorem <ref> does not guarantee that the coefficients of P_g are algebraic, unless g is holomorphic in ℍ. 




To that effect, we now study the structure of the field of modular functions of a fixed level N>1. For any field 𝔽⊆ℂ, let

    ℱ_N, 𝔽 := {Modular functions of level N whose Fourier coefficients at i ∞ are in 𝔽}.

Theorem <ref> implies that ℱ_N,ℂ is an algebraic extension of ℂ(j). 




























Following <cit.>, we consider explicit modular function of level N whose Fourier coefficients have good rationality properties. Let a=(a_1,a_2)∈1/Nℤ^2  ∖ ℤ^2, consider the function

    f_a(τ):=g_2(L_τ) g_3(L_τ)/Δ_0(L_τ) ℘_L_τ (a_1τ+a_2),

which is holomorphic on ℍ. The following properties can be checked routinely, and we leave the proof to the reader.

Let S denotes the set {(r/N,s/N): 0≤ r,s≤ N-1  and (r,s)≠(0,0)}. Let a, b∈1/Nℤ^2  ∖ ℤ^2 and f_a be as defined above. For γ = ([ p q; r s ]) and a = (a_1,a_2), let aγ=(pa_1+ra_2,   qa_1+sa_2). Then

    
  (a) f_a = f_b if and only if a≡bℤ^2,
    
  (b) for γ∈ SL_2(ℤ), f_a (γτ) = f_aγ(τ) for all τ∈ℍ.

Therefore, all elements in {f_a: a∈ S} satisfy modularity with respect to Γ(N).


In order to conclude that the functions f_a are modular functions, we study their Fourier expansions at the cusp. But first, we need to understand the behaviour of the Weierstrass ℘-function.

Fix τ∈ℍ and denote ℘_τ = ℘_L_τ. For z ∉L_τ, we have

    (2π i)^-2℘_τ(z)=1/12+1/ξ^-1-2+ξ+∑_n=1^∞ c_n q^n,

where 

    ξ=e^2π i z,  q=e^2π iτ and 
        c_n=∑_d|nd(ξ^-d-2+ξ^d) ∀ n≥ 1.



    From the definition of the ℘-function, we have 

    ℘_τ(z)   =1/z^2+∑_(m,n)∈ℤ^2
     (m,n)≠ (0,0)[1/(z-mτ-n)^2-1/(mτ+n)^2]
       =1/z^2+∑_n∈ℤ∖{0}1/(z+n)^2-∑_n∈ℤ∖{0}1/n^2+∑_m≠ 0∑_n∈ℤ1/(z-mτ-n)^2-∑_m≠ 0∑_n∈ℤ1/(mτ+n)^2
       =-2ζ(2)+∑_n∈ℤ1/(z+n)^2-2∑_m=1^∞∑_n∈ℤ1/(mτ+n)^2
                              +∑_m=1^∞∑_n∈ℤ[1/(-z+mτ+n)^2+1/(z+mτ+n)^2].

Recall the Lipschitz summation formula, which states 

    ∑_n ∈ℤ1/(z+n)^k=(-2 π i)^k/(k-1) !∑_n=1^∞ n^k-1 q^n.

For a proof, see <cit.>. Applying this, we obtain

    ℘_τ(z)=-π^2/3+(-2π i)^2∑_n=1^∞ ne^2π i n z
                        +(-2π i)^2∑_m=1^∞∑_n=1^∞ n[e^2π i n(-z+mτ)+e^2π i n(z+mτ)-2e^2π i m n τ] 
       ⇒ (2π i)^-2℘_τ(z)=1/12+∑_n=1^∞ ne^2π i n z+∑_m,n=1^∞ nq^mn[e^2π i n z+e^-2 π i n z-2 ]

This implies the result.


Using the above expansion, the interpretation of g_2(L_τ), g_3(L_τ) and Δ_0(L_τ) in terms of Eisenstein series, we get for a= ( r/N, s/N) ∈ S,

    f_a(τ) 
        =-1/2592·E_4(τ)E_6(τ)/Δ(τ)[1/12 -∑_n=1^∞ ne^2π i n s/N q_N^n r+∑_m,n=1^∞ n q^m n[e^2π i n s/Nq_N^n r+e^-2π i n s/Nq_N^-n r-2 ]].

Here q_N = e^2 π i τ/N. Since the Fourier series of Δ(τ) begins with q=q_N^N, the Fourier expansion of f_a(τ)  begins with a rational multiple of q_N^-N. Thus, f_a has pole of order N at i∞ for all a∈ S. If s is any other cusp, then there exists γ∈ S L_2(ℤ) such that s=γ(i∞). Since f_a(γτ)=f_aγ(τ), which has pole of order N at i∞, we conclude that f_a is meromorphic at all cusps, with pole of order N. Thus, f_a is a modular function of level N for all a∈ S. 


This helps us to conclude the following.

For all a∈ S, the Fourier coefficients of f_a with respect to all cusps belong to ℚ(μ_N), where μ_N=e^2π i/N. 


   Recall that Fourier coefficients of E_4(τ), E_6(τ) and Δ(τ) are integers. Hence, from (<ref>), it follows that the Fourier coefficients of f_a at i∞ lie in ℚ(μ_N) for all a∈ S. If s is any other cusp, then there exists γ∈ S L_2(ℤ) such s=γ(i∞). But f_a(γτ)=f_aγ(τ) also has Fourier coefficients in ℚ(μ_N) with respect to i∞. This completes the proof.


These modular functions, together with the j-function serve to generate all modular functions of level N. That is,

We have ℱ_N,ℚ=ℚ( j,  { f_a |a∈ S}) and ℱ_N,ℚ is a finite Galois extension of ℚ(j).














Let E_N,ℚ:= ℚ(j,  { f_a | a∈ S}). Then E_N, ℚ is a Galois extension of ℚ(j). Indeed, for each a∈ S, the modular function f_a is holomorphic on ℍ with algebraic Fourier coefficients at all cusps. Hence, by Corollary <ref> we get a polynomial P(X)∈ℚ(j)[X], which is satisfied by f_a. Thus, E_N,ℚ  /  ℚ(j) is an algebraic extension. Moreover, this extension is normal because each conjugate of f_a is of the form f_aγ_i, which is equal to f_b for some b∈ S. Since E_N,ℚ is obtained from ℚ(j) by adjoining finite number of elements, E_N,ℚ  /  ℚ(j) is a finite Galois extension.


Now we show that E_N,ℚ= ℱ_N,ℚ. To begin with, observe that ℂ and ℱ_N,ℚ are linearly disjoint over ℚ. This can be seen as follows. Suppose {c_1,c_2,…,c_r}⊂ℂ is an arbitrary ℚ-linearly independent subset of ℂ. If there exists g_i ∈ℱ_N,ℚ such that ∑_i=1^r g_i(τ)   c_i = 0 for all τ∈ℍ with

    g_i(τ)=∑_n d_i n   q_N^n,      d_in∈ℚ  for 0≤ i≤ r,

then 
    ∑_i=1^rc_i d_i n=0   for all  n.
 The linear independence of c_i over ℚ implies that all d_i n=0 for all 0≤ i≤ r, and hence, f_1=f_2=⋯=f_r=0. Thus, we have E_N,ℚ⊆ℱ_N,ℚ⊆ℂE_N,ℚ. Suppose there exists f∈ℱ_N,ℚ∖ E_N,ℚ. Since f ∈ℂE_N,ℚ, we get a ℚ-linearly independent subset {f_1,f_2,…,f_m}⊆ E_N,ℚ such that 

    f=∑_i=1^m α_i f_i,   α_i∈ℂ,

with at least one of the α_i ∈ℂ∖ℚ. Since E_N,ℚ⊆ℱ_N,ℚ, the set {f,  f_1,f_2,…,f_m} is ℚ-linearly independent subset of ℱ_N,ℚ and hence, ℂ-linearly independent. This contradicts (<ref>). Therefore, E_N,ℚ= ℱ_N,ℚ and ℱ_N,ℚ  /  ℚ(j) is a finite Galois extension.


Thus, Theorem <ref> and Theorem <ref> imply the following crucial fact.

Let g ∈ℱ_N,ℚ. Then the Fourier coefficients of g with respect to all cusps are algebraic numbers and g satisfies a polynomial over ℚ(j). 





 §.§ Algebraic independence of modular, quasi-modular forms and modular functions
 

The aim of the discussion below is to establish the algebraic independence (as functions) of the three functions arising from modular considerations - modular functions, modular forms and quasi-modular forms. For the basic theory of quasi-modular forms, we refer the reader to <cit.>. To begin with, we prove the following lemma.

A sum of meromorphic quasi-modular forms of distinct weights is not identically zero, unless, each of these quasi-modular forms is identically zero.


Suppose f_1, f_2,…,f_r are non-identically zero meromorphic quasi-modular forms of weights k_1<k_2<⋯<k_r with respect to the congruence subgroups Γ_1,Γ_2,…,Γ_r of level N_1,N_2,…,N_r respectively. Let p be the greatest depth of these quasi-modular forms. Suppose that for each i∈{1,2,…,r}, they have the following transformation formulae:

    f_i|_k_i([ a b; c d ])(τ)=∑_j=0^p f_i,j(τ)(c/cτ+d)^j
    i.e.,      f_i(aτ+b/cτ+d)=∑_j=0^p f_i,j(τ) c^j  (cτ+d)^k_i-j

for every ([ a b; c d ])∈Γ_i and τ∈ℍ. Here the functions f_i,0,f_i,1,…, f_i,p are the components of f_i, and in particular, f_i,0=f_i for all 1≤ i≤ r. Then we consider N:=∏_i=1^r N_i and S:={bN^2+1: b∈ℕ} such that for each b'=Nb^2+1∈ S, the matrix ([  1 bN;  N b' ])∈∩_i=1^rΓ_i. For all such matrices we have 

    f_i(τ+bN/Nτ+b')=∑_j=0^p f_i,j(τ) N^j (Nτ+b')^k_i-j    for all  1≤ i≤ r.

Suppose that ∑_i=1^r f_i=0. Then for τ∈ℍ and b'∈ S, we have 

    ∑_i=1^r f_i(τ+bN/Nτ+b')=0.

 From the above transformation formula for each f_i, we obtain

    ∑_i=1^r∑_j=0^p f_i,j(τ)  N^j  (Nτ+b')^k_i-j=0.

For a fixed τ∈ℍ, multiplying (<ref>) by (Nτ+b')^p', where p'=2·max{p,|k_1|,…,|k_r|},  we get 
    0   =∑_i=1^r∑_j=0^p f_i,j(τ)  N^j  (Nτ+b')^k_i+p'-j
       =∑_i=1^r N^k_i+p'∑_j=0^pf_i,j(τ)(τ+b'/N)^k_i+p'-j,

which holds for each b'∈ S. This shows that for any fixed τ∈ℍ, the polynomial 

    P(X)=∑_i=1^r N^k_i+p'∑_j=0^pf_i,j(τ)(τ+X)^k_i+p'-j∈ℂ[X]

has infinitely many roots, and hence P(X) is identically zero. Note that even if any of the k_j's are negative, the maximum exponent of X occurring in P(X) is p'+ k_r. Thus, the leading coefficient of P(X) is N^k_r+p'  f_r,0(τ). Since P(X)=0, we get f_r,0(τ)=f_r(τ)=0. This is true for any τ∈ℍ. Hence, f_r=0, which is a contradiction, proving the lemma.  


We are now ready to prove the main theorem in this context.

Let f a non-constant meromorphic modular form of non-zero weight. Let g and f be a modular function and a meromorphic quasi-modular form of positive depth and non-zero weight, respectively. Assume that f, g and f are of arbitrary level. Then the functions f, g and f are algebraically independent. 


Let f be of weight m and level N, g be of level M and f be of weight n and level N. Suppose that P∈ℂ[X,Y,Z] is such that P(f,g,f)=0. Recall that a product of a modular form of weight m and level N and a quasimodular form of weight n and level N is a quasimodular form of weight mn and level NN. Denote ℕ_0 = ℤ_≥ 0. By grouping the terms of P(f,g,f) of the same weight, we can rewrite it as

    P(f,g,f)=∑_k=-K'^K ∑_(r,s,t)∈ℕ_0^3
    mr+nt=k p_r,s,t   f^r g^s f^t ,     with    p_r,s,t∈ℂ.
 
As P(X,Y,Z) is a polynomial, p_r,s,t=0 for all but finitely many r, s and t. As noted above, for each k∈{-K',…,K} the inner sum in (<ref>) is a meromorphic quasi-modular form of weight k and level NMN. Thus, Lemma <ref> implies that

    ∑_(r,s,t)∈ℕ_0^3
    mr+nt=k p_r,s,t   f^r g^s f^t=0.

 for each -K'≤ k≤ K. 

 
Fix a k∈{-K',…,K} and consider the corresponding inner sum from (<ref>). If there exists an integer t_0≠ 0 such that p_r,s,t_0≠ 0 for at least one tuple (r,s,t_0)∈ℕ_0^3 satisfying mr+nt_0=k, then the term on the left in (<ref>)
is a meromorphic quasi-modular form of depth at least t_0. But the uniqueness of depth implies that the term on the left in (<ref>) is of depth 0. Hence, the coefficient p_r,s,t=0 when t≠ 0. If t=0, then mr=k, or in other words, r = k/m ∈ℤ. Thus, for all 0 ≠ k∈{-K',…, K} such that k/m∈ℤ, the relation in (<ref>) takes the form  

    ∑_s ∈ℕ_0 p_k/m,s,0  g^s = 0,

as f is not identically zero. If k=0, then clearly, r=0 and we get

    ∑_s ∈ℕ_0 p_0,s,0   g^s=0.

Since g is a non-constant modular function, g must have either a zero or a pole in ℍ. Say that the order of g at τ_0 is b ≠ 0. Then any function ∑_r=0^R c_r  g^r with not all c_r = 0 will have order bR ≠ 0 at τ_0. Thus, any combination of the form  ∑_r=0^R c_r  g^r cannot be identically zero. Thus, relations (<ref>) and (<ref>) imply that p_0,s,0=p_k/m,s,0=0 for all k and s. Hence P=0 and the theorem is proved.






§ PROOF OF MAIN RESULTS





 §.§ Values of modular functions when j(τ) ∉ℚ

Before proceeding with the proof of the main theorems, we establish the following intermediary observation.
 
    Let g∈ℱ_N,ℚ. If τ∈ℍ such that j(τ)∉ℚ, then g(τ) is non-zero.


By Theorem <ref>, we know that g satisfies a non-trivial polynomial over ℚ(j). On clearing denominators, we can assume that g satisfies the irreducible polynomial

    P(X)= ∑_r=0^m c_r(j)   X^r =∑_r=0^m ( ∑_s=0^d_rc_r s j^s )  X^r ∈ℚ[j][X],    m= P(X),    d_r= c_r(j),

Note that c_0(j) ∈ℚ[j] is not the identically zero polynomial. Now suppose τ∈ℍ is such that j(τ) is transcendental. If g(τ) is zero, then we get that c_0(j(τ))=0 which contradicts the transcendence of j(τ). This completes the proof. 



If g(τ)=0, then it follows from Lemma <ref> that j(τ)∈ℚ. If τ is a pole of g, we consider the modular function

    h(z):=1/g(z)∈ℱ_N,ℚ,

which vanishes at z=τ, and so j(τ)∈ℚ by Lemma <ref>.    


The proof of Theorem <ref>(b) is established in two parts. We first show that if j(τ) is transcendental, then g(τ) is as well. The converse implication is proved later.

 Let g∈ℱ_N,ℚ and τ∈ℍ be such that it is not a pole of g. If j(τ) is transcendental, then g(τ) is transcendental.


Suppose that g(τ)∈ℚ. Then the modular function 
    h(z):=g(z)-g(τ)
 belongs to ℱ_N,ℚ and vanishes at τ. This contradicts Lemma <ref>. Therefore, g(τ) is transcendental.





 §.§ Values of modular functions when j(τ)∈ℚ

From Theorem <ref>, we have 
    ℱ_N,ℚ=  ℚ({j,f_a : a∈ S}),  where S={(r/N,s/N): 0≤ r, s≤ N-1  and (r,s)≠ (0,0)}.

Hence, for any g∈ℱ_N,ℚ, the algebraic nature of g(τ) is determined by the numbers in the set {f_a(τ):a∈ S}.


For N=1, ℱ_1,ℚ = ℚ(j). Hence, for any g ∈ℱ_1,ℚ, j(τ) ∈ℚ implies that g(τ) ∈ℚ. For N>1, we have to study the nature of the values f_a(τ) for all a∈ S. We start by proving the following important lemma.

 If τ∈ℍ such that j(τ) is algebraic, then there exists a unique transcendental number (up to an algebraic multiples) ω_τ for which g_2(ω_τ L_τ) and g_3(ω_τ L_τ) are both algebraic numbers.   


Note that for any α∈ℂ^×,

    j(τ)=E_4(τ)^3/Δ(τ)=1728   g_2(L_τ)^3/Δ_0(L_τ)=1728  g_2(α L_τ)^3/Δ_0(α L_τ),

because of the homogeneity properties of the g_2, g_3 functions. Here L_τ=τ ℤ⊕ℤ and α L_τ= α τ ℤ⊕α ℤ. If we choose ω_τ such that ω_τ^4=g_2(L_τ), then g_2(ω_τ L_τ)=1. Given that j(τ) is algebraic, from (<ref>), we get that Δ_0(ω_τ L_τ) is also algebraic. Thus, the numbers g_2(ω_τ L_τ) and g_3(ω_τ L_τ) are both algebraic. Moreover, ω_τ is a period of ω_τ L_τ and both g_2(ω_τ L_τ) and g_3(ω_τ L_τ) are algebraic. By Corollary <ref> we conclude that ω_τ is transcendental.


To prove uniqueness, consider an arbitrary ω_τ'∈ℂ^× such that both g_2(ω_τ' L_τ) and g_3(ω_τ' L_τ) are algebraic. By homogeneity of g_2, we have 

    g_2(ω_τ' L_τ)= (ω_τ')^-4 g_2(L_τ)=:β∈ℚ.

Then, we get g_2(L_τ)=β· (ω_τ')^4 and hence ω_τ^4=β· (ω_τ')^4.
This completes the proof that ω_τ is unique up to algebraic multiples.


Now recall that

    f_a(τ):=g_2(L_τ) g_3(L_τ)/Δ_0(L_τ) ℘_L_τ(a_1τ+a_2),
 
where L_τ=τ ℤ⊕ℤ and a=(a_1,a_2)∈ S. Using homogeneity properties of the functions involved, we can rewrite this as follows:

    f_a(τ)=g_2(ω_τ L_τ)  g_3(ω_τ L_τ)/Δ_0(ω_τ L_τ) ℘_ω_τ L_τ(a_1ω_ττ+a_2ω_τ).


    If τ∈ℍ is such that j(τ)∈ℚ, then f_a(τ)∈ℚ for all a∈ S.


    Let a=( r/N,s/N)∈ S. Then we have

    f_a(τ)=g_2(ω_τ L_τ)  g_3(ω_τ L_τ)/Δ_0(ω_τ L_τ) ℘_ω_τ L_τ(r ω_τ τ/N+s ω_τ/N),

where ω_τ is chosen as in Lemma <ref> so that g_2(ω_τ L_τ),  g_3(ω_τ L_τ)∈ℚ.


Recall that both ℘_ω_τ L_τ(ω_τ τ/N) and ℘_ω_τ L_τ(ω_τ/N)  are algebraic by Corollary <ref>. Moreover, by the addition formula for Weierstrass ℘-function in Theorem <ref>, we get that 
    ℘_ω_τ L_τ(r ω_τ τ/N+s ω_τ/N)∈ℚ
 for all a=(r/N,s/N)∈ S. Thus, the number f_a(τ) is algebraic for all a∈ S.


Let g∈ℱ_N,ℚ and τ∈ℍ such that j(τ)∈ℚ. If τ is not a pole of g, then g(τ) is algebraic.


 From Lemma <ref>, we have   f_a(τ)∈ℚ for all a∈ S. Since 
 
    g ∈ℱ_N,ℚ=  ℚ({j,f_a |a∈ S})

 by Theorem <ref>, g(τ) is also algebraic. 



Propositions <ref> and <ref> together complete the proof of Theorem <ref>(b). We now prove the generalization of <cit.>.

Let f be a meromorphic modular form of weight k ∈ℤ. Define 
   
    h(τ):=f(τ)^12/Δ(τ)^k  ∀τ∈ℍ.

The function h ∈ℱ_N,ℚ because the Fourier coefficients of f and Δ are algebraic. Moreover, any τ∈ℍ is a zero or a pole of h if and only if it is a zero or a pole of f because Δ is non-vanishing and holomorphic on ℍ. Thus, if τ∈ℍ is a zero or pole of f, then h is non-constant and by Theorem <ref>(a), we deduce that j(τ)∈ℚ, and by Theorem <ref>(b), that either g(z) has a pole at z = τ or that g(τ)∈ℚ.





 §.§ Values of modular forms
 Recall that for any integer k and N, we denote M_k,N,ℚ, M_k,N,ℚ^w and M_k,N,ℚ^m to be the set of holomorphic, weakly holomorphic and meromorphic modular forms respectively of level N with algebraic Fourier coefficients at i∞. By Corollary <ref>, all elements of these sets have algebraic Fourier coefficients with respect to all cusps as well.



Let f ∈ M^m_k,N,ℚ. Consider the modular function 

    g(τ) := f(τ)^12/Δ(τ)^k.

If g is a non-zero constant, then f(τ) = c Δ(τ)^k/12 for some c ∈ℚ^× and all τ∈ℍ. Since 1728 Δ(τ) = E_4^3(τ) - E_6^2(τ), and E_4(τ) is algebraically independent with E_6(τ) for τ∈ℍ such that e^2 π i τ∈ℚ by Nesterenko's theorem <ref>, we deduce that f(τ) is transcendental. 


Now suppose that g is non-constant. Since f ∈ M_k,N,ℚ, g ∈ℱ_N,ℚ. Note that for τ_0 ∈ℍ such that e^2 π i τ_0 is algebraic, j(τ_0) is transcendental by Theorem <ref>. Such a τ_0 cannot be a pole of f. For if τ_0 is a pole of f, then τ_0 is a pole of g and by Theorem <ref>(a), j(τ_0) would be algebraic, leading to a contradiction. As τ_0 is not a pole of g, Theorem <ref>(b) implies that g(τ_0) is transcendental. Moreover, by Theorem <ref>, g(τ_0) is algebraic over ℚ(j(τ_0)). By Nesterenko's theorem, 

    Δ(τ_0) = E_4^3(τ_0) - E_6^2(τ_0)/1728     and      j(τ_0) = E_4^3(τ_0)/Δ(τ_0)

are algebraically independent. Therefore,

    trdeg_ ℚ  ℚ ( g(τ_0),  Δ(τ_0) ) = trdeg_ ℚ  ℚ ( j(τ_0),  Δ(τ_0) ) = 2.

Hence, f(τ_0)^12 = g(τ_0)  Δ(τ_0)^k is transcendental, proving the claim.




























































































































We remark here that the proof of Theorem <ref>(a) only requires Nesterenko's theorem and the structure of modular functions of higher level. One also immediately deduces the following. 

If f∈ M^m_k,N,ℚ, then f is algebraically dependent with E_4 and E_6 over ℚ. In particular, If f∈ M_k,N,ℚ^m and τ∈ℍ is not a pole of f, then f(τ) is algebraic over ℚ(E_4(τ),E_6(τ)). 


Let f ∈ M^m_k,N,ℚ and consider 

    g (τ) := f^12(τ)/Δ^k(τ)∈ℱ_N,ℚ.

By Corollary <ref>, there exists a polynomial P(X) ∈ℚ[j](X) such that P(g) = 0. More specifically,

    ∑_r=0^m ∑_s=0^d_r c_r,s j(τ)^s  g(τ)^r= 0     for all    τ∈ℍ.

Here c_r,s∈ℚ for all 0 ≤ s ≤ d_r and 0 ≤ r ≤ m. Multiplying by Δ^l(τ) for any positive integer l > km and substituting j(τ) and Δ(τ) in terms of E_4(τ) and E_6(τ) gives

    ∑_r=0^m ∑_s=0^d_r∑_t=0^l-k r-s(-1)^t c_r, s/1728^l-k r-s([ l-k r-s;       t ]) f(τ)^12  r E_4(τ)^3(l-k  r-t) E_6(τ)^2  t=0.

This proves the proposition.



We now consider the complementary case, namely, points τ∈ℍ such that j(τ)∈ℚ. 

Fix a τ∈ℍ such that j(τ)∈ℚ. From Lemma <ref>, we get a transcendental number ω_τ such that g_2(ω_τ L_τ) and g_3(ω_τ L_τ) are both algebraic. Moreover, we have the formulae

    E_4(τ)   =3/4π^4  g_2(L_τ)=3/4(ω_τ/π)^4 g_2(ω_τ L_τ),
     E_6(τ)   =27/8π^6 g_3(L_τ)=27/8(ω_τ/π)^6g_3(ω_τ L_τ).

From Nesterenko's Theorem <ref>, we know that at most one of E_4(τ) and E_6(τ) is algebraic. The above formulae imply that if ω_τ/π∈ℚ, then both E_4(τ) and E_6(τ) are algebraic, which is a contradiction. Hence, ω_τ/π is a transcendental number. Besides, we have 

    Δ(τ)=E_4(τ)^3-E_6(τ)^2/1728   =1/4^6(ω_τ/π)^12( g_2(ω_τ L_τ)^3-27 g_3(ω_τ L_τ)^2 )
       =1/4^6(ω_τ/π)^12Δ_0(ω_τ L_τ).

As g_2(ω_τ L_τ) and g_3(ω_τ L_τ) both are algebraic, the number Δ_0(ω_τ L_τ)∈ℚ∖{0}. Since ω_τ/π is transcendental and Δ(τ) is a non-zero algebraic multiple of (ω_τ/π)^12, we deduce that Δ(τ) is transcendental. 


Consider the modular function 
    g(τ):=f(τ)^12/Δ(τ)^k  ∀τ∈ℍ,
 which lies in ℱ_N,ℚ. Since j(τ)∈ℚ, Theorem <ref> implies that g(τ) is algebraic, say α. Thus, we get that

    f(τ)^12= α · Δ(τ)^k=α/4^6 · Δ_0(ω_τ L_τ)^k  (ω_τ/π)^12 k.

This shows that if f(τ) ≠ 0, 
 then it is a non-zero algebraic multiple of (ω_τ/π)^k and hence, is transcendental.






 §.§ Values of quasi-modular forms

Let M_k,ℚ^(p)(Γ) denote the set of all meromorphic quasi-modular forms of weight k and depth p(>0) for Γ with algebraic Fourier coefficients. We study their values at the points τ∈ℍ, where exactly one of e^2π iτ and j(τ) is algebraic.



By Theorem <ref>, we can write f in the form 
    f=∑_r=0^p f_r E_2^r,   f_r∈ M_k,N,ℚ.
 
Suppose that both e^2π iτ, f(τ)∈ℚ. By Corollary <ref>, we know that each number f_r(τ) is algebraically dependent with E_4(τ), E_6(τ). Thus, we get that E_2(τ) is algebraic over ℚ(E_4(τ), E_6(τ)). This implies that

    trdeg_ ℚ  ℚ( e^2 π i τ, E_2(τ), E_4(τ), E_6(τ) ) ≤ 2,

contradicting Nesterenko's theorem <ref>. This proves the claim.



To study values of quasi-modular forms in the complementary case (i.e., for j(τ)∈ℚ), we need the following lemma.

 Let τ∈ℍ be such that j(τ) is algebraic and ω_τ be the transcendental number determined in Lemma <ref> such that g_2(ω_τL_τ) and g_3(ω_τL_τ) are algebraic. Let η_2(ω_τ) := η_2(ω_τ L_τ) be the quasi-period 
 
    η_2( ω_τ L_τ) = ζ_ω_τL_τ(z+ω_τ) - ζ_ω_τL_τ(z).

 Then ω_τ/π and η_2/π are algebraically independent over ℚ. 


From the definition of the Weierstrass zeta-function, one gets
 ζ_ω_τ L_τ(z)= 1/ω_τ·ζ_L_τ(z/ω_τ). 
Using the definition of a quasi-period and the identity (<ref>), we obtain

    η_2= ζ_ω_τ L_τ(z+ω_τ)-ζ_ω_τ L_τ(z)= 1/ω_τ η_2(L_τ)=1/ω_τ G_2(τ)= 1/3·π^2/ω_τ E_2(τ).

Thus, we get the following formulae 

    E_2(τ)=3 ω_τ/πη_2/π,    E_4(τ)=3/4(ω_τ/π)^4 g_2(ω_τ L_τ),   E_6(τ)=27/8(ω_τ/π)^6g_3(ω_τ L_τ).

The above formulae imply that E_2(τ), E_4(τ) and E_6(τ) are algebraic over ℚ(ω_τ/π, η_2/π). Suppose that ω_τ/π, η_2/π are algebraically dependent. This implies that 
    trdeg_ ℚ  ℚ(  E_2(τ), E_4(τ), E_6(τ) )=1,
 which contradicts Nesterenko's Theorem <ref>.


Lemma <ref> and Theorem <ref> together allow us to describe values of quasi-modular forms explicitly at the points τ∈ℍ where the j-function is algebraic.

From Theorem <ref>, we have the expression 
    f(τ)=∑_r=0^p f_r(τ) E_2(τ)^r,  where  f_r∈ M^m_k,N,ℚ.

Writing the value f_r(τ) as in (<ref>) for each coefficient f_r, and using the above formula for E_2, we obtain

    f(τ)=∑_r=0^p c_r(ω_τ/π)^k-2 r(ω_τ/π·η_2/π)^r=∑_r=0^p  c_r (ω_τ/π)^k-r (η_2/π)^r,

where each c_r is an algebraic number. If  c_r ≠ 0 for some r satisfying 0 ≤ r ≤ p, then (<ref>) gives a non-trivial algebraic relation among ω_τ/π and η_2/π. This contradicts Lemma <ref>. Therefore, the number f(τ) is either zero, precisely when each f(τ)∈ℚ∖{0}, or is transcendental.     







 §.§ Algebraic independence of special values



From Theorem <ref>, we know that for any τ∈ℍ which is not a pole of g, g(τ) is algebraic over ℚ(j(τ)). Moreover, Corollary <ref> gives that f(τ) is algebraic over ℚ(E_4(τ), E_6(τ)). Since f(τ)=∑_r=0^p f_r(τ) E_2(τ)^r, we have that f(τ) is algebraic over 
    ℚ(E_2(τ), E_4(τ), E_6(τ)).
 Hence, we get that

    trdeg_ ℚ  ℚ(e^2 π i τ, g(τ), f(τ), f(τ) )     = trdeg_ ℚ  ℚ(e^2 π i τ, j(τ), E_2(τ), E_4(τ), E_6(τ) ) 
        = trdeg_ ℚ  ℚ(e^2 π i τ, E_2(τ), E_4(τ), E_6(τ) ) ≥ 3,

by Nesterenko's theorem <ref>. This establishes the claim.





§ ACKNOWLEDGMENT

The authors thank Prof. Michel Waldschmidt and Dr. Veekesh Kumar for insightful suggestions on an earlier version of the paper. They are also grateful to the referee for helpful comments.




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 Paper Title: On Hopkins' Picard group.



§ ABSTRACT 
    We compute the algebraic Picard group of the category of K(n)-local spectra, for all heights n and all primes p. In particular, we show that it is always finitely generated over _p and, whenever n ≥ 2, is of rank 2, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case n=p=2, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in p-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez–Dospinescu–Niziol.


[
    [
    
=====




  

    < g r a p h i c s >
 

René Magritte, The Voice of Space, 1931








§ INTRODUCTION




 §.§ Results

This article settles a question about the K(n)-local Picard group _n,p, relative to a prime p and an integer n≥ 1, formulated by Hopkins in the early 1990s (<cit.>). Here, K(n) denotes Morava K-theory at height n, the prime p being implicit.  Also known as Hopkins' Picard group, _n,p is the group of isomorphism classes of invertible K(n)-local spectra, under the operation of K(n)-local smash product.  The study of _n,p was initiated by Hopkins–Mahowald–Sadovsky <cit.>, who defined an approximation to _n,p in the form of a comparison map
    
    ε_n,p→_n,p^.

The target _n,p^ of this map is defined in a purely algebraic manner in terms of equivariant line bundles on Lubin–Tate space (reviewed below).  The map ε is an isomorphism for p large enough with respect to n, but generally it has a kernel κ_n,p which is a (possibly infinite) product of cyclic p-groups with bounded exponent.  Our main theorem gives a complete description of _n,p^ together with the image of ε (up to a /2 ambiguity in the case (n,p)=(2,2)). 

[<Ref>]  Let p be a prime, assume n≥ 2, and write 𝒵_n,p for the pro-cyclic group[This notation allows us to deal with all primes uniformly. When p is odd, 𝒵_n,p≅_p ⊕/(2p^n-2), while 𝒵_n,2≅_2 ⊕/(2^n-1).] lim_k /p^k(2p^n-2). 
There is an isomorphism

    _n,p^𝒵_n,p⊕_p  ⊕ (/2)^⊕ e(n,p),

where

    e(n,p)=
        
            0,     p≠ 2; 
    
            2,     p=2 and  n≥ 3; 
    
            3,     p=2 and  n=2.

In all cases except possibly (n,p)=(2,2), the map ε is surjective. 
In the case (n,p)=(2,2), the map ε is split with cokernel of order at most 2.


The case n=1 has long been understood, so we have excluded it from the statement of the theorem.  The computation for n=2 has also been known by work of Hopkins and Lader (p ≥ 5), Karamanov as well as Goerss–Henn–Mahowald–Rezk (<cit.> p=3), and Henn (p=2, unpublished). These computations rely on explicit resolutions such as those constructed in <cit.> in the case n=2 and p=3, and thus seem very difficult to extend to larger heights. Indeed, prior to the present work, there was not a single pair (n,p) with n≥ 3 for which _n,p^ had been computed. 

A precise version of <ref> appears as  <ref>, which in fact provides explicit topological generators for _n,p^. The case p=2 being rather delicate, for the moment we focus on what happens when p is odd.  As a profinite group, _n,p^𝒵_n,p⊕_p is generated by the images under ε of the following two classes in _n,p:  the suspended K(n)-local sphere Σ𝕊_K(n), and the (p-1)-fold power of the determinant sphere 𝕊_K(n)[], defined in <ref>.  (In the case n=1, 𝕊_K(1)[] and Σ^2𝕊_K(1) have the same image in _1,p^, and _1,p^𝒵_1,p is generated by the image of Σ𝕊_K(1).) Combined with the known isomorphism range of the comparison map ε, for all heights and generic primes p, we obtain a full computation of the topological K(n)-local Picard group, stated here only for the previously unknown cases:

[<Ref>]
     Let n > 2 and p>(n^2+1)/2. There is an isomorphism 
        
    _n,p≅_p ⊕_p ⊕/(2p^n-2).

   As a profinite group, _n,p is freely generated by Σ𝕊_K(n) and 𝕊_K(n)[]^⊗ (p-1).


<Ref> can be reduced to a purely algebraic result concerning deformations of formal groups, as we now explain.  Let H_n be a 1-dimensional formal group of height n over _p, and define the Lubin–Tate ring A_n as the deformation ring of H_n.  Then A_n is non-canonically isomorphic to a power series ring in n-1 variables over the ring of Witt vectors of _p.  Finally, define the Morava stabilizer group _n as the group of automorphisms of the pair (H_n,_p).  Then _n is a profinite group acting continuously on the topological ring A_n.  

The ring A_n admits a spectral incarnation E=E(H_n,_p), known as Morava E-theory or the Lubin–Tate spectrum.  This is a K(n)-local commutative ring spectrum with homotopy ring E_* = A_n[u^± 1] for a class u of degree -2.  The spectrum E is constructed functorially from the pair (H_n,_p), so that it admits an action of _n.  The completed E-homology of a spectrum is a graded E_*-module which is derived complete with respect to the maximal ideal of π_0 E=A_n, and which admits a continuous action of _n lying over the action on E_*.  We call such a structure a Morava module.

The algebraic Picard group _n,p^ is defined as the group of isomorphism classes of invertible Morava modules. The comparison map ε appearing above is given by applying completed E-homology to invertible K(n)-local spectra. The algebraic Picard group sits in an exact sequence

    0 → H^1_(_n,A_n^∗)→_n,p^→/2→ 0.

This is due to the fact that E_* is 2-periodic, so an invertible Morava module is supported in either odd or even degrees, giving the map to /2;  since the shift E_*[1] is supported in odd degrees, the map to /2 is surjective.  The kernel of this map is isomorphic to the group of isomorphism classes of invertible A_n-modules carrying an equivariant continuous action of _n.  Since A_n is a local ring, its Picard group is trivial.  Equivariant continuous actions of _n on a free A_n-module of rank 1 correspond exactly to classes in the continuous cohomology H^1_(_n,A_n^∗), where A_n^∗ is the group of units in A_n.  

The exact sequence (<ref>) allows us to reduce <ref> to a computation of H^1_(_n,A_n^∗).  
Once again we assume n≥ 2, the case n=1 being already well-understood.
 

    
    Assume n≥ 2.  There is an isomorphism
    
    H^1_(_n,A_n^∗)_p^⊕ 2⊕/(p^n-1) ⊕ (/2)^⊕ e(n,p),

    where e(n,p) is as in <ref>.    

<Ref> implies the description of _n,p^, at least up to the question of which extension of /2 by H^1_(_n,A_n^∗) is represented by (<ref>).  
What is more, we show in the course of proving <ref> that every element of H^1_(_n,A_n^∗) (or rather an index 2 subgroup of it, in the case n=p=2) lifts to an element of _n,p of even degree;  this implies the properties of ε claimed by <ref>.  

Let A_n^⊂ A_n^∗ be the subgroup of principal units (i.e., topologically unipotent elements).  
It is convenient to divide <ref> into pro-p and prime-to-p parts using the _n-equivariant decomposition A_n^∗ A_n^⊕_p^∗.  The prime-to-p part poses no difficulty (<Ref>), so <ref> is reduced to the claim H^1_(_n,A_n^)_p^⊕ 2⊕ (/2)^⊕ e(n,p).  We prove a precise statement (<Ref>) about H^1_(_n,A_n^), and show that it implies <Ref>.   




 §.§ Main ideas of the proof


An even invertible Morava module (meaning a class in _n,p^ which maps to the identity in /2 in (<ref>)) corresponds to a free A_n-module with a continuous equivariant _n-action.  We may view such modules geometrically, as line bundles on the stack [_n/_n], where _n= A_n is the Lubin–Tate deformation space, i.e., the affine formal scheme parameterizing deformations of the formal group H_n.  
The stack [_n/_n] arises naturally;  it is isomorphic to the completion of the moduli space of formal groups at the height n point corresponding to H_n (see <ref>).  

To access the Picard group of [_n/_n], we rely heavily on Faltings' isomorphism between the Lubin–Tate and Drinfeld towers, as interpreted as a relation between perfectoid spaces <cit.>.  This isomorphism is only available after passage to the generic fiber.  Therefore let K be the fraction field of the ring of Witt vectors of _p, and let _n,K be the adic generic fiber of _n, so that _n,K is isomorphic to a rigid-analytic (n-1)-dimensional open ball over K.  

In <ref> we interpret Faltings' isomorphism as a relation between stacks on Scholze's category of diamonds:

    [_n,K^♢/_n]  [^n-1,♢/_n(_p)].

On the left side of the isomorphism is the “diamond generic fiber” of the stack [_n/_n].  Appearing on the right side is (the diamond version of) Drinfeld's symmetric space ^n-1, defined as the complement in rigid-analytic ^n-1__p of all _p-rational hyperplanes.  

The isomorphism (<ref>) is indispensable for the proof of <ref>, because it trades the opaque action of _n on A_n for the transparent (indeed linear) action of _n(_p) on ^n-1.  We offer a summary of this proof.  Recall we have already reduced 
<ref> to the calculation of H^1_(_n,A_n^).  In particular, we want to show that for n≥ 2, the torsion-free part of H^1_(_n,A_n^) is _p^⊕ 2, generated by the pro-p parts of the even Morava modules ε(Σ^2𝕊_K(n)[]) and ε(𝕊_K(n)[]).  The proof brings together two important constructions, accounting for each of these two generators.

The first construction involves determinants of formal groups.  The idea is that in certain contexts, it is possible to take the determinant of a height n formal group to obtain a height 1 formal group.  See <ref> for the precise statement.  One manifestation of this phenomenon is that
there is a determinant homomorphism _n→_1 which is compatible with the natural inclusion map A_1→ A_n;  as a result there is an induced map  

    _^* H^1_(_1,A_1^)→ H^1_(_n,A_n^).

The source of this map can be calculated directly,

    H^1_(_1,A_1^) _p,     p≠ 2; 
    _2⊕ (/2)^⊕ 2,     p= 2,

and the image of the generator of the free part of this group under _^* is the pro-p part of ε(𝕊_K(n)[]), the Morava module of the determinant sphere.

The second construction is the one that leverages (<ref>).  In <ref> we define a sheaf  of principal units on an adic space, and we observe that A_n^∗∗ H^0(_n,K,^∗∗).  Let C be the completion of an algebraic closure _p of _p, and let _C^n-1 be the base change of ^n-1 to C.  Another version of (<ref>) is 

    [_n,K^♢/_n]  [^n-1,♢_C/((_p/_p)×_n(_p))].

As a formal consequence of this isomorphism, we obtain a map 

    b  H^1_(_n,A_n^)→ H^1_(_C^n-1,)^(_p/_p)×_n(_p).

The following theorem appears as <ref> combined with <ref>.


     Assume (n,p)≠ (2,2).  The maps _^* and b fit into an exact sequence
    
    0→ H^1_(_1,A_1^)  H^1_(_n,A_n^)b→ H^1_(_C^n-1,)^(_p/_p)×_n(_p)→ 0.

If (n,p)=(2,2), there is a similar exact sequence, except the first nonzero term in the sequence is H^1_(_1,A_1^)⊕/2.

Interestingly, the reason that (n,p)=(2,2) is treated differently in <ref> traces back to the fact that it is the only case where there are nontrivial continuous homomorphisms _n(_p)→_p^ (see <ref>).  

 
At this point we apply the recent advances of Colmez–Dospinescu–Niziol <cit.> on the cohomology of _C^n-1, which seem to be tailor-made for our purposes.  From their work we deduce (see <ref>) that if n≥ 2, then 
H^1_(_C^n-1,)^(_p/_p)×_n(_p) is canonically isomorphic to _p .  The final challenge (<ref>) is to show that the map b carries the pro-p part of ε(Σ^2𝕊_K(n)) onto a generator of this _p, so that b is surjective.  

Combining <ref> with (<ref>) shows that H^1_(_n,A_n^)_p^⊕ 2⊕ (/2)^⊕ e(n,p), which completes the proof of <ref>.  





 §.§ Acknowledgements


We thank Agnès Beaudry, Pierre Colmez, Ehud de Shalit, Shai Evra, Paul Goerss, Drew Heard, Hans-Werner Henn, Mike Hopkins, Wiesława Nizioł, Ori Parzanchevski, and Peter Scholze for many helpful discussions. 

TB is supported by the European Research Council (ERC) under Horizon Europe (grant No. 101042990) and would like to thank the Max Planck Institute for its hospitality. NS and TMS were supported by the US-Israel Binational
Science Foundation under grant 2018389. NS was supported by a Sloan research fellowship, NSF Grants DMS-2304781, and a grant from the Simons Foundation (MP-TSM-00002836, NS).  TMS was supported by ISF1588/18 and the
ERC under the European Union’s Horizon 2020 research and innovation program (grant agreement
No. 101125896)









§ THE K(N)-LOCAL PICARD GROUP


The purpose of this section is two-fold. First, it reviews some background material on Picard groups in chromatic homotopy theory. We then explain how to reduce the determination of Hopkins' Picard group to a cohomological computation (<ref>), which will then be tackled in the remainder of this paper. 



 §.§ Motivation: invertible spectra and degree


Let = (,⊗,) be a symmetric monoidal ∞-category. The Picard group () of  is defined as the collection of isomorphism classes of ⊗-invertible objects in , i.e., those objects X ∈ for which there exists y ∈ with x ⊗ y ≃. If  is presentably symmetric monoidal, then () is a set <cit.>. The symmetric monoidal product on  then equips () with the structure of a group, with unit .

If = (,⊗,𝕊) is the symmetric monoidal ∞-category of spectra, then we can classify invertible objects via their degree. Indeed, for any X ∈(), the Künneth isomorphism implies that H_*(X;) is concentrated in a single degree, which we define as (X). A Postnikov tower argument <cit.> then shows that the induced map
    
    () [r]

is a bijection, with inverse n ↦𝕊^n. In particular, () is generated by Σ𝕊.

Hopkins, Mahowald, and Sadofsky observed that the situation becomes more interesting when we turn to local stable homotopy theory. This article concerns the invertible K(n)-local spectra.  Here K(n) denotes Morava K-theory of height n at the (fixed) prime p. This is an associative ring spectrum with coefficients 
    
    π_*K(n) ≅   if  n = 0; 
    _p[v_n^± 1]    if  0 < n < ∞; 
    _p    if   n = ∞,

with v_n of degree 2(p^n-1). By convention, K(0) = H independently of p, while K(∞) = H_p. Our focus in this paper will be on the case of intermediate characteristic 0 < n < ∞, and we will often keep the ambient prime p implicit in the notation. The Morava K-theories are fields in , i.e., they obey a Künneth formula 
    
    K(n)_*(X ⊗ Y) ≅ K(n)_*(X) ⊗_K(n)_* K(n)_*(Y)

for any X, Y ∈, so that every module spectrum over K(n) is free. In fact, as a consequence of the nilpotence theorem due to Devinatz, Hopkins, and Smith <cit.>, this provides a full classification of all prime fields in . For more background information on chromatic homotopy theory, we refer the reader to <cit.>.

Bousfield localization constructs the category _K(n) of K(n)-local spectra, i.e., the universal localization (-)_K(n)→_K(n) which inverts all maps f of spectra for which K(n) ⊗ f is an equivalence. Localization admits a fully faithful right adjoint _K(n)⊂, so we may view a K(n)-local spectrum as a spectrum. The smash product on  descends to a symmetric monoidal structure on _K(n); explicitly, the product of two K(n)-local spectra X,Y is given by 
    
    X ⊗̂ Y = (X⊗ Y)_K(n).

Equipped with this structure, _K(n) = (_K(n), ⊗̂,𝕊_K(n)) forms a presentably symmetric monoidal stable ∞-category.


    Fix a height n≥ 1 and an implicit prime p. We define the K(n)-local Picard group as 
        
    _n,p(_K(n)).



By (<ref>), the Morava K-homology K(n)_*(X) of any invertible X ∈_K(n) is concentrated in degrees congruent to d modulo 2(p^n-1) for some 0 ≤ d < 2(p^n-1). Following the global story above, we define this number as the local degree of X: _n,p(X)  d. The next result is proven in <cit.>, relying on <cit.>.


    Let n≥ 1. The K(n)-local degree provides a surjective map 
        
    _n,p_n,p@->>[r]    /2(p^n-1)

    with kernel an infinite abelian pro-p-group. 


In particular, in contrast to the case of (), the naive notion of degree in the K(n)-local setting is only a rather coarse approximation to the Picard group. In the next subsection, we will review a refined invariant which provides a much closer algebraic approximation to _n,p.




 §.§ Morava modules


As the starting point for the construction of a good algebraic approximation to _n,p, we take the following fundamental result of Devinatz–Hopkins <cit.>. A general account of the Galois theory of commutative ring spectra was developed by Rognes <cit.>.  

Let H_n be a 1-dimensional formal group of height n over _p.  Let E_n be Morava E-theory of height n associated with H_n. This a K(n)-local commutative ring spectrum with
    
    π_*E_n = A_n[u^± 1], u=-2.

Here π_0E_n=A_n is the deformation ring of H_n, known as the Lubin–Tate deformation ring.  There is a (non-canonical) isomorphism A_n  W(_p) u_1,…, u_n-1, with W(_p) the ring of Witt vectors of _p.  Let H_n^ be the universal deformation of H_n;  then π_-2E_n= H_n^=A_nu is the Lie algebra of H_n^. The spectrum E_n admits a continuous action of the Morava stabilizer group _n=(H_n,_p).  


    The unit map 𝕊_K(n)→ E_n witnesses the target as a K(n)-local Galois extension with Galois group _n. In particular, there is a canonical equivalence 𝕊_K(n)≃ E_n^h_n.


We can twist the equivalence of this theorem by any X ∈_K(n), resulting in a new equivalence X ≃ (E_n ⊗̂ X)^h_n, where X is equipped with trivial _n-action. This in turn gives rise to a (Bousfield–Kan) descent spectral sequence of signature (see <cit.>)
    
    E_2^s,t(X):= H^s(_n,E_n,t(X))π_t-sX

that has excellent convergence properties. Indeed, since 𝕊_K(n)→ E_n is a descendable morphism of commutative ring spectra by the smash product theorem <cit.> (see also <cit.>), the spectral sequence converges strongly and collapses on a finite page with a horizontal vanishing line. When p-1 does not divide n, the stabilizer group _n is p-torsion free and thus has finite cohomological dimension (as opposed to merely finite virtual cohomological dimension), in which case (<ref>) collapses already on the E_2-page. 

The input to the spectral sequence (<ref>) is the continuous _n-cohomology of E_*(X) = π_*L_K(n)(E_n ⊗ X), which by a mild abuse of notation here refers to the completed E-homology of X. This structure is captured algebraically in the category of Morava modules, _E_*^↺_n, defined as follows: Let m= (p,u_1,…, u_n-1) ⊂π_0E_n be the maximal ideal of π_0E_n. A Morava module is a derived 𝔪-complete graded E_*-module equipped with a continuous semi-linear action of _n. Morphisms of Morava modules are continuous equivariant maps. The category of Morava modules can be modeled as (π_*E_n,E_*E)-comodules whose underlying E_*-module is derived m-complete. Completing the usual tensor product induces a symmetric monoidal structure on _E_*^↺_n, denoted ⊗̂. With these definitions, we obtain a canonical lift 
    
    _E_*^↺_n[d]^U 
    _K(n)[r]_-E_*(-)[ru]^-E_*(-)   _E_*,

where U denotes the evident forgetful functor, which forgets the action by _n on the given Morava module. Note that E_*(-) is in general not symmetric monoidal. However, when restricted to spectra X with E_*(X) projective (as E_*-module), it is symmetric monoidal. In particular, this applies to invertible K(n)-local spectra, in light of the following characterization:


    For X ∈_K(n), the following conditions are equivalent:
        
            
  * X is invertible;
            
  * E_*(X) is free of rank 1 as a E_*-module;
            
  * K(n)_*(X) is free of rank 1 as a K(n)_*-module.
        



    The algebraic Picard group _n,p^ is defined as the group of isomorphism classes of ⊗̂-invertible Morava modules. The Morava module functor (<ref>) then induces a comparison map
        
    ε_n,p⟶_n,p^(_E_*^↺_n),    X ↦ E_*(X).



The comparison map ε turns out to be a close approximation to the topological Picard group, essentially due to the excellent properties of the descent spectral sequence (<ref>). The next result summarizes the situation:



    Let p be prime such that 2p-2 > max{n^2-1,2n}, then the comparison map is bijective:
        
    ε_n,p_n,p^.
 


The injectivity part of this result was established in <cit.>, while surjectivity was proven under slightly stronger hypothesis in <cit.>; the version given here is due to Goerss and Hopkins, <cit.>.

In general, however, the map ε is not injective; non-trivial elements in the kernel are called exotic. This can only occur in the non-algebraic realm, i.e., when the descent spectral sequence (<ref>) for X ∈_K(n) does not collapse for cohomological reasons. 


    The kernel of ε is the exotic Picard group κ_n,p, fitting into an exact sequence
        
    0 [r]    κ_n,p[r]    _n,p[r]^-ε   _n,p^.




    For any n and p, the group κ_p,n is an abelian pro-p group of finite exponent.  It is pro-p as a consequence of <ref>.  The finiteness of its exponent can be seen as follows: Consider the descent spectral sequence (<ref>). Since it collapses at a finite page with a horizontal vanishing line, uniformly for any X ∈_n,p, there is an induced descent filtration on κ_n,p with subquotients E_r^r,r-1(𝕊), see for example <cit.>. By <cit.>, the latter groups are all p-torsion of finite exponent, hence so is κ_n,p. It follows from the Prüfer–Baer theorem that κ_p,n is then a (possibly infinite) product of finite cyclic p-groups. For odd p, this has already been observed by Heard <cit.>.  





 §.§ Two distinguished spheres


We discuss here the two distinguished classes in the local Picard group _n,p which make an appearance in <ref>.

The first of these classes is simply the suspension Σ𝕊_K(n).   The Morava module E_n,*(Σ𝕊_K(n)) is odd, but its tensor square E_n,*(Σ^2𝕊_K(n)) is
    
    ε(Σ^2𝕊_K(n))=π_-2E_n ≅ H_n^,

where H_n^ is the Lie algebra of the the universal deformation H_n^ of the height n formal group H_n. Since K(n)_*(Σ^2p^n-2𝕊_K(n)) ≅ K(n)_*, the local degree of  Σ^2p^n-2𝕊_K(n) is zero. In light of <ref>, this implies that the assignment k ↦Σ^(2p^n-2)k𝕊_K(n) extends to a homomorphism
    
    ι__p [r]    _n,p[r]    _n,p^.

In fact, we can do better, extending to a larger procyclic group generated by Σ^1𝕊_K(n). To do so, we will introduce an auxiliary piece of notation (inspired by <cit.>) that allows us to treat the cases of odd and even primes uniformly:


    For each prime p and height n, we set _n,plim_k /(p^k(2p^n-2)).



    For any n ≥ 1 and prime p, the group _n,p is procyclic, isomorphic to
        
    _n,p≅_p ⊕/2(p^n-1)    if  p>2; 
    _2 ⊕/(2^n-1)    if  p=2.

    For example, if p is odd, a generator for the torsion summand in this presentation is given by the element lim_k p^nk. 


Based on the periodicity theorem <cit.>, Hopkins, Mahowald, and Sadofsky <cit.>, one can use “v_n-adic interpolation” to construct an extension of ι_, which we denote by the same symbol:
    
    4pcι__n,p[r]^-1 ↦Σ^1𝕊_K(n)   _n,p^.

Here, the map _p →_n,p is induced by maps /p^k →/(p^k(2p^n-2)) determined by 1 ↦ 2p^n-2. In particular, for p odd, this splits the presentation of <ref>. However, for p=2, this embedding includes _2 as an index 2 subgroup of the _2 appearing in <ref>.

The second important class is the determinant sphere.  Let 𝕊(1) be the p-complete sphere with its natural continuous _p^∗-action. The stabilizer group then acts on 𝕊(1) through the determinant map _n →_p^∗. The determinant sphere is defined as 
    
    𝕊_K(n)[]  (E_n ⊗𝕊(1))^h_n,

where _n acts diagonally on the p-completed smash product E_n ⊗𝕊(1). As shown in <cit.>, the determinant sphere topologically realizes the Morava module A_n[], meaning the free A_n-module of rank 1 where the _n-action has been twisted by .  Thus:
    
    ε(𝕊_K(n)[]) = A_n[].



    The significance of having a concrete topological model for the determinant sphere comes from Gross–Hopkins duality, which describes the Morava module of the dualizing module for the K(n)-local category. The latter is by definition the Brown–Comenetz dual of the n-th monochromatic layer of the sphere M_n𝕊 = (L_n𝕊→ L_n-1𝕊), denoted I_n  IM_nS^0. Here, Brown–Comenetz duality I is a lift of Pontryagin duality to the category of spectra. Gross and Hopkins <cit.> prove that I_n ∈_n,p with Morava module E_*(I_n) ≃π_n-n^2E_n[], see also <cit.>. We may then restate this algebraic identification as a K(n)-local equivalence 
        
    I_n ≃Σ^n^2-n𝕊_K(n)[] ⊗̂ P_n,p,

    where P_n,p∈κ_n,p. This formula is a source for the construction of exotic elements in the Picard group, see for example <cit.>. 





 §.§ Conjectures and results


The following conjecture, dating from the early 1990s and due to Hopkins, summarizes the expected behaviour of the sequence (<ref>):


    For all heights n ≥ 1 and primes p:
        
            
  * κ_n,p is a finite p-group;
            
  * _n,p^ is finitely generated over _p.
        


Taken together, the two parts of the conjecture imply that _n,p is finitely generated over _p. In fact, this is almost an `if and only if' statement, up to the following


    Is the map ε in (<ref>) surjective?


<Ref> has been confirmed at heights 1 and 2 and all primes p through explicit computations; the following table summarizes the state of the art before the present work:


The main result of the present paper is a complete calculation of the algebraic Picard group of _K(n), for all heights n and all primes p; in particular, we recover the the computations of _n,p^ for n≤ 2 summarized in the table above. In addition, we provide explicit topological elements which under the comparison map ε generate the algebraic Picard group with the possible exception of the case n=p=2. The following is <ref> from the introduction:


    For any prime p and any height n≥ 2, the algebraic Picard group of the category of K(n)-local spectra is given by:
    
    _n,p^≅
                    
                    _n,p⊕_p    if   p>2  and  n ≥ 2;
    _n,2⊕_2 ⊕ (/2)^⊕ 2   if   p=2   and  n>2; 
    _n,2⊕_2 ⊕ (/2)^⊕ 2⊕/2    if   p=2   and  n=2,

    with _n,p as in <ref>. For all primes p, the summands _n,p and _p are generated by ε(Σ^1 𝕊_K(n)) and ε(𝕊_K(n)[]^⊗ (p-1)), respectively. For p=2, generators for the additional (/2)^⊕ 2 summand are described in <ref>. 


In the range where the comparison map ε is an isomorphism (<ref>), we thus obtain a complete computation of the Picard group of _K(n); this is <ref>.


    When 2p-2 > max(n^2-1,2n), there is an isomorphism 
        
    _n,p≅_p ⊕_p ⊕/(2p^n-2),

    with generators given by Σ𝕊_K(n) and 𝕊_K(n)[]^⊗ (p-1).


Another direct consequence of <ref> is the surjectivity of the comparison map in all cases except for possibly the anomalous one when n=p=2; see also <ref>.


    For all heights n ≥ 1 and all primes p with the possible exception of n=p=2, the comparison map ε_n,p→_n,p^ is surjective.


The goal of the remainder of this section is to reduce <ref> to a computation in p-adic geometry, stated as <ref> below, which we then carry out in the rest of this paper. 






 §.§ The even algebraic local Picard group


We now begin our reduction of <ref> to a purely cohomological statement. Observe that the K(n)-local Picard group and its algebraic counterpart are naturally /2-graded, since E_* is concentrated in even degrees and is 2-periodic. Let _n,p^,0 be the even algebraic Picard group, consisting of those invertible Morava modules whose underlying E_*-module is even, i.e., concentrated in even degrees.  Thus there is a short exact sequence
    
    0 [r]    _n,p^,0[r]    _n,p^[r]    /2 [r]     0.



The even algebraic Picard group has a simple description in terms of continuous group cohomology. 
Let M ∈_n,p^,0 be an even invertible Morava module.  We can think of M as an invertible A_n-module with continuous _n-action.  Since A_n is a local ring, the underlying A_n-module of M is free of rank 1, say with generator m.  Define a continuous 1-cocycle ϕ_M_n→ A_n^∗ by
    
    g · m = ϕ_M(g)m

for any g ∈_n. The class of ϕ_M in H^1_(_n,A_n^∗) does not depend on the chosen generator.  Conversely, a continuous cocycle in H^1_(_n,A_n^∗) determines an even Morava module by the same formula, and we obtain:


    The assignment M ↦ϕ_M induces an isomorphism
        
    _n,p^,0≅ H^1_(_n,A_n^∗).




    Let _n,p^0 be the even part of the local Picard group, i.e., those invertible K(n)-local spectra X with even Morava module E_*(X). We can translate the Morava modules of the two distinguished spheres of <ref> into a 1-cocycle description. Indeed, for each prime p and height n, under the composite 
        
    _n,p^0 [r]^-ε   _n,p^,0[r]^-ϕ_-≅    H^1_(_n,A_n^∗)

    we have:
        
            
  * since ε(Σ^2𝕊_K(n)) ≅ H, it maps under ϕ to the class of the cocycle t_0_n  → A_n^∗, where t_0 is determined by the formula g· u=t_0(g)u for g∈_n and u a generator of π_-2E_n= H.  

            
  * by construction, 𝕊_K(n)[] maps to the composition of the determinant map _n →_p^∗ with the inclusion map _p^∗→ A_n^∗.  
        


In order to better understand these elements and in particular their relation, it will be useful to separate the prime-to-p part of _n,p^,0 from its principal part. Recall that we consider Morava E-theory based on _p. As a topological ring, 
    
    A_n Wu_1,…,u_n-1,

where W=W(_p) is the ring of Witt vectors. The ring A_n is endowed with the m-adic topology, where m=(p,u_1,…,u_n-1) is the maximal ideal. The quotient map A_n→ A_n/m_p admits a multiplicative splitting via the Teichmüller lift _p→ W.  Thus there is a _n-equivariant isomorphism of topological groups:
    
    A_n^∗ A_n^∗∗⊕_p^∗,

where A_n^∗∗=1+m.  We call A_n^∗∗ the group of principal units; principal units will appear in various contexts throughout the article;  we will always use the double-star notation to refer to them. 

Accordingly, the even algebraic Picard group decomposes into prime-to-p and principal parts:
    
    _n,p^,0 H^1_(_n,_p^∗)⊕ H^1_(_n,A_n^∗∗).

With respect to this decomposition, let us write 
    
    ε = ε^p ⊕ε_p,

where ε^p, resp. ε_p, refers to the composition of ε with the projection onto the prime-to-p part, resp. the principal part. 



 §.§ The prime-to-p part and a relation


We begin by determining the prime-to-p part of _n,p^,0. 


    There are isomorphisms 
        
    H^1_(_n,_p^∗)_(_p/_p)(_p^n^∗,_p^∗) ≅/(p^n-1).



    For the proof, let us abbreviate Γ=(_p/_p).  We write σ for the Frobenius automorphism, so that σ generates Γ as a profinite group.  We recall here some facts about the structure of the Morava stabilizer group _n. It is an extension
        
    0 →_D^∗→_n→Γ→ 0,

    where _D= H_n is the ring of integers in a central simple algebra D of invariant 1/n over _p.  The ring _D has a unique two-sided maximal ideal J, with quotient ring _p^n. Let _D^∗∗=1+J be the group of principal units;  then _D^∗∗ is a normal pro-p subgroup of _n, with quotient
        
    _n/_D^_p^n^∗⋊Γ.

    The pro-p subgroup _D^∗∗⊂_n acts trivially on _p^∗, on which p acts invertibly, so we have an isomorphism for all i≥ 0:
        
    H^i_(_n,_p^∗) H^i_(_p^n^∗⋊Γ, _p^∗).

    Consider the Lyndon–Hochschild–Serre spectral sequence
        
    H^i_(Γ,H^j(_p^n^∗,_p^∗))  H^i+j_(_n,_p^∗).

    Since Γ is pro-cyclic, its cohomology vanishes in degrees ≥ 2, so we have a short exact sequence
        
    0→ H^1_(Γ,_p^∗)→ H^1_(_n,_p^∗) → H^1(_p^n^∗,_p^∗)^Γ→ 0.

    The term  H^1_(Γ,_p^∗) is the cokernel of σ-1 on _p^∗, which is 0.  The term H^1(_p^n^∗,_p^∗)^Γ is the group of Γ-equivariant homomorphisms _p^n^∗→_p^∗.  This is a cyclic group of order p^n-1, generated by the inclusion map _p^n^∗↪_p^∗.


Next, we can determine the fate of Σ^2𝕊_K(n) and 𝕊_K(n)[] in the prime-to-p part.

 

As a cyclic group of order p^n-1, H^1_(_n,_p^∗) is generated by ε^p(Σ^2𝕊_K(n)). Furthermore, the following relation holds in H^1_(_n,_p^∗):
    
    ε^p(𝕊_K(n)[])=p^n-1/p-1ε^p(Σ^2𝕊_K(n)).


  
We use the identification of <ref> and claim that this group is generated by the prime-to-p part of ε(Σ^2𝕊_K(n)).  Recall from <ref> that ε(Σ^2𝕊_K(n)) is the Morava module H_n^, where H_n^ is the universal deformation of the (unique up to isomorphism) formal group H_n  H_n^⊗_A_n_p of height n over _p.  The prime-to-p part of ε(Σ^2𝕊_K(n)) is the class in H^1_(_n,_p^∗) represented by the _n-equivariant _p-vector space H_n^⊗_A_n_p H_n.  All we need to know about the _n-action on H_n is that the subgroup _D^∗ acts on it through the composition _D^∗→_p^n^∗↪_p^∗.  Tracing through the calculation of H^1_(_n,_p^∗) in the proof of <ref>, we see that it is generated by the class of H_n^.  

With respect to the isomorphism H^1_(_n,_p^∗)_Γ(_p^n^∗,_p^∗), the 
class ε^p(𝕊_K(n)[]) corresponds to the norm map _p^n^∗→_p^∗↪_p^∗.  The norm map is (p^n-1)/(p-1) times the inclusion map, giving the relation claimed in the second part of the proposition. 


The proposition also allows us to compute the local degree of 𝕊_K(n)[], as we shall explain next. First, we will denote the even Picard group of the K(n)-local category by _n,p^0; by definition, this is the subgroup of _n,p consisting of those elements that have even Morava module. Restricting the local degree from <ref> to the even part, it factors through the inclusion /(p^n-1) ⊆/2(p^n-1) determined by 1 ↦ 2. Therefore, we obtain the even local degree homomorphism
    
    _n,p^0_n,p^,0[r]    /(p^n-1).

We observe that the target is normalized so that _n,p^0ε(Σ^2𝕊_K(n)) = 1. Moreover, the quotient map A_n^∗→ (A_n/ m)^∗≅_p^∗ is _n-equivariant, and hence induces a reduction map on continuous cohomology, q H^1_(_n,A_n^∗) → H^1_(_n,_p^∗). The next result provides an alternative perspective on the even local degree:


    There is a commutative diagram:
        
    _n,p^0[r]^-_n,p^0[d]_ϕ_(-)∘ε   /(p^n-1) [d]^≅
    
                H^1_(_n,A_n^∗) [r]^-q    H^1_(_n,_p^∗).



    Recall that the kernel of the even local degree map is a pro-p-group by <ref>, while H^1_(_n,_p^∗) is a group of order prime to p, see <ref>. It follows that the composite q ∘ϕ_(-)∘ε     factors through _n,p^0, i.e., there is a commutative square
        
    _n,p^0[r]^-_n,p^0[d]_ϕ_(-)∘ε   /(p^n-1) [d] 
    
                H^1_(_n,A_n^∗) [r]^-q    H^1_(_n,_p^∗).

    We claim that the induced right vertical map is an isomorphism. Indeed, it suffices to verify that on a generator of /(p^n-1), which we may take to be _n,p^0(Σ^2𝕊_K(n)) = 1. As already observed in the proof of <ref>, we have that  q ∘ϕ_(-)∘ε(Σ^2𝕊_K(n)) = ε^p(Σ^2𝕊_K(n)) is the canonical inclusion _p^n^∗↪_p^∗. This is our preferred generator of the cyclic group H^1_(_n,_p^∗), which verifies the claim.



    The even local degree of 𝕊_K(n)[] is p^n-1/p-1. In particular, we have 
        
    ε^p(𝕊_K(n)[]^⊗ (p-1)) = 0.



    This follows from <ref> and <ref>.





 §.§ A reduction of <ref>


The next result, whose proof will occupy the remaining sections of this paper, describes the principal part of the even algebraic Picard group. Assuming it, we can then deduce <ref> up to some special features for the prime 2 that are treated separately in <ref>.


    Let p be a prime n ≥ 2 some height. There are isomorphisms
        
    H^1_(_n,A_n^∗∗) ≅_p^2     if  p>2; 
    _2^2 ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    For all primes p, the free summands are generated by the 1-cocycles t_0 and  of <ref>, respectively. For p=2, generators for the (/2)^⊕ 2-summand are described in <ref>. 



    By <ref>, <ref>, <ref>, and (<ref>), we have isomorphisms
        
    ψ_n,p^0_n,p^,0≅ H^1_(_n,A_n^∗) ≅_p^2 ⊕/(p^n-1)    if  p>2; 
    _2^2 ⊕/(p^n-1) ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕/(p^n-1) ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    Since the 2-torsion part for p=2 is dealt with separately in <ref> below, we will ignore these summands for the remainder of this proof. Keeping in mind <ref>, it follows from the identifications of the generators in <ref> together with <ref> that the group _p ⊕/(p^n-1) is generated by ε(Σ^2𝕊_K(n)), while the other _p summand is generated by ε(𝕊_K(n)[]). 

    In order to deduce <ref> from this description, it remains to solve the extension problem of (<ref>). To this end, recall from <ref> that we have constructed a map
        
    8pcψ_n,p_n,p⊕_p [r]^-(Σ^1𝕊_K(n),𝕊_K(n)[]^⊗ (p-1))   _n,p^,
    b
 
    whose two components send the distinguished generators of _n,p and _p to ε(Σ^1𝕊_K(n)) and ε(𝕊_K(n)[]^⊗ (p-1)), respectively. This map fits into a commutative diagram of short exact sequences
        
    0 [r]     (_p ⊕/(p^n-1)) ⊕_p [r]^-(2,1)[d]_ψ_n,p^0   _n,p⊕_p [r] [d]^ψ_n,p   /2 [r] [d]     0 
     
                0 [r]    _n,p^,0[r]    _n,p^[r]    /2 [r]     0,

    where the top left map is induced by multiplication by 2 resp. the identity map on the two cyclic summands. The left vertical map is injective with cokernel (/2)^⊕ e(n,p) for 
        
    e(n,p)=
                    
                        0,     p≠ 2;
    
                        2,     p=2 and  n≥ 3;
    
                        3,     p=2 and  n=2.

    Tracing ε(Σ^1𝕊_K(n)) in the right square, we deduce that the right vertical map is an isomorphism. Therefore, the snake lemma shows that ψ_n,p is an isomorphism  with cokernel (/2)^⊕ e(n,p), as desired. 






 §.§ Addenda and anomalies for p=2


The case of the even prime has some oddities, which we discuss separately in this subsection. The description of _n,p^,0 in <ref> implies that, at least if p≠ 2, the comparison map ε_n,p→_n,p^ is surjective, since _n,p^ is generated as a profinite group by the images of Σ𝕊_K(n) and 𝕊_K(n)[] under ε.  For completeness' sake we discuss the surjectivity of ε when p=2, referring to <ref> for the anomalous case n=p=2.


    The comparison map ε_n,2→_n,2^ is surjective if n>2.


    Let n>2. In light of <ref>, it remains to find K(n)-local spectra which realize the 2-torsion in _n,2^, i.e., to find topological lifts for the non-trivial elements in /2 ⊕/2  = _n,2^[2]. This has already been achieved in <cit.>; for the convenience of the reader, we sketch the construction. There is a cyclotomic (/8)^∗-Galois extension 𝕊_K(n)→𝕊_K(n)[ω_n^8], obtained from 𝕊_K(n)[B^n/8] by splitting off a certain idempotent. By <cit.>, the cofibers of the unit maps of the three /2-Galois subextensions of 𝕊_K(n)[ω_n^8] then give the three non-zero elements in _n,2^[2].     



    Combining <ref> and <ref> show that the comparison map ε is surjective in all cases except potentially when (n,p) = (2,2). We currently do not know if the generator of the third, “anomalous” /2-summand that arises in the algebraic Picard group _2,2^ can be realized topologically. This question has also been studied in unpublished work of Hans-Werner Henn. 





 §.§ A toy case: height 1


For completeness and because we will make use of the following lemma later in the proof of <ref>, we briefly include a discussion of the case n=1.


There are isomorphisms 
    
    H^1_(_1,A_1^) ≅_(_p^∗,_p^) ≅_p    if  p>2; 
    _2 ⊕ (/2)^⊕ 2   if  p=2.



Recall that A_1≅ W(_p), equipped with its canonical action by the Galois group (_p/_p) ≅. We first claim that the inclusion _p^→ A_1^ induces an equivalence _p^≅ (A_1^)^h. Indeed, taking ϕ to be a topological generator of , what we need to prove is that there is a short exact sequence
    
    0→_p^→ W(_p)^  W(_p)^→ 0.

Using the compatible p-adic filtrations on _p and A_1, it suffices to show that the corresponding sequence of  associated graded pieces 
    
    0→_p →_p _p → 0

is short exact, which is indeed the case. It follows that there are isomorphisms
    
    H^1_(_1,A_1^) H^1((A_1^)^h(_p^∗×))  H^1((_p^)^h_p^∗) _(_p^∗,_p^)

The calculation of _(_p^∗,_p^∗∗) is routine.



    The calculation in the proof of <ref> recovers the computation of the algebraic Picard group at height 1 due to Hopkins, Mahowald, and Sadofsky <cit.>. Indeed, combined with <ref> and <ref>, we get the even algebraic Picard group _1,p^0,. The extension problem (<ref>) is then resolved as in the proof of <ref>. We obtain isomorphisms
        
    _1,p^≅_p ⊕/(2p-2)    if  p >2; 
    _2 ⊕ (/2)^⊕ 2   if  p = 2.

    For odd primes, the algebraic Picard group is generated by ε(Σ^1 𝕊_K(n)); for the even prime, the same class generates the _2-summand. Explicit generators for the 2-torsion part of _1,2^ can again be constructed as in the proof of <ref>; the original reference is <cit.>. For p>2, the comparison map gives an isomorphism _1,p≅_1,p^, while the exotic part for p=2 is a /2, see <cit.>. 






§ MODULI OF FORMAL GROUPS


We have reduced our main topological result to an algebraic one, namely <ref>, which describes the continuous cohomology group H^1_(_n,A_n^).  Its proof will require a dive into the theory of moduli of formal groups, especially those modern aspects of the theory which meet p-adic geometry and especially perfectoid spaces and diamonds.  The star player in the proof is _^n, the completion of the moduli stack of formal groups along the height n locus in characteristic p.  It enters the picture via <ref>, which states that _^n is the quotient of the formal scheme A_n by _n.  

In this section we offer the reader an exposition of the theory of moduli of formal groups and present two results required for the proof of <ref>.   Neither result is entirely original;  our work here is more or less a direct adaptation of material appearing in <cit.> and <cit.>.  Both results concern the “diamond generic fiber” of _^n, which we call 𝔐_n.  In a nutshell, 𝔐_n classifies formal groups of height n over perfectoid spaces of characteristic 0.

The first result is <ref>, which gives an alternative description of 𝔐_n in terms of linear algebra objects (“modifications of vector bundles on the Fargues–Fontaine curve”).  Faltings' isomorphism between the Lubin–Tate and Drinfeld towers appears within <ref>.

The second result is <ref>, which leverages the linear-algebraic description of 𝔐_n to construct a determinant morphism 𝔐_n→𝔐_1.  This result states that if H is a 1-dimensional formal group of height n over a perfectoid space S, there exists a functorially associated 1-dimensional formal group ⋀^n H of height 1 over S, whose associated linear algebra objects (the Tate and Dieudonné modules) are the top exterior powers of those of H.  Ultimately this result will help us understand the class of the determinant sphere in H^1_(_n,A_n^).




 §.§ The stack _^n of height n formal groups


For an introduction to the theory of moduli of formal groups aimed at the algebraic topologist, see <cit.>.  Unless specified otherwise, all formal groups in this discussion have dimension 1.

Let _ be the moduli stack of formal groups on the category of schemes with the fpqc topology.  Then _ admits a uniformization by an affine scheme L, where L=[t_1,t_2,…] is the Lazard ring.  The scheme L parameterizes formal group laws, which carry a choice of coordinate T.  For each n∈, let [n](T)∈ LT be the multiplication-by-n series in the universal formal group law over L.

Fix a prime number p.  The base change _⊗_(p) 
admits a well-known height stratification by closed subsets _FG^≥ n for 0≤ n≤∞, characterized as follows.  For n≥ 0, let v_n be the coefficient of T^p^n in [p](T).  In particular v_0=p.  For 1≤ n≤∞, the vanishing locus of (v_0,v_1,…,v_n-1) in L is coordinate-invariant, meaning that it descends to a closed substack of _⊗_(p), to wit _^≥ n.   We refer to the locally closed substack _^n_FG^≥ n+1∖_FG^≥ n as the moduli stack of formal groups of height n.  It is characterized this way:  a formal group H of height n over a characteristic p ring R is one for which the p-torsion H[p] is a locally free group scheme of rank p^n over R.  

Formal groups of height n obey an important rigidity condition with respect to isomorphisms. This is proven by Goerss in <cit.>, who attributes it to Lazard; see also <cit.>.

 
Let R be a ring of characteristic p, and let H,H' be formal groups over R of height n, where 1≤ n < ∞.  Let (H,H') denote the functor sending an R-algebra R' to the set of isomorphisms H⊗_R R'→ H'⊗_R R'.  Then (H,H') is represented by an affine scheme which is pro-finite étale over R.


For each 1≤ n≤∞, there exists a formal group H_n of height n over _p.  By  <ref> (or by choice of p-typical coordinate when n=∞), H_n is unique up to isomorphism.  H_∞ is the formal additive group, while for 1≤ n<∞, H_n is p-divisible.  Assume henceforth that 1≤ n<∞.  Recalling that _n=(H_n,_p), we have an isomorphism of stacks over _p:

    _^n  [_p/ _n].


For 1≤ n<∞, let _^n be the completion of _ along the locally closed substack _^n.  This is a formal stack over _p.  We may describe it by giving its values on rings in which p is nilpotent.  Equivalently, we may describe it by giving its values on admissible _p-algebras R, meaning that R is complete with respect to the topology defined by an ideal containing p.  For a ring R in which p is nilpotent (respectively, an admissible _p-algebra R), the R-points of _^n classify formal groups H over R such that H⊗_R R/I has height n for some nilpotent ideal (respectively, an open ideal) I⊂ R containing p.  

The stack _^n has a standard presentation as a pro-étale quotient of a finite-type formal scheme.  Let 

    _n=(H_n)

be the deformation space of H_n.  This means that for an Artinian local ring R with residue field _p, the R-points of _n classify 
pairs (H,ρ), where H is a formal group over R, and ρ H_n→ H⊗_R _p is an isomorphism. 
Note that such an R is necessarily an algebra over W=W(_p).  
Then _n is pro-representable by an affine formal scheme:  _n= A_n, where A_n is an admissible _p-algebra, isomorphic to Wu_1,…,u_n-1 with its (p,u_1,…,u_n-1)-adic topology.  The formal scheme _n admits an action of _n, which is compatible with the action of (_p/_p) on W.

    We have an isomorphism of stacks of formal schemes in the pro-étale topology over _p:

    _^n [_n/_n].



<Ref> is standard, but we discuss its proof in some detail as a means of building up the necessary machinery for what follows.  To do so, it will be important to have a moduli interpretation for the R-points of _n, where R is a general W-algebra (not Artinian or even Noetherian) in which p is nilpotent.  This interpretation, appearing in the moduli spaces studied by Rapoport–Zink <cit.>, shifts the focus from formal groups of finite height to the larger category of p-divisible groups, which we review in the next subsection.




 §.§ p-divisible groups


These were introduced by Tate <cit.>.  

 Let R be a ring.  A p-divisible group (or Barsotti–Tate group) of height n over R is an inductive system G=_ν≥ 1 G^ν of commutative group schemes G^ν over R whose underlying scheme is locally free of rank p^nν over R.  It is required that G^ν→ G^ν+1 induces an isomorphism of G^ν with the p^ν-torsion in G^ν+1.  Let _R denote the category of p-divisible groups over R, with the evident notion of morphism; then _R is a _(p)-linear category.  


The same axioms applied to an abstract group would force G=(_p/_p)^⊕ n.  Indeed, for whatever ring R, the constant group (_p/_p)_R=_ν≥ 1  (p^-ν/)_R is an example of a p-divisible group.  


There is a close link between formal groups and p-divisible groups.  Let R be an admissible _p-algebra, and let H/R be a formal group representing an R-point of _^n:  thus there is an open ideal I⊂ R containing p such that H⊗_R R/I has height n.  Locally on R we may choose a coordinate T for H;  then in (R/I)T, the multiplication-by-p series is a unit multiple of T^p^n.  It follows that the torsion H[p^ν]= RT/[p^ν]_H(T) is a connected free group scheme over R of rank p^ν h, and that (globally on R) H[p^∞]:=_ν≥ 1 H[p^ν] is a connected p-divisible group over R.  

As an example, if H is the formal multiplicative group over R, then H[p^∞]=μ_p^∞ is the group of pth power roots of unity over R.

A theorem of Tate <cit.> states that if R is a Noetherian local ring of residue characteristic p, then the functor H↦ H[p^∞] is an equivalence between the category of formal groups over R (of whatever dimension) which are p-divisible (meaning that the p-torsion is a locally free group scheme) and the full subcategory of _R whose objects are connected p-divisible groups.  

A p-divisible group G=_ν G^ν over an admissible _p-algebra R determines a functor of points valued in abelian groups:  if S is an admissible R-algebra with ideal of definition I, define G^ν(S)=_m G^ν(S/I^m) and G(S)=_ν G^ν(S).  The Lie algebra of G is defined via the formula 
    
    G=(G(R[ε]/ε^2)→ G(R)).

It is a locally free R-module, whose rank is by definition the dimension of G.  If G=H[p^∞] for a finite-height formal group H over R, then the functor of points and Lie algebra for G agree with the analogous constructions for H.  Via the functor of points it is possible to embed both formal groups and p-divisible groups as full subcategories of the category of fppf sheaves of abelian groups.  From that point of view a p-divisible formal group H is identified with its p-divisible group H[p^∞];  nonetheless we shall continue to notationally distinguish the two sorts of objects.

  Let G and G' be p-divisible groups over a ring R.  An isogeny f G→ G' between p-divisible groups over R is a morphism whose kernel is a finite locally free group scheme over R.  If f is an isogeny, its kernel f has rank p^n, where n is a locally constant function on R;  we call n=(f) the height of f.  Let _R^0=_R⊗ and call this the category of p-divisible groups up to isogeny. An isomorphism in _R^0 is called a quasi-isogeny.


Thus the multiplication-by-p map on G is an isogeny whose height equals the height of G.  Note that if f G→ G' is an isogeny of height n, then f factors through multiplication by p^n, so that f is also a quasi-isogeny.  Conversely, a quasi-isogeny becomes an isogeny when multiplied by a sufficient power of p.  There is an extension of the notion of height to quasi-isogenies, so that if f is a quasi-isogeny, its height (f) is a (possibly negative) integer, and if the composition f'∘ f of isogenies is defined, then (f'∘ f)=(f')+(f).  

Quasi-isogenies satisfy a rigidity property similar to what appears in <ref>.  For p-divisible groups G and G' over a ring R, let (G,G') be the functor which associates to an R-algebra S the set (G_S,G'_S), meaning the set of quasi-isogenies G⊗_R S→ G'⊗_R S.  Then (G,G') is formally étale over R:


     Let R be a ring in which p is nilpotent, and let I⊂ R be a nilpotent ideal.  Let G and G' be p-divisible groups over R.  The natural map (G,G')→(G_R/I,G'_R/I) is a bijection. 


Consequently, given p-divisible groups G and G' over R, an isogeny G⊗_R R/I→ G'⊗_R R/I will lift to an isogeny G→ G' once multiplied by a sufficiently high power of p. 



 §.§ Lubin–Tate space as a moduli space of p-divisible groups


Recall that H_n is the (unique up to isomorphism) 1-dimensional formal group of height n over _p.  Let

    G_n = H_n[p^∞],

so that G_n is a p-divisible group over _p of height n.  Let _D_n= H_n G_n, and let D_n=_D_n[1/p].  Then D_n is a division algebra over _p of invariant 1/n, and _D_n is its ring of integers.  Therefore the group of automorphisms of G_n (as an object in __p) is _D_n^∗, and the group of quasi-isogenies of G_n (that is, automorphisms in ^0__p) is D_n^∗.  By <ref>, the functor (G_n,G_n) is formally étale over an algebraically closed field; it is therefore the constant group scheme associated to D_n^∗.  

We can view the Morava stabilizer group _n as the group of automorphisms of the pair (G_n,_p).  Let _n^0 denote the group of automorphisms of the pair (G_n,_p) in the isogeny category.  Explicitly, an element of _n^0 is a pair (τ,u), where τ∈Γ__p, and u G_n→τ^*G_n is a quasi-isogeny.

The groups discussed here fit into the diagram

    0 [d]     0 [d]       
    
    0 [r]    _D_n^∗[r] [d]     D_n^∗[r]^[d]    [r]     0 
    
    0 [r]    _n [r] [d]    _n^0 [r]^[d]    [r]     0
       Γ__p[d] [r]_=   Γ__p[d]       
        0     0

in which rows and columns are exact, and Γ__p=(_p/_p).  

The following definition is a special case of the constructions appearing in <cit.>. 

 Let _n be the functor which inputs a ring R in which p is nilpotent and outputs the set of triples (G,ι,ρ), where G/R is a p-divisible group, ι is a ring homomorphism _p→ R/p, and 

    ρ G_n ⊗__p,ι R/p→ G⊗_R R/p

is a quasi-isogeny. 


The homomorphism ι determines a homomorphism W→ R, so that there is a natural transformation _n→ W.  
There is a decomposition of functors over W:

    _n=∐_h∈_n^(h),
 where _n^(h) classifies those (H,ι,ρ) where ρ has constant height h.  

There is an action of the group _n on _n lying over the action of Γ__p on W:    An element g=(τ,u)∈ D_n^∗ sends (G,ι,ρ) to (G,ι∘τ^-1,ρ∘ u^-1).

Under this action, an element g∈ D_n^∗ induces an isomorphism _n^(h)_n^(h-(g)).  Therefore all components _n^(h) are isomorphic, and in order to describe _n with its _n^0-action, it suffices to describe _n^(0) with its _n-action.  

Recall that A_n is the universal deformation of H_n.  Let 𝔪 be the maximal ideal of A_n, and let H_n^ be the universal deformation of H_n, so that there is an isomorphism H_n H_n^⊗_A_n A_n/𝔪.  Let G_n^=H_n^[p^∞], so that we have an isomorphism 

    G_n G_n^⊗_A_n A_n/𝔪.

For each k≥ 1, we apply <ref> over the ring A_n/(p,𝔪^k);  the above isomorphism lifts to a quasi-isogeny

    ρ_k G_n⊗_A_n A_n/(p,𝔪^k) → G_n^⊗_A_n A_n/(p,𝔪^k)

of height 0.  Thus one obtains a morphism A_n/𝔪^k→_n^(0) for all k≥ 1, or all the same, a morphism _n= A_n→_n^(0).  This morphism is _n-equivariant.  In fact:


  
 
    The morphism _n→_n^(0) is an isomorphism.
 


We can now complete the proof of  <ref>, which claims that _^n [_n/_n]. The universal deformation of H_n determines a _n-equivariant morphism _n→_^n.  Composing with the isomorphism in <ref>, we obtain a _n-equivariant morphism _n^(0)→_^n, which extends to a _n^0-equivariant morphism _n→_^n.  To complete the proof, we need to show that this morphism is a pro-étale _n^0-torsor.


Suppose R is a ring in which p is nilpotent, and that we are given a morphism R→_^n, corresponding to a formal group H over R
such that H⊗_R R/I has height n for a nilpotent ideal I containing p. The fiber F:= R×__^n_n is the functor which sends an R-algebra S to the set of pairs (ι,ρ), where ι_p→ S/p is a ring homomorphism, and 

    ρ G_n⊗__p,ι S/p → H[p^∞]⊗_R S/p

is a quasi-isogeny.  Our objective is to show that F is a pro-étale _n^0-torsor over R.

First we show that F→ R has pro-étale local sections.  Let R'=R⊗__p W, so that R'→ R is pro-étale.  By <ref>, 

    (H_n⊗__p R'/I, H⊗_R R'/I)

is represented by an affine scheme which is pro-finite étale and surjective over R'/I;  let S be the lift of this affine scheme to R'.    
Let H_n⊗__p S/I→ H⊗_R S/I be the universal isomorphism. By <ref> applied to the ring S/p, the associated isomorphism of p-divisible groups lifts to a quasi-isogeny G_n⊗__p S/p→ H[p^∞]⊗_R S/p of height 0, determining a morphism S→ F.  Therefore F→ R admits pro-étale local sections.

Any two sections of F over an R-algebra S differ by a pair (τ,u), where τ∈Γ__p and u G_n⊗__p S→τ^*G_n⊗__p S is a quasi-isogeny.  The quasi-isogeny u represents an S-point of the _p-scheme Q(G_n,τ^*G_n).  By <ref>, Q(G_n,τ^*G_n) is formally étale over _p, which (since the latter is algebraically closed) must be the constant scheme associated to the set of quasi-isogenies G_n→τ^*G_n.  Thus any two sections of F differ by an S-point of the constant scheme associated with _n^0. We have shown that F→ R is a _n^0-torsor, which completes the proof.




 §.§ Dieudonné modules


The theory of Dieudonné modules <cit.> provides a functor from p-divisible groups to linear algebra objects.  It is roughly analogous to the functor that sends an abelian variety to its first de Rham cohomology group.  We give here a brief summary.  

Let R be a p-adically complete ring, and let G be a p-divisible group of height n over R.   We can think of G as defining an fppf sheaf on the category of R-algebras.  Let R'→ R be a p-adically complete PD thickening, meaning that R' is p-adically complete and R'→ R is a surjection of rings whose kernel is equipped with a divided power structure.  Let G' be any lift of G to R', and let EG'→ G' be the universal vector extension of G'.  (A vector extension of G' is a surjective morphism of sheaves of abelian groups whose kernel is isomorphic to a direct sum of copies of the additive group.)  The (covariant) Dieudonné crystal is defined as
    
    M(G)(R'→ R) =  EG'.

It is a projective R'-module of rank n.  The Dieudonné crystal does not depend on the choice of G', in the strong sense that the formation of M(G)(R'→ R) is functorial in both G and R'.

We will only deal with the case that R/p is a semiperfect ring, which means that the  Frobenius endomorphism ϕ R/p→ R/p is surjective.  In this case there is a universal p-adically complete PD thickening A_(R)→ R.  The construction goes this way:  let R^♭=_ϕ R/p be the “tilt” of R;  there is a natural surjective map W(R^♭)→ R, and then A_(R) is defined as the PD-hull of this surjection.  In this situation we simply write 

    M(G)=M(G)(A_(R)→ R),

and call this the Dieudonné module of G.  Considering the quotient EG'→ G', we find an R-linear surjective map

    β_G  M(G)⊗_A_(R) R→ G.

Evidently A_(R) only depends on R/p, and M(G) only depends on G⊗_R R/p.  We also write B_^+(R)=A_(R)[1/p], so that M(G)[1/p] is a projective B_^+(R)-module.

The Dieudonné module M(G) comes with two important structures:  Frobenius and Verschiebung.  To define these, we first assume that R is a semiperfect ring of characteristic p, with Frobenius map ϕ.  Let G^(p)=G⊗_R,ϕ R, and let F G→ G^(p) be the relative Frobenius morphism.  Let V G^(p)→ G be Cartier dual to the Frobenius morphism G^∨→ G^∨,(p);  we have the fundamental relations VF=p in G and FV=p in G^(p).

The Frobenius endomorphism induces by functoriality an endomorphism of A_(R) which we continue to call ϕ.  The morphisms F and V induce maps on Dieudonné modules as in the following definition. 

 Let R be p-adically complete ring such that R/p is semiperfect.  The category of Dieudonné modules _R is the category of projective A_(R)-modules M together with A_(R)-linear maps
F M⊗_A_(R),ϕ A_(R) → M and V M→ M⊗_A_(R),ϕ A_(R) satisfying VF=p and FV=p.  We let _R^0=_R⊗ be the corresponding isogeny category.

We write G↦ M(G) (resp., M^0(G)) for the Dieudonné module functor _R→_R (resp., _R^0→_R^0). 


  In fact F G→ G^(p) induces the map V on M(G) and vice versa.  This notational awkwardness is avoided when one uses the contravariant Dieudonné module.   For G=(_p/_p)_R the constant p-divisible group, the relative Frobenius on G is induced from the identity on G, and so M(G)=A_e for a basis element e satisfying F(e⊗ 1)=pe and V(e)=e⊗ 1.  
 


    

In the special case that R=k is a perfect field of characteristic p, we have A_(k)=W(k), since the kernel of W(k)→ k is (p) and p^n/n!∈_p for all n≥ 0.  In this case ϕ is an automorphism of A_(k).  A classical result of Fontaine <cit.> states that the Dieudonné module functor _k→_k is an equivalence.   In retrospect we may define the p-divisible group G_n∈__p by its Dieudonné module:  M(G_n)=_p^⊕ n⊗__p W(_p), and F=F_0⊗ϕ, where F_0∈_p^⊕ n is the matrix

    [         1; p        ;   p      ;     ⋱;   p   ].



For more general semiperfect rings R, the Dieudonné module functor can fail to be an equivalence, but under a weak assumption on R the functor is at least fully faithful up to isogeny.

  Assume R is the quotient of a perfect ring of characteristic p by a finitely generated ideal.  Then the Dieudonné module functor 

    M^0_R^0→_R^0
 
is fully faithful.





 §.§ The Fargues–Fontaine curve


Let R be a perfectoid algebra over _p.  This means that R is a Banach _p-algebra whose subring of power-bounded elements R^∘ is p-adically complete, and there exists an element p^1/p∈ R^∘ which is (up to a unit) a pth root of p, such that the pth power map R^∘/p^1/p→ R^∘/p is an isomorphism of rings.  It follows from this that R^∘/p is semiperfect.  For instance, if C is a complete algebraically closed field containing _p, then C is a perfectoid algebra.  

If a p-divisible group G over R^∘ is given, its Dieudonné module M(G) is a projective A_(R^∘)-module equipped with a ϕ-semilinear action of Frobenius.  Therefore it ought to descend to a vector bundle over the quotient of A_(R^∘) by ϕ.  The Fargues–Fontaine curve is morally something like this quotient, although one must invert p.  

Define a graded _p-algebra by
    
    P_R = ⊕_d≥ 0 B_^+(R^∘)^ϕ=p^d,

and a scheme X_R by 
    
    X_R =  P_R.

The quotient map B_^+(R^∘)→ R induces a closed immersion i R→ X_R.  


The foundational work <cit.> is concerned with the “absolute curve” X_C appearing in the case that C is an algebraically closed perfectoid field.  Despite not being finitely generated over a field, X_C was discovered to share properties with the Riemann sphere:  it is a Noetherian scheme of dimension 1, and furthermore the complement in X_C of any closed point is the spectrum of a PID.  

The construction of the Fargues–Fontaine curve globalizes, so that we have a scheme X_S for any perfectoid space S.  The reader is warned that there is no morphism X_S→ S.  Nonetheless, it is still something like a family of curves, in that one gets an absolute curve for each rank 1 geometric point of S.

What we have presented here is the “schematic” curve;  there is also an “adic” version which is more suitable to applications in p-adic geometry.  The two curves have the same category of quasi-coherent sheaves, by “GAGA for the curve” <cit.>.


There is a functor M↦(M) from rational Dieudonné modules over R to quasi-coherent sheaves on X_R.  For an object M of ^0_R, think of F as a ϕ-semilinear map ϕ M→ M, and then define ℰ(M) as the quasi-coherent sheaf on X_R associated to the graded P_R-module

    ⊕_d≥ 0 M^ϕ=p^d+1.

The definition has been normalized so that if M=M^0(_p/_p) is the rational Dieudonné module of the constant p-divisible group, then (M) is the trivial line bundle _X_R.  If M is the rational Dieudonné module of our fixed p-divisible group G_n over _p, and if a ring homomoprhism _p→ R^∘/p is given, we obtain a vector bundle (M) of rank n over X_R, and we denote this _X_R(1/n), or simply (1/n) if the base curve X_R is understood.

A major accomplishment of <cit.> is the complete classification of vector bundles on the absolute curve X_C, by means of a Harder–Narasimhan formalism.  Every vector bundle is isomorphic to a direct sum of vector bundles (λ) parameterized by rational numbers λ (“slopes”).  For each λ, the degree and rank of (λ) are the numerator and denominator of λ in lowest terms, and the automorphism group of (λ) is the group of units in a central division algebra over _p of invariant λ.  In particular, every automorphism of (1/n) arises from an automophism of G_n in the isogeny category:  (1/n) D_n^∗. 

The study of families of vector bundles on the Fargues–Fontaine curve (that is, vector bundles over X_S for a general perfectoid space S) is very rich, and has been shockingly fruitful in attacking the local Langlands correspondence <cit.>.  An important first result <cit.> is that vector bundles form a stack for the pro-étale topology (in fact this is true for the finer v-topology). Another result concerns families whose geometric fibers are all “isoclinic” (admitting only one slope); these are vastly simpler.

  Let S be a perfectoid space over W(_p).  Let λ be a rational number whose denominator d≥ 1 (in lowest terms) is a divisor of n.  The following groupoids are equivalent:
 

  * Vector bundles of rank n over X_S which are everywhere on S isoclinic of slope λ, which is to say isomorphic to _0=(λ)^⊕ (n/d).

  * Pro-étale J_λ-torsors over S, where J_λ is the group of units in the unique central simple algebra over _p of degree n and invariant λ.  

The equivalence sends a vector bundle  to the sheaf (_0,) whose R'-points (for a perfectoid R-algebra R') parameterize isomorphisms between the vector bundles after pullback to X_R'.


We spell out the consequences of Proposition <ref> in the two cases relevant to us:

In the case λ=0, we have J_λ=_n(_p).  Note that a pro-étale _n(_p)-torsor is the same thing as a _p-local system of rank n.  If T is a _p-local system of rank n, let us write T⊗__p_X_S for the vector bundle on X_S corresponding to the _p-local system  T[1/p].  

In the case λ=1/n, we have J_λ=D_n^∗.  As a corollary of the proposition, suppose S is a general perfectoid space, with no structure morphism to W(_p).  Then there is an equivalence of groupoids between (a) vector bundles of rank n on X_S which are everywhere of slope 1/n, and (b) pro-étale _n^0-torsors over S admitting a map to (K,W); or all the same, sections of the stack [(K,W)^♢/_n^0] over S.  




 §.§ p-divisible groups over a perfectoid ring


Keep the assumption that R is a perfectoid _p-algebra, and let S=(R,R^∘).  Let G be a p-divisible group of height n over R^∘.  We gather here some of the structures associated with G which live over the Fargues–Fontaine curve X_S.  

First we have the 
Tate module

    TG = _ν G[p^ν](R^∘) __R(_p/_p,G).

Here we write _p/_p for the constant p-divisible group over R^∘.  
Since R^∘ is integrally closed in R and each G[p^ν] is finite over R^∘, the natural map G[p^ν](R^∘)→ G[p^ν](R) is an isomorphism.  On the other hand G[p^ν]_R is an étale group scheme of rank p^nν over R.  This shows that TG can be given the structure of a _p-local system of rank n over S_.  Let TG⊗__p_X_S be the corresponding vector bundle on X_S which is everywhere on S isomorphic to ^⊕ n.
 
On the other hand we have the rational Dieudonné module M^0(G), which gives rise to a quasi-coherent sheaf on X_S, which we notate (G). 
If G arises from an R^∘-point of _^n, it means there exists pro-étale locally on R^∘/p a quasi-isogeny G_n→ G, and thus an isomorphism M^0(G_n)→ M^0(G).  Translating to vector bundles, we find that (G) is a vector bundle on X_S which is everywhere on S isomorphic to (1/n).  

A morphism _p/_p→ G representing a section of TG induces a map of vector bundles (_p/_p)→(G), which is just a global section of (G) since (_p/_p) is the trivial line bundle.  We have defined a map TG→ H^0((_p/_p)) of sheaves of groups on the pro-étale site of S;  this is adjoint to a map of vector bundles

    α_G    TG⊗__p_X_S→(G).


  Let R be a perfectoid algebra over _p, and let S=(R,R^∘).  Let G be a p-divisible group over R^∘.  The morphism α_G fits into an exact sequence of _X_S-modules:
    
    0→ TG⊗__p_X_Sα_G→ℰ(G) → i_* G[1/p]→ 0,

where i R→ X_S is the closed immersion defined above.



The injectivity of the map α_G comes from the faithfulness part of <ref>.  

The surjective map (G)→ i_* G[1/p] arises via adjunction from the surjective map i^*(G)=M^0(G)⊗_B_^+(R^∘) R → G[1/p] from (<ref>).  

Next we claim that the sequence in (<ref>) is actually a complex.  A section s of TG⊗__p_X_R is represented by a morphism _p/_p→ G.  The image of s in (G) is the image of the canonical generator of (_p/_p)= under (_p/_p)→(G).  We now use the commutativity of

    (_p/_p) [r] [d]     i_*_p/_p [d] 
    (G) [r]     i_* G

together with _p/_p=0 to conclude that the image of s in i_* G is zero.  

The exactness of (<ref>) can be checked on every geometric point of S, so let us assume R=C is an algebraically closed field.  The result follows from a numerical argument:  since α_G is an injective map of vector bundles, its cokernel is torsion of degree equal to G;  for this to surject onto i_* G must mean that the cokernel is exactly i_* G.





 §.§ The diamond stack _n


We return now to the study of the stack _^n over _p.  
Our next order of business is to pass from _^n to its generic fiber to obtain an analytic object living over _p.  

 Suppose X is a presheaf of sets or groupoids on the category of admissible _p-algebras.  The diamond generic fiber X^♢__p of X is the sheafification of the presheaf

    (R,R^+)↦ X(R^+)

on perfectoid (_p,_p)-algebras with respect to the pro-étale topology.

Thus for a perfectoid space S over (_p,_p), an object of X__p^♢(S) is given by a pro-étale cover of S by affinoids (R_i,R^+_i), and for each i an object of X(R^+_i), together with a descent datum to S.

 
Let 
    
    _n = (_FG^n)^♢__p,

and call this the diamond stack of 1-dimensional formal groups of height n.  


Suppose S is a perfectoid space over (_p,_p), and let H be an S-point of _n.  To specify H is to give the data of a covering of S_ by affinoid perfectoid spaces (R,R^+) and a formal group over each R^+ representing an R^+-point of _^n.  We shall refer to H as a formal group of height n over S_.  

We remark here on the following “partial properness” property of _n.  Suppose (R,R^+) is an affinoid perfectoid space, so that R^+ is an open integrally closed subring of the p-adically complete ring R^∘.  We claim that the natural map _^n(R^+)→_^n(R^∘) is an equivalence.  By <ref>, it is enough to check that _n(R^+)→_n(R^∘) is a bijection.  This is true because _n(R^∘) is the (n-1)-fold cartesian product of the set of topologically nilpotent elements of R^∘, and these all lie in R^+ because R^+ is integrally closed in R^∘.  We conclude that _n is partially proper, meaning that its groupoid of (R,R^+)-points depends only on R and not on the subring R^+⊂ R.

Suppose H is a formal group of height n over S_.  Since quasi-coherent -modules satisfy pro-étale descent, H[1/p] defines an invertible -module on the analytic site S_.  

  The following stacks are equivalent:
    
        
  * _n;
        
  * the stack _n whose points over S=(R,R^∘) classify modifications
            
    0→ T⊗_X_S→ℰ→ i_*L → 0,

        where:
            
                
  * T is a _p-local system on S_,
                
  * ℰ is a vector bundle on X_S which is isomorphic to (1/n) everywhere on S,
                
  * L is a locally free _S-module of rank 1;
            
        
  * the quotient stack [_n,K^♢/_n];
        
  * the quotient stack [^n-1,♢/_n(_p)], where ^n-1 is Drinfeld's symmetric space.
    
The equivalence between _n and _n sends a formal group H to the exact sequence associated in <ref> to G=H[p^∞].  Furthermore, let H^ be the universal formal group over (_^n)^♢__p, so that H^[1/p] is an invertible -module on this stack.  With respect to the isomorphism to [^n-1,♢/_n(_p)], H^[1/p] corresponds to the restriction of the _n(_p)-equivariant tautological line bundle (1) from ^n-1__p to ^n-1.  



  The equivalence between [_n,K^♢/_n] and [^n-1,♢/_n(_p)] is the “isomorphism between the Lubin–Tate and Drinfeld towers”.  We phrase it here through the medium of modifications of vector bundles on the Fargues–Fontaine curve.  Detailed proofs can be found in <cit.>. 

The equivalence between (1) and (3) is a consequence of <ref>.  
  
Let W=W(_p) and K=W[1/p].  Let _n be the sheaf whose value on an affinoid perfectoid space S=(R,R^+) over (K,W) is the set of modifications
    
    0→ T⊗__p_X_S→_X_S(1/n)→ i_*L→ 0,

    where T is a pro-étale _p-local system on S, and L is a locally free _S-module of rank 1.  
Then _n is an example of a local shtuka space as introduced in <cit.>.  There is a morphism _n^♢→_n lying over (K,W), described as follows:  if R is a perfectoid K-algebra, and if (G,ρ) represents a (R,R^∘)-point of _n^♢, then ρ induces an isomorphism _X_S(1/n)→ℰ(G), and then <ref> produces a (R,R^∘)-point of _n.  The map _n^♢→_n is an equivalence:  this is a special case of <cit.>.

There is an obvious morphism _n→_n, which is a torsor for the automorphism group of the pair (_p,(1/n)), which is exactly _n^0.  Therefore

    _n [_n/_n^0] [_n^♢/_n^0] [_n^♢/_n],
 
where the last isomorphism derives from <ref>.  This is the equivalence between (2) and (3).  


For the equivalence between (2) _n and (4) [^n-1/_n(_p)]:  We interpret S-points of projective space ℙ^n-1 to mean invertible _S-modules ℓ which are direct summands of _S^⊕ n.  Observe that such an ℓ determines
a vector bundle  containing _X_S^⊕ n, whose sections are the same except they are allowed to have a simple pole along the hyperplane determined by ℓ.  Thus ℓ determines a modification 
    
    0→_X_S^⊕ n→ℰ→ i_*L → 0,

where L is an invertible _S-module.  (In fact L is a twist of ℓ by the tangent line to the curve X_C along the point i; this tangent line is canonically a Tate twist of C.)  The kernel of _S^⊕ n=i^*_X_S^⊕ n→ i^*ℰ is exactly ℓ.  

Let (C,C^∘) be a geometric point of S, and let ℓ_C⊂ C^n and ℰ_C be the pullbacks of those objects to C.  The following statements are all equivalent:


    
  * The line ℓ_C is contained in a _p-rational hyperplane.
    
  * There exists a decomposition of _p^n as the direct sum of two nonzero _p-linear subspaces, such that ℓ is contained in one of them.
    
  * There exists a decomposition of _X_C^⊕ n as the direct sum of two nonzero sub-bundles of slope 0 (i.e., they are both direct sums of _X_C), such that ℓ is contained in the fiber of one of them.  (The equivalence of this statement with the last one boils down to H^0(X_C,_X_C)=_p.)
    
  * The modified vector bundle  admits a trivial summand.
    
  * The modified vector bundle  is not isomorphic to (1/n).     
For the equivalence of the last two statements:  a priori  has degree 1 and rank n, and all its slopes are nonnegative because it contains ^⊕ n as a maximal rank sub-bundle.  Therefore  must be isomorphic to (1/i)⊕^⊕ (n-i) for some i=1,…,n.  This contains a nontrivial -summand if and only if i<n.       

A morphism S→ℙ^n-1 determines a line ℓ, which determines in turn a modification;  the above discussion shows that the modified vector bundle  is everywhere isomorphic to (1/n) if and only if S→ℙ^n-1 factors through Drinfeld symmetric space ^n-1, defined as the complement in ℙ^n-1 of all _p-rational hyperplanes.  

Thus we find a morphism ℋ^n-1→_n carrying a line ℓ to its corresponding modification and forgetting the trivialization of the _p-local system T of rank n.  This morphism a torsor for the group _n(_p).





 §.§ The determinant map _n→_1.


To access the determinant sphere 𝕊_K(n)[] as a Picard class on _n, we will need to understand the role of the determinant in the study of 1-dimensional formal groups.  

  
There exists a determinant morphism

    _n→_1

making the following diagram commute:
    
    [_n,K^♢/_n] [rr]^-[d]__      _n [rr]^-[d]_       [^n-1,♢/_n(_p) ] [d]^_
    
            [_1,K^♢/_1] [rr]_-      _1 [rr]_-       [^0,♢/_p^∗].

In the diagram:

    
  * the map labeled _ is induced from:
    
        
  * the structure map _n,K→_1,K(K,W);
        
  * the determinant map _n→_1, explained in the remarks below,
    
    
  * the map labeled _ is induced from:
    
        
  * the structure map ^n-1→^0(_p,_p);
        
  * the determinant map _n(_p)→_p^∗.
    
    


Before beginning the proof, we offer some remarks.  For a perfectoid space S over (_p,_p), the S-points of the morphism _n→_1 should input a formal group H of dimension 1 and height n over S_, and output a formal group ⋀^nH over S_ of dimension 1 and height 1. The crucial computation that makes this work comes from the case of Dieudonné modules over _p:  if M(G_n) is the Dieudonné module of G_n as described in <ref>, then the top exterior power ⋀_W(_p)^n M(G_n) is isomorphic to M(G_1).  Consequently we have an isomorphism of line bundles on X_S for any perfectoid space S:  ⋀^n _X_S(1/n)_X_S(1).  

Considering the action of _n=(G_n,_p) on ⋀_W(_p)^n M(G_n) M(G_1), we find a determinant map
    
    _n→_1.

This map is isomorphic to the profinite completion of the reduced norm map D_n^∗→_p^∗, where D_n= G_n ⊗__p_p.  

 One might wonder whether general exterior powers exist beyond the determinant.  
    For a general p-divisible group G of height n over a perfect field k of characteristic p, the exterior powers ⋀^i_W(k) M(G) are Dieudonné modules if and only if G≤ 1, so one can only expect exterior powers of G itself to exist in this case.  A  construction of exterior powers of p-divisible groups of dimension ≤ 1 over the ring of integers in a nonarchimedean local field appeared previously in <cit.>.


 
Let S be a perfectoid space over (_p,_p), and let H be a formal group of dimension 1 and height n over S_, representing an S-point of _n.  Let 

    0 → T⊗__p_X_Sα_H→(H)→ i_* L→ 0

represent the corresponding S-point of _n.  

The determinant of α_H is an injective morphism of line bundles:

    α_H⋀^n__pT ⊗__p_X_S→⋀^n (H),

and ⋀^n(H) is locally isomorphic to _X_S(1).  The cokernel of α_H has degree 1 at every geometric point of S and is supported on the image of i, so it can be none other than the line bundle i_*i^*⋀^n.  Therefore α_H determines an S-point of _1, corresponding to a formal group ⋀^n H of height 1 over S_.  

For any n≥ 1, the stack [(_p,_p)^♢/_n(_p)] classifies _p-local systems of rank n. The commutativity of the right square of the diagram is exactly the statement that there is a functorial isomorphism of _1(_p)=_p^∗-torsors

    T(⋀^n H)⋀^n__p TH,

and this is in turn an artifact of our construction of ⋀^n H.  

Similarly, the commutativity of the left square of the diagram traces to the statement that there is a functorial isomorphism of vector bundles

    (⋀^n H)⋀^n (H);

once again this is an artifact of our construction.






§ THE FUNDAMENTAL EXACT SEQUENCE


The goal of this section is to construct what we dub the fundamental exact sequence, describing the group H^1_(_n,A_n^∗∗) we are interested in in terms of H^1_(_1,A_1^∗∗) and H^1_(_C^n-1,^∗∗)^Π_n. The key inputs are the isomorphism of towers <ref>, the determinant map, as well as a logarithm exact sequences for the sheaf of principal units (<ref>) to compute the terms. Throughout, special care is needed for the “anomalous” case n=p=2. 

The objects of study in this section are closely related to those considered by Ertl, Gilles, and Nizioł in <cit.>, whose methods inspired the approach in this section. 




 §.§ The sheaf of principal units ^∗∗


As a general reference for the material we recall in this subsection, we refer the interested reader to <cit.> and <cit.>.

For a topological ring A, let A^∘ be the subset of power-bounded elements, and let A^∘∘⊂ A^∘ be the subset of topologically nilpotent elements. Let K be a nonarchimedean field of characteristic (0,p).  This means that K is complete with respect to a real-valued ultrametric  which satisfies 0<p<1.  
In particular K contains _p as a subfield.  The subset K^∘ (also called K^+ or _K) is a valuation subring of K, equal to the set of all f∈ K with f≤ 1.  

Recall the notion of a Huber pair (A,A^+) over (K,K^+).  In this situation A is a topological K-algebra containing an open subring A_0 (a ring of definition) which carries the p-adic topology, and A^+⊂ A is an open and integrally closed K^+-subalgebra contained in A^∘.  The pair (A,A^+) is complete if A contains a p-adically complete ring of definition.

 Let (A,A^+) be a complete Huber pair over (K,K^+).  

    
  * For f∈ A, there exists n≥ 0 such that p^nf∈ A^+.
    
  * pA^∘⊂ A^+.
    
  * A^∘∘ = (pA^+), the radical of pA^+.
    
  * 1+A^∘∘ is an open subgroup of A^∗.  



 


  * Let f∈ A.  Since p^n→ 0 in A_0⊂ A, and multiplication in A is continuous, we have p^nf→ 0.   Since A^+ is open we have p^nf∈ A^+ for n large enough. 

  * If f∈ A^∘, the sequence f^n is bounded, so that (pf)^n→ 0.  For n large enough (pf)^n∈ A^+, and then pf∈ A^+ since A^+ is integrally closed.

  * If f∈ A^∘∘, then f^n→ 0, so for n large enough f^n∈ pA^+.  (Note that pA^+⊂ A is open since multiplication by p^-1 is continuous on A.) This implies f∈ A^+ (once again since A^+ is integrally closed) and so f∈(pA^+).  For the reverse inclusion, it is enough to show pA^+⊂ A^∘∘, and indeed pA^+⊂ pA^∘⊂ A^∘∘ since p is topologically nilpotent.

  * The subset 1+A^∘∘ is clearly closed under multiplication.  To see that it contains inverses, let x∈ A^∘∘, and note that (1+x)^-1=1-x+x^2-⋯ converges to an element of 1+A^∘∘.  It is open because it contains the neighborhood 1+pA_0 of 1.  


In the context of the lemma we write 

    A^∗∗=1+A^∘∘,

and we call this the group of principal units of A.  

Let X be an adic space over (K,K^+).  Thus X is covered by affinoid adic spaces (A,A^+), where (A,A^+) is a complete Huber pair over (K,K^+).  For x∈ X, let K_x denote the completed residue field of x, and let K_x^+ denote its valuation ring.  Then (K_x,K_x^+) is an affinoid field.  If f is an analytic function defined in a neighborhood of x, let f(x) denote the image of f in K_x.

Recall the topologies X_,X_,X_ on X, listed here in increasing order of fineness.  Each topology has a basis consisting of affinoid objects U→ X which are open immersions in the case of X_, étale morphisms in the case of X_, and pro-étale morphisms from perfectoid affinoids in the case of X_.  

Each of the three topologies carries a structure sheaf, written _X or simply .  In the case of X_, what we mean is the completed structure sheaf (written  in <cit.>).  Let ^∗⊂ be the sheaf of invertible elements.  For each topology t∈,,, the first cohomology group _t X:=H^1_t(X,^∗) classifies invertible -modules on X_t.  As one has descent of -modules in the étale topology, the natural map _(X)→_(X) is an isomorphism.  However, _(X) is generally strictly larger than _(X).  A systematic treatment of Picard groups in finer topologies is given in <cit.>; see also <cit.> for the case that X is a Stein space.

Let ^+⊂ be the subsheaf of integral elements;  that is, for every U as above we have

    ^+(U)=f∈(U)|f(x)≤ 1, all x∈ U.

Finally, define the sheaf
of principal units ^∗∗⊂^+∗ by 

    ^∗∗(U)=f∈(U) |f(x)-1<1, all x∈ U.


 If U=(A,A^+) is an affinoid object of X_, X_, or X_, then ^∗∗(U) = A^∗∗.


 The lemma can be restated as follows:  Given f∈ A, we have that f(x)<1 for all x∈ U if and only if f∈ A^∘∘.  Consider the rational subset 
    
    U(1/f)=x∈ U |f(x)≥ 1.

We have U(1/f)=(R,R^+), where R=R_0[1/p], and R_0 is the completion of A_0[1/f].  We have that U(1/f) is empty if and only this completion is the zero ring, which is true if and only if f is 
topologically nilpotent.  


This lemma allows us to use the notation ^∗∗(U) and A^∗∗ interchangeably, and we will do so below. In particular, we note:

  Let A=_KT_1,…,T_d, and let D= A.  Let D_K be the adic generic fiber of D, so that D_K is the nonvanishing locus of p in (A,A).  Then (in whatever topology) H^0(D_K, ^∗∗)  1+M, where M⊂ A is the maximal ideal.  Applied to Lubin–Tate space, we find H^0(_n,K,) A_n^ = 1 + 𝔪.  





 §.§ The logarithm exact sequence


The following logarithm exact sequence will allow us to access the cohomology of ^∗∗ on the pro-étale site.  

Since ^∗∗(U) is a topological group for all U in a basis for the topology, we can think of ^∗∗ as a sheaf of condensed abelian groups.

 Let X be a rigid-analytic variety.  There is an exact sequence of sheaves of condensed abelian groups on X_ (and hence on X_):
    
    0 →μ_p^∞→^∗∗log→→ 0.

Here μ_p^∞ is the sheaf of pth power roots of 1.  


 Suppose U→ X is an affinoid object of X_, with U=(A,A^+).  The description of ^∗∗ in the previous lemma shows that the usual logarithm series converges on ^∗∗(U) with values in (U).  

We claim first that the kernel of log^∗∗(U)→(U) is μ_p^∞(U).  Given f∈^∗∗(U), there exists n≥ 0 large enough so that f^p^n-1∈ p^2A_0.  Then 

    p^n log f =log f^p^n =∑_i≥ 1 (-1)^i-1(f^p^n-1)^i/i∈ (f^p^n-1)(1+pA_0).

(Here we have used 2(i-1)≥ v_p(i)+1 for all i≥ 2, where v_p is the p-adic valuation on integers.)  
Thus if log f = 0, we must have f^p^n=1.

Now we claim that log^∗∗→ is surjective.  Let U=(A,A^+) as above.  For f∈(U)=A, let n≥ 0 be large enough so that p^nf ∈ p^2 A^+.  Then exp(p^nf) converges to an element e∈ A^∗∗ such that log e = p^n f.  Let B=A[e^1/p^n];  since e∈ A^∗ and p is invertible in A, we have that B is a finite étale A-algebra.  We find that if V=(B,B^+) (with B^+= integral closure of A^+ in B), then V→ U is an étale cover for which ^∗∗(V) contains a preimage of f under log, namely e^1/p^n.  





 §.§ The fundamental exact sequence


In this section we leverage the isomorphisms between stacks in <ref> to control the H^1_(_n,A_n^∗∗).  By <ref>, A_n^∗∗^∗∗(_n,K).  

We repeatedly apply the following general formalism:   Suppose a stack 𝔛 on the pro-étale site of __p admits a presentation 𝔛 [Y/G] for a diamond Y and a profinite group G which acts on Y.  Let ℱ be a sheaf of condensed abelian groups on 𝔛. Then we have a low-degree exact sequence 
    
    0→ H^1_(G,ℱ(Y)) → H^1_(𝔛,ℱ) → H^1_(Y,ℱ)^G

arising from the descent spectral sequence of E_2-signature
    
    H^i_(G,H^j_(Y,ℱ))  H^i+j_(𝔛,ℱ).

(For background on sheaves of condensed abelian groups on the pro-étale site, see <cit.>, and especially Proposition 3.6.3 for an example of how continuous G-cohomology appears naturally in the context of a pro-étale G-torsor.)

The formation of the exact sequence in (<ref>) is functorial in the following sense:  If a morphism of pairs (Y_1,G_1)→ (Y_2,G_2) is interpreted to mean a homomorphism ϕ G_1→ G_2 and ψ Y_1→ Y_2 such that ψ(gy)=ϕ(g)ψ(y) for g∈ G_1, y∈ Y_1, then such a morphism induces a morphism of descent spectral sequences and hence a morphism between the corresponding low-degree exact sequences.

Now suppose there are two such presentations for 𝔛:
    
    𝔛 [Y_1/G_1] [Y_2/G_2].

The corresponding low-degree exact sequences assemble to give a diagram

    0 [d]    
        P [r] [d]     H^1_(G_1,ℱ(Y_1)) [d]    
    
                0 [r]     H^1_(G_2,ℱ(Y_2)) [r]     H^1_(𝔛,ℱ) [r] [d]     H^1_(Y_2,ℱ)^G_2
           H^1_(Y_1,ℱ)^G_1,

where P is the indicated pullback in the diagram. In particular, there is an exact sequence

    0 → P → H^1_(G_1,ℱ(Y_1)) → H^1_(Y_2,ℱ)^G_2

as well as a similar one which swaps the roles of the two presentations.  The formation of the diagrams in (<ref>) and (<ref>) are functorial on the category of data consisting of (Y_1,G_1),(Y_2,G_2), and an isomorphism [Y_1/G_1] [Y_2/G_2] and with morphisms consisting of a pair of maps of pairs and a 2-commuting square of stacks.
  
Let C be the completion of an algebraic closure of _p, and let 
    
    Γ__p=(_p/_p),

so that (_p,_p)(C,C^∘)/Γ__p.  <Ref> shows that for each n≥ 1 there is a pair of presentations of the stack _n:

    _n  [_n,K^♢/_n]  [_C^n-1,♢/(Γ__p×_n(_p))].

Here, we have replaced ^n-1,♢ with its equivalent presentation _C^n-1,♢/Γ__p. It will be convenient to use the notation 
    
    Π_n = Γ__p×_n(_p).


Recall from <ref> that the determinant morphism _n→_1 is compatible with the two presentations of _n given in (<ref>).  Applying (<ref>) functorially to the determinant morphism, we obtain a commutative diagram:

    scale=0.55,center          0 
       P_1      H^1_(Π_1,(_C^0))    0 
          P_n      H^1_(Π_n,(_C^n-1))
    
    	0    H^1_(_1,(_1,K))      H^1_(_1,)      H^1_(_1,K,)^_1
        0    H^1_(_n,(_n,K))      H^1_(_n,)      H^1_(_n,K,)^_n
             H^1_(^0_C,)^Π_1
                H^1_(_C^n-1,)^Π_n.["q"', from=2-2, to=3-3]
    	[from=3-5, to=5-5]
    	[from=2-5, to=3-5]
    	[from=5-5, to=7-5]
    	[from=3-3, to=3-5]
    	[from=2-4, to=4-4]
    	["_^∗", from=2-4, to=3-5]
    	[from=2-2, to=2-4]
    	[from=1-4, to=2-4]
    	[from=4-4, to=5-5]
    	[from=4-4, to=6-4]
    	[from=6-4, to=7-5]
    	[from=3-3, to=5-3]
    	[from=5-3, to=5-5]
    	[from=5-5, to=5-7]
    	["_^∗",from=4-2, to=5-3]
    	["r", from=2-2, to=4-2]
    	[from=4-2, to=4-4]
    	[from=4-4, to=4-6]
    	["_^∗", from=4-6, to=5-7]
    	
    	[from=4-1, to=4-2]
    	[from=5-2, to=5-3]

We proceed with six lemmas concerning objects in the diagram (<ref>).


    Considering _p^=1+p_p with its trivial _n(_p)-action, we have:
        
    H^1_(_n(_p),_p^) 
                        0,     (n,p) ≠ (2,2); 
    /2     (n,p) = (2,2).

In the case (n,p) = (2,2), the group H^1_(_n(_p),_p^) is generated by the homomorphism given by the composition 
    
    _2(_2)_2(/2) S_3{± 1}⊂_2^.


 Since _p^ is abelian, H^1_(_n(_p),_p^) is isomorphic to the group of continuous homomorphisms _n(_p)^→_p^, where 
    
    _n(_p)^ab = _n(_p) / [_n(_p),_n(_p)].

Since _n() is dense in _n(_p), the image of _n()^ab→_n(_p)^ab is also dense.  We apply the well-known result <cit.>:
    
    _n()^ab = 
    0,     n ≠ 2; 
    /12     n =2.

This settles the claim for n≠ 2, and shows that in all cases _n(_p)^ab is a (possibly trivial) torsion cyclic group. For p ≠ 2, _p^≅_p is torsion free, so this settles the claim for p ≠ 2.

We are left with the case (n,p) = (2,2). As _2^≅_2⊕/2 and _2(_2)^ab is torsion cyclic we see that H^1_(_2(_2),_2^) is of order at most 2. To conclude we observe that the map _2(_2)→_2^ described above determines a non-trivial element in H^1_(_2(_2),_2^).


 
For (n,p)≠ (2,2), the map

    _^∗ H^1_(Π_1,(_C^0))
    → 
    H^1_(Π_n,(_C^n-1))

is an isomorphism.  For (n,p) = (2,2), _^∗ is an inclusion of a direct summand with complement /2 generated by the class α∈ H^1_(Π_2,C^) defined as the composite 
    
    Π_2 →_2(_2) _2(/2) ≃ S_3 {± 1}⊂ C^,

where the first map is the canonical projection.



We first recall from <cit.> that every bounded-by-1 function on ^n-1__p is constant:  H^0(^n-1__p,^+) ≅_p.  This follows from the existence of an integral model __p^n-1 for ^n-1, whose special fiber __p^n-1 is a connected scheme each of whose irreducible components is projective (see <cit.>). Indeed, this shows that H^0(__p^n-1,)≅_p, which suffices because H^0(^n-1__p,^+) H^0(^n-1__p,) is p-adically complete. Consequently H^0(_C^n-1,)≅ C^.

The map _^∗ agrees with the one appearing in the inflation-restriction sequence for C^ and the map Π_n →Π_1 induced by the determinant map _n(_p)→_p^∗ with kernel _n(_p):
    
    0→ H^1_(Π_1, (C^)^_n(_p))  H^1_(Π_n,C^) → H^1_(_n(_p),C^)^Π_1.

Using that _n(_p) acts trivially on C^, Tate's theorem <cit.> which shows that C^Γ__p=_p, and postcomposing with H^1_(_n(_p),C^)^_p^∗⊆ H^1_(_n(_p),C^), we obtain an exact sequence 
    
    0→ H^1_(Π_1, C^)  H^1_(Π_n,C^) → H^1_(_n(_p),_p^).

For (n,p) ≠ (2,2), the claim follows from <ref>. In the case (n,p) = (2,2), it follows from <ref> and the fact that the class α∈ H^1_(Π_2,C^) is a lift of the nontrivial class in H^1_(_2(_2),_2^) with 2α=0, that the above sequence is split short exact.




The map
    
    r P_1→ H^1_(_1,(_1,K))

is an isomorphism.


    Consider the following exact sequence which is an instance of (<ref>) applied to (<ref>):
        
    0 → P_1  H^1_(_1,(_1,K)) → H^1_(^0_C,)^Π_1.

    Since _C^0 = (C,_C), we have H^1_(^0_C,)^Π_1 = 0.

















 
    The map 
        
    _^∗ H^1_(_1,K,)^_1→ H^1_(_n,K,)^_n

is injective.

  
It is enough to show that the map 
    
    _^∗ H^1_(_1,K,)→ H^1_(_n,K,)

is injective. This follows from the observation that any point (K,W)→_n,K induces a retract for ^∗.



 
Assume that (n,p)=(2,2). The map
    
    H^1_(Π_2,(_C^1)) → H^1_(_2,K,)

of the low degree exact sequence (<ref>) sends the class α from <ref> to 0.  Therefore, the class α lifts to P_2.


Consider the short exact sequence of sheaves:
    
    0→μ_2 →→ 0.

This short exact sequence gives rise to the following diagram with exact right vertical column:
    

    (_2,K)
          (_2,K)
    H^1_(Π_2,μ_2(_C^1))   H^1_(_2,μ_2)   H^1_(_2,K,μ_2)
    H^1_(Π_2,(_C^1))   H^1_(_2,)   H^1_(_2,K,).["( )^2"', from=1-3, to=2-3]
    	["∂"',from=2-3, to=3-3]
    	[from=3-1, to=3-2]
    	[from=3-1, to=4-1]
    	[from=3-2, to=3-3]
    	[from=3-2, to=4-2]
    	[from=3-3, to=4-3]
    	[from=4-1, to=4-2]
    	[from=4-2, to=4-3]

Let α̃∈ H^1_(Π_2,μ_2(_C^1)) be the class defined as the composite 
    
    Π_2 →_2(_2) _2(/2) ≃ S_3 {± 1}≃μ_2(_C^1).

The image of α̃ under the map H^1(Π_2,μ_2(_C^1)) → H^1(Π_2,(_C^1)) is α.
Let α∈ H^1(_2,K,μ_2) be the image of α̃  under the composition of the top horizontal maps. It now suffices to show that α lies in the image of ∂(_2,K) → H^1(_2,K,μ_2).   

The group H^1(_2,K,μ_2) classifies étale μ_2-torsors (i.e., double covers) of the rigid-analytic space _2,K, and the map ∂ sends a principal unit in A_2^ to the double cover of _2,K obtained by adjoining its square root.  The μ_2-torsor corresponding to α can be described this way:  Let H^_2 be the universal deformation of H_2 over A_2.  The 2-torsion subgroup H^_2[2] is a group scheme of order 4 over _2= A_2, whose generic fiber H^_2[2]_K is étale over _2,K.  Trivializations of H^_2[2] over _2,K (or, what is the same, markings of its three nonzero sections) form a torsor under the group _2(/2) S_3.  Finally, α is the quotient of this torsor corresponding to the sign homomorphism S_3→μ_2, which is to say it is the torsor classifying markings of the three nonzero sections of H^_2[2] up to cyclic permutation.  

Let E be a supersingular elliptic curve over _2, with universal deformation E^.  By Serre–Tate theory, there is an isomorphism between _2 and the formal scheme parameterizing deformations of E, which identifies H_2^ with the completion of E^ along its origin.  In particular H_2^[2] is identified with the 2-torsion group E^[2], and α is the double cover of _2,K which classifies markings of the three nonzero sections of E^[2] up to cyclic permutation.

A choice of nonvanishing differential ω on E^ determines a Weierstrass model for E^ over the local ring A_2.  Over A_2[1/2], the model can be brought into reduced form, with affine equation y^2=f(x), for a monic cubic polynomial f(x)∈ A_2[1/2][x].  The nonzero 2-torsion sections of E^ correspond to roots e_1,e_2,e_3 of f(x).  Therefore α is the double cover of _2,K obtained by adjoining the square root of Δ(E^,ω), where Δ(E^,ω)=∏_i<j (e_i-e_j)^2∈ A_2^∗ is the discriminant of the Weierstrass model determined by ω.  For a scalar λ∈ A_2^∗ we have Δ(E^,λω)=λ^12Δ(E^,ω);  since every element is a 12th power in _2^∗, we can find an ω with Δ(E^,ω)∈ A_2^∗∗.    We conclude that α lies in the image of ∂;  indeed it is the image of Δ(E^,ω). 




For (n,p) ≠ (2,2), the map q P_1→ P_n is an isomorphism. When (n,p) = (2,2), the map q is an inclusion of a direct summand with complement /2 generated by α.
    

Formation of the exact sequence (<ref>) is functorial, so we find a diagram

    0  [r]     P_1 [r] [d]_q     H^1_(Π_1,(_C^0)) [d]__^∗[r]     H^1_(_1,K,)^_1[d]__^∗
    
    0  [r]     P_n [r]      H^1_(Π_n,(_C^n-1)) [r]     H^1_(_n,K,)^_n.

When (n,p) ≠ (2,2), the map _^∗ is an isomorphism by <ref> and _^∗ is injective by <ref>, so q is an isomorphism. In the case (n,p) = (2,2), the map q is injective since _^∗ is injective by <ref>. The induced map from the cokernel of q to the cokernel of _^∗ is injective so has size at most /2, again by <ref>. By <ref>, α is a 2-torsion element of P_n that is not in the image of q. Since _^∗ is injective by <ref>, this class thus generates the /2 cokernel.


When p=2, we denote the image of α under the map P_2 → H^1(_2,A_2^∗∗) by α̂.
Combining the lemmas above, we deduce the following theorem.


    For (n,p) ≠ (2,2), there is an exact sequence
        
    0 → H^1_(_1,A_1^∗∗)  H^1_(_n,A_n^∗∗) b→ H^1_(_C^n-1,^∗∗)^Π_n,

    where b is the composite H^1_(_n,A_n^∗∗) → H^1_(_n,^∗∗) → H^1_(_C^n-1,^∗∗)^Π_n in (<ref>). 
    
    When (n,p) = (2,2) there is an exact sequence
        
    0 → H^1_(_1,A_1^∗∗) ⊕/2  H^1_(_2,A_2^∗∗) b→ H^1_(_C^1,^∗∗)^Π_2,

    where b is the composite H^1_(_2,A_2^∗∗) → H^1_(_2,^∗∗) → H^1_(_C^1,^∗∗)^Π_2 in (<ref>).     
         

    Consider the following commutative diagram, which is part of (<ref>):
        
    P_n [r] [d]     H^1_(Π_n,(_C^n-1)) [d] 
    
                H^1_(_n,A_n^∗∗) [r] [rd]_b     H^1_(_n,) [d] 
        H^1_(_C^n-1,^∗∗)^Π_n.

    As an instance of (<ref>), we obtain an exact sequence
        
    0 → P_n → H^1_(_n,A_n^∗∗)  H^1_(_C^n-1,^∗∗)^Π_n.

    <Ref> and <ref> provide isomorphisms
        
    P_n ≅
                        H^1_(_1,A_1^∗∗)    if  (n,p) ≠ (2,2); 
    
                        H^1_(_1,A_1^∗∗) ⊕/2    if  (n,p) = (2,2).

    Together with the commutativity of the left square in (<ref>) and the class α, this gives the two exact sequences of the theorem. 


When n=1, the theorem is uninteresting:  _^∗ is an isomorphism, and the target of b is 0 for dimension reasons.  The next objective is to identify H^1_(_C^n-1,^)^Π_n for n≥ 2.  
In the next section we will prove that this group is a free _p-module of rank 1 and the map b is surjective, generated by the image of ε_p(Σ^2𝕊_K(n)).  This claim implies <ref>.   





§ THE COHOMOLOGY OF DRINFELD SYMMETRIC SPACE _C^N-1


Via the fundamental exact sequence in <ref>, we have reduced the crucial <ref> to a question about H^1_(^n-1_C,), the cohomology of  on the pro-étale site of Drinfeld's symmetric space _C^n-1.  Recall that _C^n-1 is the complement in rigid-analytic ℙ^n-1_C of all _p-rational hyperplanes;  as such it admits an action of _n(_p).   In particular _C^1=ℙ^1_C\ℙ^1(_p) is analogous to the (upper and lower) half-plane ℙ^1(ℂ)\ℙ^1(ℝ).  

The cohomology of _C^n-1 was studied first by Drinfeld <cit.> in the case n=2, and then by Schneider–Stuhler <cit.> for general n.  The main theorem of <cit.> is that if H is a cohomology theory on rigid-analytic varieties satisfying certain reasonable axioms (for instance, homotopy-invariance relative to the unit disc D), then H^i(_C^n-1) is a generalized Steinberg representation of _n(_p) for 0≤ i≤ n, and is 0 otherwise.  (The construction of the generalized Steinberg representations are reviewed below.)  The result applies to ℓ-adic étale cohomology and to de Rham cohomology.

In contrast, étale cohomology with _p-coefficients does not satisfy homotopy-invariance; indeed, H^1_(D,_p) is very large.  One therefore might expect that there cannot be a meaningful description of H^i_(_C^n-1,_p), let alone H^i_(_C^n-1,_p).  Nonetheless, both groups were identified by Colmez–Dospinescu–Niziol <cit.>, <cit.>.  In the latter setting, H^i_(_C^n-1,_p) is an extension of a space of differential forms by a general Steinberg representation.  

Combining these results with the logarithm exact sequence (<ref>), we gain access to H^i_(_C^n-1,), see <ref> below.  Specializing to the case i=1, we find that H^1_(^n-1_C,)^Π_n is (for n≥ 2) isomorphic to the space of _n(_p)-invariants in a generalized Steinberg representation with coefficients in _p/_p.  This space of invariants is shown to be a free rank 1 _p-module in <ref>.




 §.§ Generalized Steinberg representations: definitions
 The following is a rephrasing of the constructions appearing in <cit.>.  For 0≤ r≤ n, let X_r be the set of flags
    
    V_1⊂ V_2⊂⋯⊂ V_r⊆_p^⊕ n,

where V_i is a _p-vector space of dimension i. Note that we in particular allow the empty flag. For  1≤ j≤ r, let X_r,j be set of flags as above with the jth vector space omitted.  The set X_r (resp., X_r,j) is a quotient of _n(_p) by a parabolic subgroup.  As such, it is a profinite set admitting a continuous action of _n(_p); note also the quotient map X_r→ X_r,j.  

For a profinite set S= S_i and a ring A, let (S,A) denote the ring of locally constant functions on S valued in A.  If A carries a topology, we give (S,A)=(S_i,A) the colimit topology, where each (S_i,A) (a finite free A-module) has the natural topology.  Define the generalized Steinberg (or special) representation by

    _r(A)=(X_r,A)/∑_j=1^r (X_r,j,A).

Each _r is a topological group admitting a continuous action of _n(_p). 
As a special case,

    _1(A)=(ℙ^n-1(_p),A)/A.

For any topological A-module M, let

    M^∗ = _(M,A),

meaning continuous A-module homomorphisms M→ A.  One can think of _r(A)^∗ as the module of distributions on X_r whose value is 0 on functions pulled back from any X_r,j.  

In the case r=1, identify ℙ^n-1(_p) with the set of linear forms in n variables up to scaling by _p^∗.  Let δ_ℓ denote the Dirac distribution at ℓ∈ℙ^n-1(_p).  Then any difference δ_ℓ-δ_ℓ' determines an element of _1(A)^∗.  

We give an ad hoc definition of the _n(_p)-module _r(_p/_p)^∗, defining this to be the quotient:

    0 →_r(_p)^∗→_r(_p)^∗→_r(_p/_p)^∗→ 0

Thus for instance elements of _1(_p/_p)^∗ are _p/_p-valued distributions on ^n-1(_p) with total measure 0.


  Assume n≥ 2.  The subset of _1(_p/_p)^∗ fixed by _n(_p) is a free _p-module of rank 1.  


 Let μ be a _p/_p-valued distribution on ^n-1(_p) = ^n-1(_p). For each m≥ 1, we have a _n(_p)-equivariant surjective map 
    
    π_m^n-1(_p)→^n-1(/p^m).

As m varies, the fibers of the π_m form a basis of open subsets of ^n-1(_p), so that μ is determined by its values on the fibers.  Now suppose μ is fixed by _n(_p).  Since _n(_p) acts transitively on ^n-1(/p^m), the value of μ is the same on all fibers of π_m;  let a_m denote the common value.  Then by the additivity of μ:
    
    μ(^n-1(_p)) = p^n-1/p-1a_1=p^n-1/p-1p^n-1a_2 = p^n-1/p-1p^2(n-1)a_3 = ⋯

Via μ↦ (a_1,a_2,…), the subgroup of _1(_p/_p)^∗ fixed by _n(_p) may be identified with the subgroup of ∏_m≥ 1_p/_p satisfying 
    
    0 = a_1 = p^n-1a_2 = p^2(n-1)a_3 = ⋯.

This group is _m /p^(n-1)m_p.


Taking invariants under _n(_p) in the exact sequence (<ref>), we find an injective  connecting map

    ∂ H^0(_n(_p),_1(_p/_p)^∗) → H^1(_n(_p),_1(_p)^∗).

Let μ∈ H^0(_n(_p),_1(_p/_p)^∗) be the generator of this group with a_m=1/p^(n-1)(m-1) for all m≥ 2.  


  Let ℓ∈^n-1(_p) be arbitrary.  Then ∂(μ) is represented by the cocycle
    
    g↦δ_ℓ-δ_g(ℓ).



The measure μ can be lifted to a (non-_n(_p)-invariant) measure μ̃∈_1(_p)^∗ defined by the formula (for m≥ 2)

    μ̃(π_m^-1(x)) = 1/p^(n-1)(m-1)-1,   π_m(ℓ)=x; 
    1/p^(n-1)(m-1),   otherwise.

Then ∂(μ) is represented by the cocycle sending g∈_n(_p) to g(μ̃)-μ̃, and this is easily seen to agree with δ_ℓ-δ_g(ℓ).




 §.§ Results of Colmez–Dospinescu–Niziol


We turn now to some crucial results of Colmez–Dospinescu–Niziol on the étale and pro-étale cohomology of Drinfeld symmetric space _C^n-1 over a complete algebraically closed field C/_p.  

Recall the difference between étale and pro-étale cohomology:  whenever X is a scheme (resp. stack, rigid-analytic space, etc.), one has the constant sheaf /p^n on the étale site of X, and then one defines

    H^i_(X,_p)   =    H^i_(X,/p^n) 
    
    H^i_(X,_p)   =    H^i_(X,_p)⊗__p_p.

The latter is not the cohomology of any sheaf “_p” on X_.  On the other hand, one has the constant sheaf _p on the pro-étale topology X_, defined by sending an object U∈ X_ to the ring of continuous maps U→_p, and then H^i_(X,_p) really is the sheaf cohomology of _p on X_.  The projection X_→ X_ induces a map 

    H^i_(X,_p)→ H^i_(X,_p)

which can fail to be an isomorphism if X is not quasi-compact (for instance, if X is the discrete infinite union of points).  

The cohomology groups H^i_(_C^n-1,_p) ≃ H^i_(_C^n-1,_p) and H^i_(_C^n-1,_p) were computed in <cit.> and <cit.>, respectively.  We now review these results, starting with H^i_.  
We need one more ingredient before we can give the statements, namely the Kummer map

    κ^∗(_C^n-1)→ H^1_(_C^n-1,_p(1)).

This map arises as follows: Consider the sequence of short exact sequences
    
    0 [r]    μ_p^n+1[r] [d]    ^∗[r]^-(-)^p^n+1[d]_(-)^p   ^∗[r] [d]^-id    0 
    
            0 [r]    μ_p^n[r]    ^∗[r]^-(-)^p^n   ^∗[r]      0.

Taking limits results in another short exact sequence, 
    
    0 [r]    _p(1) [r]    _(-)^p^∗[r]    ^∗[r]     0,

and κ is the boundary map in cohomology of this short exact sequence.


   There is a Γ__p×_n(_p)-equivariant isomorphism
    
    r_i_i(_p)^∗ H^i_(_C^n-1,_p(i))
    
such that (in the i=1 case) the relation
    
    r_1(δ_ℓ_1-δ_ℓ_2) = κ(ℓ_1/ℓ_2)

holds for all ℓ_1,ℓ_2∈ℙ^n-1(_p).  On the right side of the relation, ℓ_1 and ℓ_2 are interpreted as _p-rational linear forms in n variables, so that ℓ_1/ℓ_2∈^∗(_C^n-1) is well-defined up to _p^∗, and κ(ℓ_1/ℓ_2) is a well-defined class in H^1_(_C^n-1,_p(1)).


 The map r_i is defined in <cit.>;  for i=1 it is characterized by the relation r_1(δ_ℓ_1-δ_ℓ_2)=κ(ℓ_1/ℓ_2).  (There is a characterization of r_i for general i in terms of a “regulator map.”) The isomorphy of r_i is <cit.>.


Let us turn now to pro-étale cohomology.  For a rigid space X over C, the short exact sequence of abelian sheaves on X_ from <Ref>,
    
    0 →μ_p^∞→^∗∗log→→ 0,

gives rise, after taking sequential limits along the multiplication by p map to a short exact sequence of sheaves of _p-vector spaces on X_
    
    0 →_p(1)→_× plog→→ 0.

We thus get a connecting map ∂_log[-1]→_p(1) in the derived category of sheaves on X_.  The pro-étale cohomology of  was studied in <cit.>:  If ν X_→ X_ is the projection, then R^iν_*Ω^i(-i), where Ω^i=Ω^i_X/C is the sheaf of differential i-forms.  As a result there is a spectral sequence

    H^i_(X,Ω^j(-j)) H^i+j_(X,).

In the special case that X is a Stein space, the coherent sheaves Ω^j_X/C are acyclic, and so 
    
    H^i_(X,)Ω^i(X)(-i).

Therefore the connecting map [-1]→_p(1) induces a map
    
    expΩ^i-1(X)(1-i) → H^i_(X,_p(1)).
  
The map exp is related to the Bloch–Kato exponential, see <cit.> for a discussion.   When X admits a semistable formal model over the ring of integers of a local field, <cit.> identifies the kernel of exp with the module of closed forms d, and identifies the cokernel of exp in terms of the Hyodo–Kato cohomology of the special fiber of the model.  
We are now ready to state the theorem of Colmez–Dospinescu–Niziol on the pro-étale cohomology of _C^n-1.

  There is a Γ__p×_n(_p)-equivariant exact sequence of _p-vector spaces
    
    0→Ω^i-1(_C^n-1)/ dexp→ H^i_(_C^n-1,_p(i)) →_i(_p)^∗→ 0,

compatible with <ref>, in the sense that the evident diagram involving H^i_(_C^n-1,_p(i))→ H^i_(_C^n-1,_p(i)) commutes.


We readily deduce the pro-étale cohomology of μ_p^∞ on _C^n-1:


    There is a Γ__p×_n(_p)-equivariant exact sequence
        
    0→Ω^i-1(_C^n-1)/ dexp→ H^i_(_C^n-1,μ_p^∞(i-1)) →_i(_p/_p)^∗→ 0.



    <Ref> and <ref> together supply a map of exact sequences
        
    0 [r] [d]      H^i_(_C^n-1,_p(i)) [r]^-≅[d]    _i(_p)^∗[r] [d]     0 
    
                0 [r]    Ω^i-1(_C^n-1)/ d[r]^-exp     H^i_(_C^n-1,_p(i)) [r]    _i(_p)^∗[r]     0.

    Note that the right vertical map is injective with quotient _i(_p/_p)^∗, see (<ref>), while the middle vertical map comes from (i-1)-fold twist of the short exact sequence
        
    0 →_p(1)→_p(1) →μ_p^∞→ 0.

    The snake lemma then furnishes the short exact sequence of the corollary.




 §.§ Consequences for H^i_(_C^n-1,)

We gather here some consequences of <ref> and <ref>. The next results concern the -cohomology of _C^n-1.  

 

For i≥ 0 there is a short exact sequence
    
    0 →_i(_p/_p)^*(1-i)→ H^i_(_C^n-1,)→Ω^i,(_C^n-1)(-i) → 0,

where Ω^i, denotes the sheaf of closed differential i-forms on _C^n-1.


    Let ∂_log→μ_p^∞[1] be the connecting homomorphism  of the logarithm exact sequence (<ref>), and write
        
    ∂_log^i H^i_(_C^n-1,) → H^i+1_(_C^n-1,μ_p^∞)

    for the induced map on cohomology. It follows from the long exact sequence associatd to the logarithm sequence that there is a short exact sequence
        
    0 →(∂_log^i-1) → H^i_(_C^n-1,) →(∂_log^i) → 0.

    Using <ref>, we can identify the kernel and cokernel of ∂_log^i. To this end, first note that there is a lift
        
    [d]^∂_log@–>[dl]    
    _p(1)[1] [r]    μ_p^∞[1] [r]    _p(1)[2],

    where the bottom exact sequence is a rotation of the triangle coming from (<ref>). Indeed, this lift exists because the composite →μ_p^∞[1] →_p(1)[2] is trival, since the target is p-complete while the source has p acting invertibly. On cohomology, this lift is precisely the map (<ref>).

    Combining this observation with <ref> and (<ref>), we thus obtain a map of exact sequences
        
    0 [r]    Ω^i(_C^n-1)(-i) [r]^-≅@->>[d]     H^i_(_C^n-1,) [r] [d]^-∂_log^i    0 [d] 
    
                0 [r]    Ω^i(_C^n-1)/ d(-i) [r]^-exp    H^i+1_(_C^n-1,μ_p^∞) [r]    _i+1(_p/_p)^∗(-i) [r]     0

    By our discussion above, the left vertical map is the canonical quotient map. The snake lemma then implies that
        
    (∂_log^i) ≅Ω^i,(_C^n-1)(-i)     and    (∂_log^i) ≅_i+1(_p/_p)^∗(-i),

    as desired.


 
    Consider the short exact sequence of pro-étale sheaves on _C^n-1:
        
    0→_p(1) →lim_p→ 0.

    The associated long exact sequence for cohomology has zero boundary maps. 
    Therefore for every m ≥ 0 we obtain a short exact sequence of Π_n-modules
        
    0→ H^m_(ℋ_C^n-1,_p(1)) →  H^m_(ℋ_C^n-1,lim_p ) →  H^m_(ℋ_C^n-1, )→ 0.

    

The claim is equivalent to showing that the map
    
    H^m_(ℋ_C^n-1,lim_p ) →  H^m_(ℋ_C^n-1, )
 
is surjective for all m. Note that this map factors as
    
    H^m_(ℋ_C^n-1,lim_p ) →lim_p H^m_(ℋ_C^n-1, ) → H^m_(ℋ_C^n-1, ).

By the Milnor sequence, the first map  
    
    H^m_(ℋ_C^n-1,lim_p ) →lim_p H^m_(ℋ_C^n-1, )

is a surjection. The second map is surjective as well, since 
H^m_(ℋ_C^n-1,) is p-divisible. This follows from <ref>, which exhibits H^m_(ℋ_C^n-1,) as an extension of p-divisible abelian groups.  


Finally, we identify the Π_n-invariants in H^1_(_C^n-1,).  


  
There is a canonical isomorphism
    
    H^1_(_C^n-1,)^Π_n(_1(_p/_p)^∗)^_n(_p).

Thus if n≥ 2, then by <ref>, H^1_(_C^n-1,)^Π_n is a free _p-module of rank 1 with distinguished generator. When n=1 we have H^1_(_C^0,)^Π_1=0.


 Take Π_n-invariants in the exact sequence in the i=1 case of <ref>. Since ^n-1 is defined over _p we have an isomorphism
    
    Ω^1,(_C^n-1)(-1)Ω^1,(^n-1)⊗__p C(-1),
 
so that
    
    H^0(Γ__p,Ω^1,(_C^n-1)(-1))Ω^1,(^n-1)⊗__p H^0(Γ__p,C(-1))=0.

Therefore we have an isomorphism
    
    H^0(Π_n,_1(_p/_p)^∗) H^0(Π_n,H^1_(_C^n-1,)).

The group Π_n acts on _1(_p/_p)^∗ through its quotient _n(_p), and we have seen in <ref> that the module of invariants is rank 1 over _p for n≥ 2. For n=1, _C^0 = (C,_C) and thus H^1_(_C^0,
ℱ) is zero for any sheaf ℱ.




 §.§ A primitivity result
 Assume throughout that n≥ 2. 
Let us recall the fundamental exact sequence from <ref>, applying what we have learned from <ref>:

    0 [r]      H^1_(_1,A_1^∗∗) [r]     H^1_(_n,A_n^∗∗) [r]^-b    H^1_(_C^n-1,)^Π_n[d]_
             _p

Our current goal is to prove that the map labeled b is surjective.  To do this, it suffices to show that the image under b of the class ε_p(Σ^2𝕊_K(n)) is primitive, i.e. that it is a generator of the _p.  More precisely, we will prove:


 The map b in <ref> sends ε_p(Σ^2𝕊_K(n)) to the distinguished generator of the _p-module H^1_(_C^n-1,)^Π_n. 




The players in the proof of <ref> fit into the following commutative diagram:

    H^1_(_n,A_n^) [d]_g_0@/^-4pc/[dd]_b    H^1_(_n,^∗) 
    
        H^1_(_n,) [d]_e_1[ru]^-g_1[rd]_f_1    H^1_( Π_n, ^∗(_C^n-1)) [u]_g_2[d]^f_2
    
        H^0(Π_n,H^1_(_C^n-1,)) [r]_-ϕ_2[d]_i_1^≅    H^1_(Π_n, H^1_(ℋ_C^n-1,_p(1))) [d]^i_2
    
        H^0(_n(_p),_1(_p/_p)^∗) [r]^-∂    H^1_(_n(_p),_1(_p)^∗).

We explain the maps appearing in (<ref>). The map labeled g_0 arises from the equivalence _n [_n,K^♢/_n], g_1 arises from the inclusion →^∗, and g_2 arises from the equivalence _n [_C^n-1/Π_n]. The map labeled ϕ_2 appears as a connecting map when applying the long exact sequence in continuous Π_n-cohomology to the short exact sequence of Π_n-modules in 
<ref>.  The bottom map labeled ∂ is the connecting map from (<ref>); it is injective as noted in the discussion around (<ref>).  The map i_1 is the isomorphism from <ref>, and i_2 arises from the identification of H^1_(_C^n-1,_p(1)) with _1(_p)^∗ in <ref>. Finally, e_1 is the map from (<ref>) arising in the edge sequence, and b = e_1 ∘ g_0 and f_1 = ϕ_2 ∘ e_1 are the composites. 

For the construction of f_2, start with the short exact sequence of Π_n-equivariant sheaves 
on _C^n-1:

    0 →_p(1) →lim_p ^* →^* → 0.

Consider the boundary map 

    ^*(ℋ_C^n-1) → H^1_(ℋ_C^n-1,_p(1))

which is a map of Π_n-modules. Applying H^1_(Π_n,-) gives

    f_2  H^1_( Π_n, ^∗(_C^n-1)) → H^1_(Π_n, H^1_(ℋ_C^n,_p(1))).


 Let a_1=g_0ε(Σ^2(p^n-1)𝕊_K(n)), and let a_2∈ H^1_(Π_n,^∗(_C^n-1)) be the class of the cocycle g↦ (g(ℓ)/ℓ)^p^n-1, where ℓ∈^∗(_C^n-1) is a _p-rational linear form.  Then g_1(a_1)=g_2(a_2).



 Recall that H is the universal formal group over _n, and H[1/p] is its corresponding line bundle.  Also recall that by <ref>, under the isomorphism _n [_C^n-1/Π_n], the line bundle H[1/p] corresponds to (1)|__C^n-1.  Since ℓ is a nowhere vanishing section for (1)|__C^n-1, the image of a_2 in H^1_(_n,^∗) is ( H[1/p])^⊗ (p^n-1).  This equals g_1(a_1), ultimately because ε(Σ^2𝕊_K(n))∈ H^1_(_n,A_n^∗) is the class of H as a _n-equivariant invertible A_n-module.
 


    Consider two classes x_1∈ H^1_(_n,) and x_2 ∈ H^1_( Π_n, ^∗(_C^n-1)).
If g_1(x_1) = g_2(x_2), then f_1(x_1)  = f_2(x_2).


The proof of <ref> is postponed to the next subsection.

  By <ref> and <ref>, we have

    f_1g_0ε(Σ^2(p^n-1)𝕊_K(n))=(p^n-1)f_2(c),

where c the class of the cocycle g↦ g(ℓ)/ℓ in H^1_(Π_n,^∗(_C^n-1)).  The prime-to-p part of H^1_(_n,A_n^∗) has order (p^n-1), and so (p^n-1)ε_p(Σ^2𝕊_K(n))=ε(Σ^2(p^n-1)𝕊_K(n)).  Since (p^n-1) acts invertibly on the _p-module H^1_(Π_n,H^1_(_C^n-1,_p(1))), we must have f_1g_0ε_p(Σ^2𝕊_K(n))=f_2(c).  Apply i_2 and use the commutativity of the diagram in (<ref>) to find that ∂ i_1bε_p(Σ^2𝕊_K(n)) equals the cocycle g↦δ_g(ℓ)-δ_ℓ from <ref>.  Since ∂ is injective and i_1 is an isomorphism, we conclude that bε_p(Σ^2𝕊_K(n)) is the distinguished generator of H^1_(_C^n-1,)^Π_n.  





 §.§ Proof of <ref> and <ref>

We begin with some preparatory lemmas.


Assume that C and D are complexes in D() such that C ≃τ^[0,1]C and D ≃τ^[0,1]D (i.e., C,D are concentrated in cohomological degrees 0 and 1). Assume that H^*(C) is p-divisible and H^*(D) is derived p-complete.   Then 
every map f  C → D factors as 

    C → H^1(C)[-1] → H^0(D) → D,

where H^1(C) and H^0(D) are viewed as complexes concentrated in degree 0.


Since the cohomology groups H^*(D) are derived p-complete, D is derived p-complete, so the map f factors as
    
    C →C→ D,

where C is the derived p-completion of C. Since H^*(C) is p-divisible, we have H^i(C) = H^i+1(C). In particular, we have C≃τ^[-1,0]C. We thus get a decomposition 
    
    C →C→ H^0(C) → H^0(D) → D

and  H^0(C) ≅H^1(C)[-1]. We get that the composition C →C→ H^0(C) factors
as 
    
    C → H^1(C)[-1] →H^1(C)[-1]≃ H^0(C).

Combining the last two displayed factorizations gives the claim.


When A is a p-divisible sheaf (for any topology) there is a short exact sequence of sheaves

    0 → T_pA →_p A → A → 0,

where T_pA=_m A_p^m. This induces a map A → T_p A[1]. Now, the map τ^[0,1]RΓ(A) →τ^[0,1]RΓ(T_pA[1]) satisfies the requirements of <ref>. This provides us with a factorization:

    τ^[0,1]RΓ(A) [r] [d]     H^1(A)[-1] [d] 
    τ^[0,1]RΓ(T_pA[1])     H^1(T_pA)[0]. [l]


  In the derived category of abelian groups with Π_n-action,
    the map 
    
    H^1_(_C^n-1, _p(1))[0] →τ^[0,1]RΓ(_C^n-1,  _p(1)[1])

    is sent to an isomorphism  via the functor
    
    A↦ H^1_(Π_n,A):= H^1(A^hΠ_n).



    Consider the cofiber sequence 
        
    H^1_(ℋ_C^n-1, _p(1))[0] →τ^[0,1]RΓ_(_C^n-1,  _p(1)[1]) → H^2_(_C^n-1, _p(1))[-1].

    It is enough to show that 
        
    H^0(Π_n,H^2_(_C^n-1, _p(1))) = 0.

From <ref> we have H^2_(_C^n-1,_p(1))(_p)^∗(-1), which has no nonzero Π_n-fixed vectors because _p(-1)^Γ__p=0.



Consider the following diagram, in which all cohomology is computed on the pro-étale site of _C^n-1:


    τ^[0,1]RΓ() [ddl] [ddd] [rr]        H^1()[-1] [ddd] 
             
    τ^[0,1]RΓ(^∗) [dr]        H^0(^∗)[0] @-'[l][ll] [dr]    
       τ^[0,1]RΓ(_p(1)[1])        H^0(_p(1)[1]). [ll]


The diagram is commutative: the front rectangle commutes by <ref>, the bottom parallelogram commutes by naturality of the edge maps, and the left triangle arises from the composition ^∗∗→^∗→_p(1)[1].  

Applying the functor A↦ H^1(A^hΠ_n) to the above diagram yields: 


    !    H^1(_n, ) [ddl]_g_1[ddd]_ϕ_1[rr]^e_1       H^1(_C^n-1,)^Π_n[ddd]_ϕ_2
             
    
    H^1(_n, ^∗) [dr]_δ       H^1(Π_n,^∗(_C^n-1)) @-'[l][ll]_      g_2[dr]_f_2   
        H^1(Π_n,RΓ(_C^n-1,_p(1)[1]))        H^1(Π_n,H^1(_C^n-1,_p(1)[1])). [ll]_e_2^∼

Here e_2 is an isomorphism by <ref>.
We can now prove <ref>, in which f_1=ϕ_2∘ e_1.  Given that g_1(a_1)=g_2(a_2), we have 
    
    f_1(a_1)=ϕ_2e_1(a_1)=e_2^-1δ g_1(a_1) =e_2^-1δ g_2(a_2)=f_2(a_2),

as desired.


<Ref> is now proved.  We explain how it implies <ref>, which in turn implies our main theorem (<ref>), see the discussion following <ref>.  


    We first consider the case (n,p)≠ (2,2). <Ref> together with <ref> state that there is an exact sequence
        
    0→ H^1_(_1,A_1^)  H^1_(_n,A_n^)→_p → 0

    in which ε_p(Σ^2𝕊_K(n)) is sent to the generator of the _p.  Since _p is free, the exact sequence splits:
        
    H^1_(_n,A_n^)_p⊕ H^1_(_1,A_1^).
 
    Similarly for (n,p)= (2,2), we have an isomorphism
        
    H^1_(_2,A_2^)_2⊕ H^1_(_1,A_1^)⊕/2,
 
    where in both cases the _p summand is generated by ε_p(Σ^2𝕊_K(n)). The term H^1_(_1,A_1^) has been computed in <ref>; substituting this into (<ref>) resp. (<ref>) yields isomorphisms
        
    H^1_(_n,A_n^∗∗) ≅_p^2     if  p>2; 
    _2^2 ⊕ (/2)^⊕ 2   if  p=2  and  n>2; 
    _2^2 ⊕ (/2)^⊕ 3   if  p=2  and  n=2.

    As explained in <Ref>, a generator of the torsion-free part of H^1_(_1,A_1^) is sent to the determinant in H^1_(_n,A_n^), thus finishing the proof.




alpha


 Paper Title: Fontaine-Laffaille Theory over Power Series Rings.




§ ABSTRACT 
Let k be a perfect field of characteristic p > 2. We extend the equivalence of categories between Fontaine-Laffaille modules and ℤ_p lattices inside crystalline representations with Hodge-Tate weights at most p-2 of <cit.> to the situation where the base ring is the power series ring over the Witt vectors R  W(k) t_1, ⋯ , t_d and T a p-adically complete ring that is étale over the Tate Algebra W(k)⟨ t_1^± 1, ⋯ , t_d^± 1⟩. 

Fontaine-Laffaille Theory over Power Series Rings
    Christian Hokaj
    August 5, 2024
=================================================



Introduction
Let k be a perfect field of characteristic p > 2, W(k) the ring of Witt-vectors over k with φ its Frobenius automorphism, K its fraction field, and R  W(k) t_1, ⋯ , t_d the ring of power series over W(k) in d variables. Let T be a “small base ring," a ring that is p-adically completed étale over the Tate algebra, W(k)⟨ t_1^± 1, ⋯ , t_d^± 1⟩ (i.e. the ring of restricted Laurent series over W(k)), for some d. Let r be an integer such that 0 ≤ r ≤ p-2. When the base ring is W(k), Fontaine and Laffaille introduced in <cit.> strongly divisible W(k)-lattices, which we now call Fontaine-Laffaille modules, to study ℤ_p-lattices in crystalline representations of G_KGal(K/K) with Hodge-Tate weights in [0, r]. Denote this category by Rep_ℤ_p, [0,r]^cris(G_K). We first recall the definition of a Fontaine-Laffaille module in the classical setting:



 A finite free Fontaine-Laffaille module over W(k) is a triple (M, ^i(M), φ_i) where:


  * M is a finite free module over W(k)

  * {^i(M)}_i=0^∞ is a decreasing filtration of M such that ^i+1 (M) is a direct summand of ^i(M) for all i, ^0(M) = M, ^r+1(M) = 0. 

  * φ_i are φ-semilinear maps ^i(M) → M whose restriction to ^i+1(M) is pφ_i+1.  

  * ∑_iφ_i (^iM)  = M.



Let MF^ff, [0,r](W(k)) denote the category of finite free Fontaine-Laffaille modules over W(k). These modules classify lattices within crystalline representations as in the following theorem:


When 0 ≤ r ≤ p-2, the functor given by 
    T_cris(M)  (^r(A_cris⊗_W(k) M))^φ_r=1
 is an equivalence of categories 
    MF^ff, [0,r](W(k)) ↔Rep_ℤ_p, [0,r]^cris(G_K).
 


The work of Faltings in <cit.> extended this theory to more general base rings. Our goal is to extend the above theorem to the setting where the base ring is R and to the setting of a small base ring T. 

Let us first recall the representation category in the relative case. Let G_R denote Gal(R[p^-1]/R[p^-1]) = π_1^ét(SpecR[1/p], η) where η is a fixed geometric point and  R is the union of finite R-subalgebras R' of a fixed algebraic closure of Frac(R) such that R'[p^-1] is étale over R[p^-1]. Let Rep_ℤ_p, [0,r]^cris(G_R) denote the category of crystalline ℤ_p representations of G_R with Hodge-Tate weights in [0,r] and we make the analogous definitions over the base ring T. 

Using recent results of Du, Liu, Moon, and Shimizu in <cit.> and the category of finite, free Fontaine-Laffaille modules in the relative case first introduced by Faltings in <cit.> and more recently studied by <cit.>, which we will denote MF^ff, [0,r]_∇(R), we obtain the following result (see Definition <ref> for the definition of Fontaine-Laffaille modules in the relative case, and see Section <ref> for a brief review of the period rings and crystalline representations in the relative case): 


Let r be an integer satisfying 0 ≤ r ≤ p-2 and let  denote either the base ring R or T. The functor 
    T_cris: MF^ff, [0,r]_∇()   →Rep_ℤ_p, [0,r]^cris(G_)
 given by 
    T_(M)  (^r(A_() ⊗_ M))^φ_r=1
 is an equivalence of categories.
 Showing this functor is fully faithful is routine, and the difficulty lies in showing essential surjectivity and specifically associating a Fontaine-Laffaille module to a given lattice inside a crystalline representation. 

The key input of <cit.> is the existence of a Kisin module  over the ring u associated to an object of Rep_ℤ_p, [0,r]^cris(G_
). The ring  comes equipped with a Frobenius map φ_ extending the Frobenius on . We can then define the associated Fontaine-Laffaille module as /u⊗_, φ_, but the Kisin module alone is not enough to obtain the full data of a Fontaine-Laffaille module over .

In the classical case, Breuil in <cit.> studied ℤ_p lattices in crystalline representations of G_K using strongly divisible S-modules. To obtain the full data of our Fontaine-Laffaille module, we define S in the relative setting to be the p-adic completion of the divided power envelope of  taken with respect to (E(u)) with E(u) = u-p or u+p. The ring S comes with a canonical divided power filtration and additional structure which will descend to our proposed Fontaine-Laffaille module. 


To show the main theorem holds over R, we will verify that the data properly descends through the use of two base change maps relating our R-module back to the classical theory. Let k_g be the perfection of Frac(R/pR) and R_g  W(k_g), which gives rise to a flat embedding R ↪ R_g. Let b: R → W(k) denote reduction moduo (t_1, ⋯, t_d). A result of <cit.> allows us to verify that M is an object of MF^ff, [0,r]_∇(R) by checking that M ⊗_R R_g and M ⊗_W(k), b W(k) are Fontaine-Laffaille modules over W(k_g) and W(k) in the classical sense.

To show the main theorem holds over T, we obtain a result analogous to that of <cit.> where we can check that M is Fontaine-Laffaille at the p-adic completion of localizations at maximal ideals. This enables us to use a base change T → R as described in Section <ref> and reduce to the setting where the base ring is R. 

Recently Würthen in <cit.> proved this result in the setting of a smooth p-adic formal scheme over a mixed characteristic complete discrete valuation ring with perfect residue field. Our work uses different methods to focus on the case where the base ring is more limited, and it is routine to extend the results from a small base ring to accommodate the setting of a smooth, p-adic formal scheme.  

The  work of Imai, Kato, and Youcis in  <cit.>,developed simultaneously, uses different methods to obtain a different fully faithful functor between lattices inside crystalline representations and Fontaine-Laffaille modules in a relative setting. Their functor has the advantage of being extendable beyond the Fontaine-Laffaille range, and it yields objects in a nicer category. Our functor has the advantage of being essentially surjective. They also place additional assumptions on the prismatic F-crystal, and their technique has not yet been applied to the case where the base ring is a power series ring. 

Acknowledgements: The author would like to thank his Ph.D advisor Tong Liu for suggesting the work in this paper, and for his extensive and invaluable guidance, comments, and conversation throughout the production of this paper and its earlier drafts. This paper includes and generalizes the content of the author's Ph.D. thesis at Purdue University when he was partially supported by the Ross Fellowship of Purdue University and a Summer Research Grant during the summer of 2021. 


Preliminary Results on some Categories and Functors in the Relative Setting
Base Ring Conventions
Much of integral p-adic Hodge Theory has been established in the relative case in far more generality than Fontaine-Laffaille theory. We present the most general setting we will consider here. 

Recall k is a perfect field of characteristic p > 2. Let K^tr be a totally ramified extension of K = Frac(W(k)) with ring of integers 𝒪_K^tr. Fix π a uniformizer of K^tr and let E(u) ∈ W(k)[u] denote the monic minimal polynomial of π. Write W(k)⟨ T_1^± 1, ⋯ , T_d^± 1⟩ for the p-adic completion of the Laurent polynomial ring W(k)[ T_1^± 1, ⋯ , T_d^± 1]. 

In order to reference results with maximal generality, we will say ℛ is a general base ring if ℛ is a p-adically complete 𝒪_K^tr-algebra which is of the form ℛ = 𝒪_K^tr⊗_W(k)ℛ_0 where ℛ_0 is an integral domain obtained from W(k)⟨ T_1^± 1, ⋯ , T_d^± 1⟩ by a finite number of iterations of the operations:


  * p-adic completion of an étale extension

  * p-adic completion of a localization;

  * completion with respect to an ideal containing p.



Our new results will hold for a subset of these general base rings which we describe next: 



We will say a general base ring T is small if it is p-adically completed étale over W(k)⟨ T_1^± 1, ⋯ , T_d^± 1⟩ for some d. 


The key strategy for showing our results over small base rings will be to reduce to the case of the power series ring R = W(k) t_1, ⋯ , t_d due to the local structure of small base rings described in the following lemma:



Let T be a small base ring that is p-adically compaeted étale over W(k)⟨ T_1^± 1, ⋯ , T_d^± 1⟩ and let 𝔪 be a maximal ideal (note that p ∈𝔪). Then T_𝔪, the p-adic completion of the localization of T at 𝔪, is isomorphic to W(k) t_1, ⋯ , t_d.

Let 𝔪 be a maximal ideal of T, and let  denote the p-adic completion of the localization of T at 𝔪. The ring T is a regular Noetherian ring, and thus ≅ W(k) X_1, ⋯ , X_d≅ R by Cohen's Structure Theorem.



We will try to reserve the letter R in usual font for referring to such a power series ring, the letter T in usual font for a small base ring, and the stylized ℛ for a general base ring. We will use the stylized 𝒯 to concisely state results that hold for a small base ring or the power series ring R. 

When we refer to the “classical" theory we will be referring to the setting where the base ring is K: the historically typical setting of p-adic Hodge Theory. 

We will now describe the Frobenius structure and the category of Galois representations for a general base ring. We will let φ_ℛ denote the lift of Frobenius on ℛ/W(k) uniquely determined by φ(T_i) = T_i^p. Let ℛ denote the union of finite ℛ-subalgebras ℛ' of a fixed algebraic closure of Frac(ℛ) such that ℛ'[p^-1] is étale over ℛ[p^-1]. Set G_ℛ = Gal(ℛ/ℛ[p^-1]). Let Rep_ℚ_p(G_ℛ) denote the category of finite dimensional ℚ_p vector spaces with continuous G_ℛ action, and let Rep_ℤ_p^ff(G_ℛ) denote the full category of finite free ℤ_p modules equipped with a continuous action of 𝒢_ℛ. 


This describes the Frobenius structure for a general base ring, but we will actually take φ_R(t_i) = (t_i+1)^p -1 for our base ring R in order to use the results of <cit.> as stated. This is not significant, as we will show the relevant category of Fontaine-Laffaille modules defined below is independent of the choice of Frobenius. 

Important Base Change Maps

Our main technique for developing the Fontaine-Laffaille Theory for new base rings will be to base change to rings where the Fontaine-Laffaille Theory is already known. We describe the important base change maps here.

Shilov Point Base Changes

Let k_g be the perfection of Frac(ℛ/pℛ) and let ℛ_g  W(k_g). We have a φ-compatiable, flat embedding ℛ↪ℛ_g. Set _ℛ,g W(k_g) u,  and let S_ℛ,g  be the p-adically completed divided power envelope of _ℛ,g with respect to E(u) which comes equipped with a φ structure and PD-filtration. As the Shilov point above forgets the data of the connection ∇, we will also need to consider a second Shilov point with imperfect residue field. Let ℛ_L denote the p-adic completion of the localization ℛ_(p). Then ℛ_L is a complete discrete valuation ring with imperfect residue field. We define _L = ℛ_L  u and S_L to be the p-adically completed divided power envelope of _L with respect to (E(u)). 

Closed Fiber Base Change of a Power Series Ring

In the case of our base ring R, we also have a natural projection π: R → W(k) given by t_i ↦ 0 for every i. Showing that the Fontaine-Laffaille Theory holds for R will amount to checking it is compatible with the Shilov point base change and the closed fiber base change.

Small Base Ring Base Changes

We will now discuss a base change map from a small base ring T to R suggested by Lemma <ref>. Let 𝔪 = (p, f_1, ⋯ , f_d) be a maximal ideal of T (recall T is regular), and let  denote the p-adic completion of the localization of T at 𝔪 which is isomorphic to ≅ W(k) X_1, ⋯ , X_d≅ R by Lemma <ref>. 

This gives us a flat base change ring homomorphism b_𝔪: T →≅ R. In order to make this map Frobenius equivariant, though, we must consider φ_R to be the lift of the Frobenius on W(k) which acts on the X_i as φ_R(X_1) = b_𝔪(φ_T(f_i)). For example, if 𝔪 = (p, t_1-1, ⋯, t_d - 1), then we must take φ_R(X_i) = (X_i+1)^p-1. 

By Lemma <ref> below, there is a canonical equivalence between the categories of finite, free Fontaine-Laffaille modules over R with different Frobenii, so this choice will not impact our results.  


Review of relative period rings and crystalline representations

In this section we briefly review the crystalline period rings and the functor D_ in the relative case. For full details of their constructions, see <cit.> for a treatment of the period rings and <cit.> for a treatment of relevant functors. Let ℛ be a general base ring as in Convention <ref>. Let ℛ be the p-adic completion of ℛ. Set ^♭ = _φ/p. Then we define A_inf() = W(^♭) and write [π^♭] for the Teichmüler lift of π^♭ where π^♭ = (π^1/p^n)_n ≥ 0 is a compatible sequence of p-power roots of π. Let θ: () → be the unique surjective W(k)-algebra homomorphism lifting the first projection.  

Let () be the p-adic completion of the divided power envelope of () with respect to the kernel of θ. Let ϵ_n ∈ denote a (non-trivial) sequence of compatible p-power roots of unity (i.e. such that ϵ_0 = 1), ϵ_1 ≠ 1, ϵ_n = ϵ_n+1^p. Let ϵ = (ϵ_n)_n ∈^♭ and t = log[ϵ] ∈(). Let () ()[p^-1, t^-1].  

Let θ_0 be the extension of θ to _0 ⊗_W(k)() → and set () to be the p-adic completion of the divided power envelope of _0 ⊗_W(k)() with respect to the kernel of θ_0. Let () ()[p^-1, t^-1].

For V ∈Rep_ℚ_p(G_ℛ), we have a functor 
    D_(V)  (OB_(ℛ) ⊗_ℚ_p V)^G_ℛ
 which is a finite projetive ℛ[p^-1] module of rank at most dim_ℚ_pV equipped with a φ and an integrable connection ∇  induced by that on OB_(ℛ). We say that V is crystalline if the natural map 
    OB_(ℛ) ⊗_ℛ[p^-1] D_(V) →OB_(ℛ) ⊗_ℚ_p V
 is an isomorphism. An object N ∈Rep_ℤ_p^ff(G_ℛ) is said to be crystalline if N ⊗_ℤ_pℚ_p is crystalline. There is also a contravariant version of D_ defined to be 
    D_^∨(V) _G_ℛ(V, OB_(ℛ))
 dual to the above covariant version. 


Finite free Fontaine-Laffaille modules
Here we introduce the various categories of Fontaine-Laffaille modules in the relative case, following the work of <cit.> and <cit.>. Here we will need to distinguish between the case of the base ring R and a small base ring T for the statement of some theorems. 



Let MF_big^f(R) be the category whose objects consist of a an R-module M, a sequence of R-modules F^i(M), and sequences of R-linear maps ι_i: F^i(M) →F^i-1(M), π_i: F^i(M) → M, and R-semi-linear maps φ_i: F^i(M) → M, satisfying the following conditions: (1) The composition F^i(M) F^i-1(M)  M is the map π_i: F^i(M) → M(2) The map π_i : F^i(M) → M is an isomorphism for i ≪ 0. (3) The composition of φ_i-1 with ι_i: F^i(M) →F^i-1(M) is pφ_i. Morphism between these objects are compatible collections of R-linear maps between M's and F^i(M)'s.


We write MF_∇, big^f(R) when we consider the category of such modules equipped with an integrable connection ∇: M → M ⊗_RΩ_R for which the φ_i are parallel and that satisfies Griffiths transversality. Specifically, when the following conditions hold:

    
  * Griffiths-transversality holds. i.e. the following diagram commutes: 
    scale=1,center
  [scale=1, column sep=large]
    F^i(M) d∇_i rι_i     F^i-1M d∇_i-1

    F^i-1M⊗_R Ω_R rι_i-1 ⊗_R Ω_R     F^i-2(M) ⊗_R Ω_R
  


    Note that this simplifies to ∇ (F^i(M)) ⊂F^i-1(M) ⊗_R Ω_R when the ι_i are injective. 
    
  * The φ_i are parallel: ∇∘φ_i = (φ_i-1⊗_R dφ_1) ∘∇ as a map from F^i(M) to M ⊗_R Ω_R where dφ_1: Ω_R ⊗_R, φ R →Ω_R is dφ /p.


We write MF_∇, big^f, [a,b](R) when F^a(M) = M and F^b+1(M) = {0}. 

We will replace “f" with “tor" when working in the setting of p-power torsion modules as in <cit.>. We define these categories analgously when working over small base rings.



Let MF^ff(R) denote the full subcategory of MF^f(R) such that M and F^i(M) are finitely generated and free, each F^i+1(M) is a direct summand of F^i(M), F^i(M) = {0} for i ≫ 0, and ∑_iφ_i(F^i(M)) generate M.  We call an object in this category a “finite free Fontaine-Laffaille module (over R)" or sometimes just a “Fontaine-Laffaille module (over R)" for convenience. The tags ∇, [a, b], and “tor" are used in the same way as for the big category, and we define the categories analagously for small base rings.
 

We next recall a main theorem of <cit.> which we will refine to our finite free setting over the base ring R, and we will develop an analogue for small base rings. Recall the Shilov point R_g from Section <ref> and that here is a φ-compatible, flat embedding b_g: R ↪ R_g. Recall from Section <ref> that there is a natural projection b: R → W(k) determined by t_i ↦ 0 for each i. A main theorem of <cit.> is as follows:  <cit.>

Suppose M ∈MF_big^tor, [0, r](R) such that each ^i(M) is finite as an R-module. Then M is in MF^ff(R) if and only if: (1) Both M_0  M ⊗_R, b W(k) and M_g  M ⊗_R R_g are objects in MF^tor, [0,r](W(k_g)) and MF^tor, [0,r](R_g), respectively. (2) M_g ≅ M_0 ⊗_W(k) R_g as R_g-modules.
 

Lt MF_big^ff, [0,r](R) denote the full subcategory of MF_big^f, [0,r](R) where M and each ^i(M) are finite, free R-modules. Then we have:



Theorem <ref>, holds if each “tor" is replaced with “ff."

If M ∈MF_big^ff, [0,r](R), by tensoring with W(k) and R_g and noting that M and ^i(M) are finite free R-modules, it is clear that (1) and (2) hold. 

We can show the reverse direction by reducing to the torsion case. Let n be a positive integer. It is clear that M/p^nM ⊗_R, b W(k) ≅ M ⊗_R R/p^nR ⊗_R, b W(k) ≅ M_0/p^nM_0 and M/p^nM ⊗_R R_g ≅ M_g/p^nM_g are elements of MF^tor, [0,r](W(k)) and MF^tor, [0,r](R_g), respectively. 

By tensoring M_g ≅ M_0 ⊗_W(k) R_g with R/p^nR, we also see that M_0/p^nM_0 and M_g/p^nM_g are of the same type. Then by Theorem <ref>, we conclude that M/p^nM is an element of MF^tor, [0,r](R). Since M =  M/p^nM and F^i(M) = F^i(M)/p^nF^i(M) and each is free, we deduce that each is finite free and we have the direct summand condition. Similarly, as φ_i(F^i(M/p^nM)) generates M/p^nM, the same condition lifts to M.



 

We will now build to a theorem analogous to Theorem <ref> but which holds for small base rings. First we will show that the category MF_∇^ff(R) is independent of the lift of Frobenius. 


Let a,b be such that 0 ≤ b-a ≤ p-1. Let φ, ψ be two lifts of the natural Frobenius on W(k) to R which coincide modulo p. We write R_φ (resp. R_ψ) when we are considering R with the Frobenius lift φ (resp. ψ). Then there is an equivalence between the corresponding categories MF_∇^ff,[a,b](R_φ) and  MF_∇^ff,[a,b](R_ψ), and up to canonical isomorphism, MF_∇^ff, [a,b](R) is independent of the lift of Frobenius.  


We show the result for the torsion categories, and the result for the finite free categories follows via projective limits. 

The proof of <cit.> applies almost identically in the torsion case, but we do not have an étale map from W(k)[t_1, ⋯ , t_d] → R, but we still have that {∂_i}_i=1^d {∂/∂ X_i}_i=1^d gives the dual basis of R-derivations. From there we proceed identically to Faltings. By shifting if necessary, we can assume that a = 0, b = p-1. The ∂/∂ X_i act on M via ∇. Given a multi-index I = (i_1, ⋯, i_d), we get an endomorphism ∇(∂)^I of M. Let (φ(X)-ψ(X))^I denote ∏_j=1^d(φ(X_j)-ψ(X_j))^i_j, and |I| = ∑_j=1^d i_j, and I! = ∏_j=1^d i_j!. Comparing divisibility by p, we can see this gives a well defined element. 

Note that an object M ∈MF_big^tor, [a,b](R) is in MF^tor, [a,b](R) if and only if the Frobenius ϕ on M induces an isomorphism M⊗_R, φ→ M where M is the right exact functor defined explicitly as the cokernel of the map 
    θ_M: ⊕_i=a+1^b F^i(M) →⊕_i=a^b F^i(M),
 where θ_M is given by 
    θ_M((x_a+1, ⋯, x_b)) = (ι_a+1(x_a+1), -px_a+1+ι_a+2(x_a+2), ⋯, -px_b-1 + ι_b(x_b) , -px_b).
  Equivalently, M can be defined as the colimit of the following diagram:

    ⋯→F^i+1(M) ←F^i+1(M) →F^i(M) ←F^i(M) →F^i-1(M) ←⋯
 where the right arrows denote the ι maps and the left arrows denote multiplication by p.  

Now we define a map α: M⊗_R, φ R →M⊗_R, ψ R. Let m ∈ F^i(M), which defines an element of M, and it suffices to show α is an isomorphism. We define: 
    α(m ⊗ 1) = ∑_I∇(∂)^I(m) ⊗(φ(X) - ψ(X))^I/I! · p^min(|I|, i).
 Noting that p^|I| divides (φ(X)-ψ(x))^I, we can confirm that (φ(X) - ψ(X))^I/I! · p^min(|I|, i) gives a well-defined element of R and that this sum converges to 0 in the p-adic topology. If |I| ≤ i, the fraction is obviously well-defined as it forces |I| ≤ p-1 and v_p(I!) = 0. If |I| > i, this can be done by writing each i_j in its base p representation as i_j = ∑_k=0^∞ a_j,kp^k and using the well-known identity 
    v_p(i_j!) = 1/p-1∑_k=0^∞ a_j,k(p^k-1).
 Then the p-adic valuation of the fraction becomes at least 

    |I| - min(|I|, i) - ∑_j=1^d 1/p-1∑_k=0^∞ a_j,k(p^k-1)
       = -i +∑_j=1^d∑_k=1^∞ a_j,k[p^k - p^k-1/p-1] 
       = -i +∑_j=1^d∑_k=1^∞ a_j,k[ p^k(p-2) +1/p-1]

We can then see that for p > 2, p^k(p-2)+1 > (p-1)^k. Since i ≤ p-1 < |I|, the p-adic valuation is nonnegative and thus the original fraction is well-defined, and we can see that |I| - v_p(I!) grows without bound as |I| does. 

As in Faltings, Taylor's formula φ(r) = ∑_Iψ(∇(∂)^I(r)) ⊗ (φ(X)-ψ(X))^I/I! shows us that α gives a well-defined map. The α's satisfy transitivity for three different Frobenius lifts by the binomial formula. Applying this to φ, ψ, φ gives us that the α are isomorphisms. It is easy to verify that α is parallel for the connections. 



We are now ready to state the analogue of Theorem <ref> for a small base ring T: 


    Suppose M is in MF_big^f(T) such that each F^i(M) is finite as a T-module. Then M is in MF^ff(T) if and only if (M_𝔪, F^i(M_𝔪))  (M ⊗_b_𝔪, T , F^i(M) ⊗_b_𝔪,T) ∈MF^ff(T_𝔪) for all maximal ideals 𝔪 of T.  


Again we show the result in the torsion case and the result in the finite free case follows from taking projective limits. 

For the forward direction, it is clear that M_𝔪, F^i(M_𝔪) are finitely generated p-power torsion -modules with F^i(M_𝔪) = {0} for i ≫ 0 since M and F^i(M) have these properties over T. The axioms of MF_big() are verified directly by properties of the tensor product and the fact that b_𝔪 is φ-equivariant with our choice of Frobenius. Recalling that M is a cokernel and again using that b_𝔪 is φ-equivariant, we obtain an isomorphism M ⊗_b_𝔪, T≅M⊗_b_𝔪, T. Thus φ_M_𝔪: M_𝔪⊗_, φ_𝔪 induces an isomorphism, and M_𝔪 is an object in MF(). 

For the reverse direction, we are assuming that each F^i(M) is a finite T module, so we need only check that φ_M : M⊗_T, φ T → M is an isomorphism. It suffices to check that this is an isomorphism locally at maximal ideals. Since T_𝔪 is a Noetherian local ring and (p) is an ideal contained in 𝔪, the ring map T_𝔪→T_𝔪 is faithfully flat. 
Thus it suffices to check that T_𝔪⊗φ_M is an isomorphism for every maximal ideal 𝔪, which is true by assumption since, as in the previous paragraph, M_𝔪≅M ⊗_b_𝔪, T≅M⊗_b_𝔪, T.



Étale (φ_𝒪_ℰ, 𝒪_ℰ)-modules
In this section and the next we will work over a general base ring  as in Convetion <ref>. Here we review the theory of étale (φ_𝒪_ℰ, 𝒪_ℰ)-modules in the relative case. 

Let _ℛℛ u with Frobenius φ__ℛ obtained by setting φ__ℛ(u) = u^p. Let 𝒪_ℰ, ℛ = be the p-adic completion of _ℛ[u^-1] with Frobenius φ_𝒪_ℰ extending that on _ℛ. 


An étale (φ_, )-module is a pair (ℳ, φ_ℳ) where ℳ is a finitely generated -module and φ_ℳ: ℳ→ℳ is a φ-semilinear endomorphism such that the linearization 1 ⊗φ_ℳ: φ^*ℳ→ℳ (i.e. the map c ⊗ m ↦ cφ_ℳ(m) for c ⊗ m ∈⊗_φ_, ℳ) is an isomorphism. 


Let Mod^pr_𝒪_ℰ denote the category of projective étale (φ_, )-modules whose morphisms are φ_-compatible -linear maps and Mod^ff_𝒪_ℰ the category with the projective condition replaced by finite free. We now recall how étale (φ_, )-modules relate to Galois representations. 

Recall that π_n ∈K^tr is chosen compatibly so that π_0 = π and π^p_n+1 = π_n. Set K_∞ to be the p-adic completion of ∪_n=0^∞ K^tr(π_n) and K_∞^♭ its tilt. Let E_ℛ_∞^+ = /p and Ẽ_ℛ_∞^+ denote the u-adic completion of _φ E_R_∞^+. Write ℛ̃_∞ for W(Ẽ_ℛ_∞^+) ⊗_W(K_∞^♭)𝒪_K_∞.


While we have described ℛ̃_∞ in full generality for completeness, for our base ring R we will only need to use R̃_∞ which has a much more explicit description as 
    R̃_∞ = ⋃_n ≥ 1, 1 ≤ i ≤ d R(π_n, √(1+t_i))
 for a fixed choice of compatibly chosen p^nth roots of 1+ t_i.


Then we have the following relationship: 

<cit.>, <cit.>

There is a functor ℳ from the category Rep_ℤ_p^ff(G_ℛ̃_∞) of finite free ℤ_p-modules with continuous G_ℛ̃_∞ action to the category Mod^pr_𝒪_ℰ which is an exact equivalence of categories. The inverse of ℳ is given by 
    T_(ℳ)  (^ur⊗_ℳ)^φ=1.
 This functor behaves well with respect to base change in the following sense: Let ℛ' be another base ring equipped with Frobenius satisfying the same conditions as ℛ with a φ-equivariant map ℛ_0 →ℛ_0'. If T ∈Rep_ℤ_p^ff(G_R̃_∞), then T can be considered as a G_ℛ̃_∞'-representation via the map G_ℛ̃_∞'→ G_ℛ̃_∞ and we have the isomorphism 
    𝒪_ℰ, ℛ'⊗_ℳ(T) ≅ℳ_ℛ̃'(T)
 as étale (φ_𝒪_ℰ, ℛ', 𝒪_ℰ, ℛ')-modules. 


Kisin Modules

We next review Kisin modules in the relative case and the main result of <cit.> which provides the key input to the proof of our main theorem. 

First we recall the the ring  considered in <cit.>. Let ℛ_ denote the absolute prismatic site of ℛ. In full generality,  is the ring ℛ_0 u and E is an choice of minimal Eisenstein polynomial of π (in our setting, E(u) = u-p or u+p). We write (, (E)) for the Breuil-Kisin prism. Then (, (E)) is defined to be the self-product of (, (E)) in ℛ_ and that it satisfies a universal property as follows: 

If (B, I) ∈ℛ_ and if we are given maps f_1, f_2: (, (E)) → (B, I) such that the maps ℛ→ B/I induced by f_1, f_2 agree, then there is a map (, (E)) → (B, I) uniquely determined by f_1, f_2. We will write p_1, p_2 for the maps (, (E)) → (, (E)). We will also utilize the triple-self product of (, (E)) denoted (^(2), (E)). 

The details of the construction of (, (E)) and a justification that this self-product exists can be found in <cit.>. We now introduce the category of free Kisin modules with descent data. 



Let DD_ denote the category consisting of triples (, φ_, f) where

    
  *  is a finite -module that is projective away from (p,E) and saturated.
    
  * φ_: → is a φ-semi-linear endomorphism such that (, φ_) has finite E-height.
    
  * f: ⊗_p_1, →⊗_p_2, is an isomorphism of -modules that is compatible with Frobenii and satisfies the cocycle condition over ^(2).
    
An object of this category is called an integral Kisin descent datum. If we replace the third piece of data with a map f: [p^-1] ⊗_p_1, →[p^-1] ⊗_p_2, that is an isomorphism of [p^-1]-modules that is compatible with Frobenii and satisfies the cocycle condition over ^(2)[p^-1], we call such an object a rational Kisin descent datum. Let DD_, [0,r]^ff denote the full subcateogry consisting of objects with E-height ≤ r and which are finite free over . 


The category DD_ is equivalent to the category of completed prismatic F-crystals on ℛ introduced in <cit.>. 


The main theorem of <cit.> is the following:



    There is an equivalence of categories between DD_, [0,r] and Rep_ℤ_p, [0,r]^.


A main result of <cit.> is the association of a Kisin module to a crystalline representation:

<cit.>

Let V be a crystalline ℚ_p representation of G_ℛ with Hodge-Tate weights in [0,r], and let Λ be a finite free ℤ_p lattice of V stable under the G_ℛ action. Let ℳ be the  étale (φ_, )-module associated to Λ as in Proposition <ref>. Then there exists a  submodule  of ℳ stable under Frobenius such that:


  *  with the induced Frobenius is a Kisin module over  of E-height ≤ r.

  * ⊗_ = ℳ 

  *  is equipped with the data of ∇_, a map 
    ∇_: (ℛ_0 ⊗_φ, ℛ_0/u)[p^-1] → (ℛ_0⊗_φ, ℛ_0/u)[p^-1] ⊗_ℛ_0Ω_ℛ_0
 which is a topologically quasi-nilpotent connection commuting with Frobenius and satisfying rational S-Griffiths transversality. (See <cit.> and Definition <ref> below.)
 


If , ℳ, and Λ are as above, we call  a Kisin module associated to Λ. On ocassion we will want to forget the ∇ structure on the Kisin module.



In the setting of our base ring R, <cit.> shows that the Kisin module associated to a lattice in a crystalline representation is projective. As R is a local ring, the Kisin module is thus a free -module. Furthermore, in the setting of a small base ring T, the Kisin module associated to a lattice in a crystalline representation is projective (but not necessarily free).




Base Change of the Kisin Module

Here we will again work over a general base ring ℛ to state the base change theorems in full generality. 

We recall the following standard fact from commutative algebra:

<cit.>

Let A be a ring, Q a flat A-module, and N_1, N_2 submodules of an A-module N. Then as submodules of Q ⊗_A N, we have 
    Q ⊗_A (N_1 ∩ N_2) = (Q ⊗_A N_1 ) ∩ (Q ⊗_A N_2).

 

Shilov point base change of the Kisin module

We will now recall the base change properties of the Kisin module. Let ℛ' be another another general base ring for which there exists a map ℛ→ℛ' compatible with Frobenius which makes ℛ' into an ℛ-module.



Let (, φ_) be the module and φ_ data of the Kisin module associated to the lattice Λ inside the crystalline representation V as in Theorem <ref> with ∇ structure forgotten. Let (', φ_') be the module and φ_' data of the Kisin module associated to Λ|_G_ℛ' with ∇ structure forgotten, and assume  and ' are finite free. Then ⊗_ℛℛ' gives the module and φ_' data of the Kisin module associated to Λ|_G_ℛ', i.e. ⊗_ℛℛ'≅'. 


Let ℳ denote the étale φ-module associated to Λ as in Theorem <ref>.  Let ' denote the Kisin module associated to Λ|_G_ℛ' and ℳ' its associated étale φ-module. We have injections ↪ℳ and ' ↪ℳ'. By Proposition <ref>, we know that ℳ' ≅𝒪_ℰ, ℛ'⊗_𝒪_ℰ, ℛℳ and so we also have an injection 
    ⊗__ℛ_ℛ'↪ℳ'.


Then ⊗__ℛ_ℛ' and ' are both the module data of Kisin modules lying inside of ℳ'. We claim there is unique such module data. By  <cit.> and Proposition <ref> there is a unique Kisin module _L associated to the étale φ_𝒪_ℰ, L-module ℳ_L =ℳ' ⊗_𝒪_ℰ, ℛ'𝒪_ℰ, ℛ_L' and thus 
    ⊗__ℛ_ℛ'⊗__ℛ_L'_ℛ'_L= '⊗__ℛ'_ℛ_L'
 as submodules of ℳ_L. We also know that 
    ⊗__ℛ_ℛ'⊗__ℛ'𝒪_ℰ, ℛ'≅ℳ' ≅' ⊗__ℛ'𝒪_ℰ, ℛ'.
 Thus we have ⊗__ℛ_ℛ' contained in the intersection of ' ⊗__ℛ'_L' and ' ⊗__ℛ'𝒪_ℰ, ℛ'. By Lemma <ref> and the fact that _ℛ_L'∩𝒪_ℰ, ℛ' = _ℛ', this intersection is ', concluding the proof as ' and  have the same rank and our maps are all compatible with the φ structures. 


Working in the setting of R, set _0  W(k) u and let S_0 be the p-adically completed divided power envelope  of _0 with respect to E(u). We have induced maps _R →_0 and 𝒪_ℰ, R→𝒪_ℰ, W(k) which confirm that if ℳ is an étale (φ_𝒪_ℰ, R, 𝒪_ℰ, R)-module corresponding to a ℤ_p-stable lattice inside a crystalline representation of G_R then ℳ_0 ℳ⊗𝒪_ℰ, W(k) is an étale (φ_𝒪_ℰ, W(k), 𝒪_ℰ, W(k))-module corresponding to a ℤ_p-stable lattice inside a crystalline representation of G_K. Then by <cit.> or as a corollary of Theorem <ref>, we have 

If  is a Kisin module associated to a ℤ_p-stable lattice in a crystalline representation of G_R, then ⊗__0 is, in the classical setting of <cit.>, a Kisin module associated to a ℤ_p stable lattice in a crystalline representation of G_K.



    Here we are using that  is free by Remark <ref> and that r ∈ [0, p-2], which is why the corollary has only been stated for the base ring R.



Breuil Modules

In this section we will work over  which is either a small base ring or the power series ring R. We define S_ to be the p-adically completed divided power envelope of _ u with respect to E(u) a choice of minimal polynomial of p. The ring S_ comes equipped with a Frobenius map φ_S_ that extends φ__. It also comes with its divided power filtration that we will denote ^iS_. Note that for 1 ≤ i ≤ p-1, φ_S_(^i S_) ⊂ p^iS (just consider the action of φ_S_ on E(u)^i/i!), so we can set φ_S_,i = φ_S_/p^i: ^i S_→ S_. The ring S_ also comes equipped with an integrable connection ∇_S_: S_→ S_⊗_Ω_.



For r ≤ p-1 a positive integer, let Mod_S, ∇^ff, r() be the category whose objects are quadruples (, ^i(), ∇_, φ_, i) where:
 

  *  is a finite, projective S_-module.

  * ^i() is a decreasing filtration of  with ^0() =, ^r+1() ⊂ (^1 S_ ), and (^iS_)^i⊂^i+1.

  * φ_, i are φ_S-semilinear maps φ_, i: ^i() → so that the composite ^i() →^i-1() is pφ_, i. Also, φ_r(^r) generates .

  * ∇_ is a topologically quasi-nilpotent integrable connection which satisfies S-Griffiths transversality (see Definition <ref> below) and commutes with each φ_, i as before. 


We refer to objects of this category as Breuil modules.


Given a Kisin module , we define the associated Breuil module to be 
    () ⊗__, φ__ S_.
 If such a Kisin module exists for a given Breuil module , we will say that “ arises from a Kisin module." In the next few sections, we show how the full data of a Breuil module can be constructed on ⊗__, φ__ S_. 

The filtration from the Kisin module

 Let  be a Breuil module that arises from a Kisin module  which arises from a crystalline representation. We define a decreasing filtration on ℳ[p^-1] via:
    ^i([p^-1]) {m ∈[p^-1]  |  (1 ⊗φ_)(m) ∈^i(S_[p^-1]) ⊗__}
 and a filtration on  via 
    ^i() ∩^i([p^-1]).
  We have the following lemma which shows that the filtration structure descends integrally:



Let  be a Breuil module that arises from a Kisin module  and let 0 ≤ i < p-1. We can express the filtration on  as in Definition <ref> as 
    ^i() = {m ∈ | (1 ⊗φ_)(m) ∈^i(S_) ⊗_}.
 







Observe that ^i() ⊂^i([p^-1]), so we can realize an element of ^i() as an element m ∈ such that (1⊗φ_)(m) is contained in 
    (^i(S[p^-1]) ⊗__) ∩ (S_⊗__) = (^i(S_[p^-1]) ∩ S) ⊗_
 where we are relying on Lemma <ref> (note  is projective and hence flat). By comparing elements term-by-term we can show ^i(S_[p^-1]) ∩ S_ = ^iS_, as needed. Specifically, an element in S_[p^-1] can be written as 
    x = ∑_k=0^∞E(u)^k/k!∑_j=0^deg(E)-1 a_kj u^j
 with a_kj∈[p^-1]. For this element to be in ^i(S_[p^-1]) we must have a_kj = 0 whenever 0 ≤ k < i. But x ∈ S_ with each a_kj = 0 whenever 0 ≤ k < i, so x ∈^i(S_), as well.



The φ_i structure 

 Let  be a Breuil module that arises from a Kisin module . We can also define a φ_, i structure by setting φ_, i to be the composition: 
    φ_, i : ^i() ^i(S_) ⊗_.
 
 
Note the first arrow is well-defined by the definition of ^i(). It is then clear that the composite ^i() →^i-1() is pφ_, i because φ_S_,i = φ_S_/p^i. 
The integrable connection 
Here we discuss where the connection 
    ∇_[p^-1]: [p^-1] →[p^-1] ⊗_R Ω_R
 arises from, and then we will show it descends to the integral setting. 

Let  be a Breuil module that arises from a Kisin module with descent datum . By Theorem <ref> there exists Λ, a lattice inside a crystalline representation V Λ⊗_ℤ_pℚ_p corresponding to  under the equivalence of categories. Then we consider D_^∨(V) which is a finite projective [p^-1]-module equipped with an integrable connection ∇_D, a φ, and filtration structure which we will denote ^i((V)). Set M = ⊗_φ, /u. Then by <cit.> there is a φ-compatible isomorphism M[p^-1] →(V). By <cit.> we can identify 𝒟 S_[p^-1] ⊗_ M with [p^-1] as modules. Also see Lemma <ref>.

The connection ∇_D can be further extended to an integrable connection on 𝒟⊗_[p^-1] S_[p^-1] defined as ∇_[p^-1]∇_D ⊗ 1 + 1 ⊗∇_S_[p^-1], and 𝒟 comes equipped with a filtration defined inductively by setting ^0𝒟 = 𝒟 and: 
    ^i𝒟{x ∈𝒟 | N_u(x) ∈^i-1𝒟, q(x) ∈^i((V))},
 where N_u: 𝒟→𝒟 is the -linear derivation N_u, S_⊗ 1 and q(x) is the projection → M as in <cit.>. By <cit.>, the filtration ^i([p^-1]) defined in Definition <ref> using the φ-structure on  and the filtration ^i(𝒟) defined above agree and, consequently, ∇_[p^-1] satisfies rational S_-Griffiths transversality as we define below:



 We say the connection ∇_[p^-1] satisfies rational S_-Griffiths transversality if for every i 
    ∂_u(^i+1([p^-1]) ⊂^i([p^-1])
 and 
    ∇_[p^-1](^i+1([p^-1]) ⊂^i([p^-1]) ⊗_R Ω_R
 where ∂_u: [p^-1] →[p^-1] is the derivation given by ∂_u, S_[p^-1]⊗ 1. 

If ∇_[p^-1] restricts to a connection ∇_ : →⊗_Ω_ where the same condition on the filtration holds with ^∙ ([p^-1]) replaced by ^∙( ), we say that ∇_ satisfies integral S_-Griffiths transversality.


We now show this connection descends to the integral setting. 

 Let  be a Breuil module that arises from a 
Kisin module . Then  is stable under ∇_ and satisfies integral S-Griffiths transversality. 


We consider the base change _L ⊗_S S_,L to the Shilov point with imperfect residue field (defined in Section <ref>. Let V be the crystalline representation corresponding to the étale (φ_𝒪_ℰ, 𝒪_ℰ)-module ⊗__. By <cit.> there is a Breuil module _L' associated to V|_G_L. 

By <cit.> there exists a Kisin module _L' such that _L' = _L' ⊗_φ__,L, _,L S_,L. By its uniqueness and Theorem <ref> we must have _L' ≅⊗___,L and thus _L' ≅⊗_φ, _ S_,L≅_L. 

Thus _L is stable under its integrable connection ∇__L = ∇_⊗ 1 + 1 ⊗∇_S_L. Thus the image of  under ∇_[p^-1] is contained in [p^-1] ∩_L. By Lemma <ref>:

    [p^-1] ∩_L    = (⊗_S S_[p^-1]) ∩ (⊗_S_ S_,L) 
       = ⊗_S_ (S_[p^-1] ∩ S_,L) 
       =

where S_[p^-1] ∩ S_,L = S_ can be checked term-by-term as in the proof of Lemma <ref> below. 

For integral S-Griffiths Transversality, we know ∇_(^i+1()) ⊂^i ⊗_R Ω_R as it satisfies rational S-Griffiths transversality and that ∇_(^i+1()) ⊂⊗_R Ω_R because  is stable under ∇_. We conclude that ∇_ satisfies integral S-Griffiths Transversality from Definition <ref>.




The functor from Fontaine-Laffaille modules to Breuil modules
In the classical setting, it has already been established as the main result of <cit.> that there is a “direct" equivalence of categories between Fontaine-Laffaille modules and Breuil modules. We rely on this equivalence when reducing the relative setting to the classical setting. It is also well-known that the same story holds in the classical case as in the above section. There is an equivalence of categories between Kisin modules and lattices inside of crystalline representations. If Λ is a lattice associated to Kisin module , there is a functor D_ such that (Λ⊗_ℤ_pℚ_p) ≅ (/u)[p^-1] where M /u is a Fontaine-Laffaille module in the classical sense of <cit.>. 

In the relative setting there is an analogous “obvious" functor from Fontaine-Laffaille modules to Breuil modules. 

  Let M ∈MF_∇, [0,r]^ff. Then we can define the functor (M) = M ⊗_ S_, the p-adic completion of M ⊗_ S. The additional structure is defined in the natural way as follows:


  * φ_φ_M ⊗φ_S_ 

  * ∇_∇_M ⊗ 1 + 1 ⊗∇_S_

  * ^r() ∑_i=0^r ^i(M) ⊗_R^r-i(S_)

  * φ_r ∑_i=0^r φ_i, M⊗φ_r-i, S_

where we've added an “M" tag to a structure if it arises from that appropriate structure on M and an “S" tag if it arises from the appropriate structure on S_. Much of our focus will be on trying to reverse this functor: obtaining Fontaine-Laffaille module data from Breuil module data. 

 In the classical setting, this functor is well-known to be an equivalence of categories and Hui Gao established a direct quasi-inverse in <cit.>

 
     Our results will imply that this functor M ↦(M) is an equivalence of categories in the setting with base ring , as well. 
 



Functors to Galois Representations
Definition of 

Let M ∈MF^ff, [0,r]_∇(R). Now we recall how the Galois action on (R) ⊗_R M and hence 
    (M)  (^r(A_cris⊗_R M))^φ_r=1
 is defined. This is discussed in <cit.>. Set φ on R to extend the natural φ on W(k) with φ(t_i) = (1+t_i)^p -1. Choose 1+t_i∈R^♭ to be a fixed sequence of compatible p^ith roots of 1+t_i and denote by [1+t_i] its Teichmüller lift. For any g ∈ G_R we define β_i(g) = g([1+t_i]) -[1+t_i] ∈^i(R), and we have for a ⊗ x ∈(R) ⊗_R M: 
    g(a ⊗ x) = ∑_I g(a)γ_I(β(g)) ⊗∇ (∂)^I(x)
 where γ_I(β(g)) is defined as ∏_ℓβ_ℓ(g)^i_ℓ/∏_ℓ i_ℓ! and I = (i_1, ⋯ , i_d) is a multi-index with i_k ≥ 0 for all k and ∇(∂)^I = [∇(∂/∂ t_i)]^i_1⋯[∇(∂/∂ t_d)]^i_d. It follows from <cit.> that this formula gives a well-defined -semilinear action of G_R which is compatible with the filtration structure (i.e. g(⊗_R ^rM) ⊂^r(⊗_R M)) and commutes with the Frobenius structure.  This ensures (M) is well-defined and that (M) ∈Rep_cris, ℤ_p^[0,r](G_R). 

(M) can be defined analogously for M ∈MF^ff, [0,r]_∇(T), but φ(t_i) is set to be t_i^p and we use t_i∈T^♭ instead of 1+t_i.

The definition of T_()

Write  for _R. Let (, φ_, f) be a Kisin module with descent data associated to a crystalline representation. Here we will work over our base ring R and explain how the map f as in Definition <ref> is used to construct a Galois action on ⊗_ as in <cit.>. First recall that we have an embedding R ↪ via t_i ↦ [t_i+1]-1. 

Let f_1: (, (E)) → (, (E)) be the map determined by u ↦ [π^♭] which extends the above embedding of R into . Let g ∈ G_R and let f_2: (, (E)) → (, (E)) be the map determined by u ↦ g([π^♭]) and t_i ↦ g([t_i+1]-1). Then the universal property described in Section <ref> gives a unique map f_g: (, (E)) → (, (E)) and since f satisfies the cocycle condition there exists a map d making the following diagram commute: 

scale=1,center
  [scale=1]   
  ⊗_p_1,  rf df_1    ⊗_p_2,  df_g

  ⊗_ rd_g     ⊗_g,   

and from this we have a G_R action on ⊗_. Alsom the G_R action commutes with the φ-structure. 


We will make use of this G_R action on two separate functors. We define 
    T_()  (⊗_ W(R^♭[1/u]) )^φ=1
  and 
    T^r_()  (^rφ^*⊗_)^φ_r=1
 where we write φ^* = ⊗_φ, and we set ^r φ^* = {x ∈φ^*| (1 ⊗φ)(x) ∈ E(u)^r} and φ_r: ^rφ^*→φ^* defined by 
    φ_r(x) φ(x)/φ(E(u)^r).


These functors are related via 
    T^r_() ≅ T_()(r)
 as in <cit.>, where (r) denotes the rth Tate twist. Their proof only establishes the isomorphism in the classical case, but the proof works mutatis mutandis in the relative case. 


We are using the covariant versions of these functors rather than the contravariant versions used in <cit.> (defined below) which are dual to the covariant versions. With this in mind, if we take a lattice Λ∈Rep_cris, ℤ_p^[0,r](G_R), we should first consider Λ^∨(r)∈Rep_cris, ℤ_p^[0,r](G_R), the dual with the rth Tate twist applied. Then using Theorem <ref> as stated for the contravariant versions of the functors, there exists a Kisin module with descent data (, φ_, f) such that T_^∨() = Λ^∨(r) Then taking duals and applying the rth Tate twist, we obtain T_^r() = Λ, so we do have a Kisin module with descent data (, φ_, f) with T_^r() = Λ for our covariant version, too.




    We can define T_ and T_^r analogously when working over a small base ring T using the analogous embedding of T →(T) via t_i ↦ [t_i]. Note that the embedding T → induces an embedding (T) →() which gives an isomorphism __T() → T__(⊗_T) when  is a Kisin module over _T.


Equivalence between T_ and T_
Again let  be a Kisin module associated to a crystalline representation. By <cit.>, the natural G_R̃_∞-equivariant map 
    T_^∨() _, φ(, ^ur) →_, φ(ℳ, ^ur) = T_^∨(ℳ)
 is an isomorphism, though we need to make slight adjustments because we are using the covariant versions of these functors as defined above rather than the contravariant versions sometimes used in <cit.>. 



The natural map 
    T_(ℳ) → T_()
 is an isomorphism of G_R̃_∞ representations. 

Recall ℳ = ⊗_. 

The embedding f_1: → = W(R^♭) above extends to an embedding ^ur→ W(R^♭[1/u]) which gives us a natural embedding 
    T_(ℳ) = (ℳ⊗_^ur)^φ=1 = (⊗_^ur)^φ=1↪ (⊗_ W(R^♭[1/u]))^φ=1 = T_().
 The last embedding is known to be an isomorphism of G_R̃_∞ representations as a consequence of <cit.>, completing the proof. 



The definition of T_S()

Write S For S_R. Let  be a Breuil module. We can define a Galois action on (R) ⊗_S is defined using ∇ in the same way as the Galois action on a Fontaine-Laffaille module was defined above. First embed S into (R) where u maps to [π^♭], the Teichmuller lift of a compatible sequence of p-power roots of π. If we let g ∈ G_R and a ⊗ x ∈⊗_S we set 
    g(a ⊗ x) = ∑ g(a) ∂_u^j_0∂_t_1^j_1⋯∂_t_d^j_d(x) ·γ_j_0(g[π^♭]-[π^♭])∏_i=1^d γ_j_i(g([t_i^♭])-[t_i^♭])             (*)
 with notation as above in the previous section and the sum over multi-indices (j_0, ⋯ , j_d) of nonnegative integers. Similarly to Section <ref>, this gives us an (R)-semilinear action of G_R which preserves the filtration structure (g(^r(⊗_S ) ⊂^r(⊗_S)) and commutes with Frobenius. It is also clear that if = M ⊗_R S, then 
    T_S() ^r(⊗_S )^φ_r=1 = ^r(⊗_S (M ⊗_R S))^φ_r=1 = (M).



We can also view this Galois action on (R) ⊗_S determined by ∇ as one arising from the descent data. We first recall that there are rings S^(1) and S^(2) analogous to  and ^(2). The details of the construction can be found in <cit.>. We will write p_1 and p_2 for the maps from S to S^(1) analogous to the p_1 and p_2 defined in Section <ref> 


Given (, φ_, f) a
Kisin module associated to a lattice in a crystalline representation as in Theorem <ref> , <cit.> constructs a descent datum f_S on [p^-1] = S[p^-1] ⊗_R M as follows: 

Let ∂_u: S[p^-1] ⊗_R M → S[p^-1] ⊗_R M be the derivation given by ∂_u, S⊗ 1 and let ∂_t_i be the derivation given by ∂_t_i, S⊗ 1 + 1 ⊗∂_ti, M. Then we can define f_S: S^(2)⊗_p_1, S[p^-1] → S^(2)⊗_p_2, S[p^-1] as 
    f_S(x) = ∑_J ∂_u^j_0∂_t_1^j_1⋯∂_t_d^j_d(x) γ_j_0(p_2(u)-p_1(u)) ∏_i=1^d γ_j_i(p_2(t_i) -p_1(t_i)),              (**)
 where the sum is over multi-indices J = (j_0, ⋯ , j_d) of nonnegative integers. 

This descent datum determines a Galois action on (R) ⊗ in the same way as the descent datum f determined a Galois action on ⊗_φ^* just with  and  replaced by S^(2) and . Specifically, let g ∈ G_R. The maps f_1 and f_2 from Section <ref> extend to embeddings S → and by the analogous universal property as discussed at the beginning of Section <ref>, there exists a unique map f_S, g: (S^(1), (E)) → ((R), (E)) and since f_S satisfues the cocycle condition, there exists a map d, depending on g, making the diagram commute: 

  scale=1,center
  [scale=1]   
  S^(1) ⊗_p_1, S rf_S df_1    S^(1) ⊗_p_2, S df_S,g

  ⊗_S rd_g     ⊗_g, S which establishes the desired Galois action on ⊗_S. 

So f_S determines a Galois action of G_R on ⊗_S and hence on 
    T_S() ^r(⊗_S )^φ_r=1.


We have seen two ways of defining a Galois action on T_S(): the first via the descent datum f_S and the second via the connection ∇. Via the explicit formulas (*) and (**) above, we see that the two ways of defining the Galois action are the same.


We define T_S analogously when working over a small base ring T instead of R. Writing S for S_T and letting  be a Breuil module over S, we have T_S() ^r((T) ⊗_S )^φ_r=1.



Equivalence between T_^r and T_S
We will now explore the relationship between T_^r() and T_S() defined in the preceding sections. Continue to assume that (, φ_, f) is a Kisin module arising from a ℤ_p-crystalline representation of G_R. Let  be a Breuil module arising from (, φ_, f) as in Section <ref> (recall this means = () ⊗_, φ_ S).

Now consider the natural map 
    α: ⊗_φ^*→⊗_S .
 
  
  By <cit.>, the map f_S as defined in the previous section (originally constructed in <cit.>) satisfies 
    f_S = S^(1)⊗_φ, ^(1) f.
 Thus we see that f_S and S^(1)⊗_φ, ^(1) f define the same Galois action on ⊗_S, as we discussed how f defines a Galois action on ⊗φ^* using the following diagram:

scale=1,center
  [scale=1]   
  ⊗_p_1,  rf df_1    ⊗_p_2,  df_g

  ⊗_ rd_g     ⊗_g,   

  and f_S defines a Galois action on ⊗ via the same diagram mutatis mutandi: 

  scale=1,center
  [scale=1]   
  S^(1) ⊗_p_1, S rf_S df_1    S^(1) ⊗_p_2, S df_S,g

  ⊗_S rd_g     ⊗_g, S 

Thus we see that the Galois action established on T_^r() is compatible with the Galois action on T_S() via the natural map α.
  
  As = (), it is standard to see from the definitions of the filtration structure on  and  that the map α respects the filtration and φ_r structures. This gives us a natural injective map 
    T_^r() → T_S()
 compatible with the G_R-action, filtration, and φ_i, structure. 

We will later prove the following theorem:



The natural injection 
    T_^r() → T_S() ^r(⊗_S )^φ_r=1
 induced by the embedding → is an isomorphism of G_R representations.  

but we will need information about the Fontaine-Laffaille module associated to  which won't be established until the next section. Nevertheless, we state the result here in our section on Galois representations for completeness.



A summary of categories
Here we include a brief diagram summarizing the relationship between the many categories introduced above in the setting of our base ring R:

 scale=1,center
  [scale=1]    
       Rep_ℤ_p, [0,r]^(G_R)    l[above] MF_∇,[0,r]^ff d-  ⊗_R S

    Mod_^ff urT_    l[below]⊗_ - DD_, [0,r]^ff r[below]- ⊗_, φ S  uT_^r[right]≅     Mod_S, ∇^ff, r ul[above]T_S

  
 

T_ is fully faithful
To conclude Section 2, we will show that  is fully faithful. First we work over the base ring R. 

Let M_1 and M_2 be Fontaine-Laffaille modules over R so that there exists a morphism 
    ψ: (M_1) →(M_2)
 of G_R representations. By Proposition <ref> there is an equivalence of categories between ℤ_p stable lattices in finite free G_R_∞ representations and étale (φ_ , )-modules and thus exists a unique morphism ψ_ℳ: ℳ_1 →ℳ_2 where ℳ_i is the étale (, φ_)-module associated to the G_ representation (M_i). Moreover, there is a unique morphism ψ_: _1 →_2 between the associated Kisin modules with descent data by <cit.>. Proposition 3.25 establishes an equivalence of categories between DD^f_ and a category of prismatic F-crystals, and Theorem 3.28 establishes that this category of prismatic F-crystals is equivalent to Rep_ℤ_p, [0,r]^cris(G_). So it suffices to show that ψ_ induces a map ψ_M: M_1 → M_2 such that T_cris(ψ_M) agrees with T_^r(ψ_) on (M_1) ≅ T_^r(_1), which are isomorphic as established above. 

Set ψ_ = ψ_⊗_, φ_ S: _1 →_2 which is a morphism of the associated Breuil modules. We can then obtain a morphism on the associated Fontaine-Laffaille modules using the unique sections guaranteed by Theorem <ref>. Let s_1 be the section M_1 →_1 and s_2 the section M_2 →_2 as in the following diagram: 

 scale=1,center
  [scale=1]    
      _1 rψ_     _2

      M_1 rψ_M us_1    M_2 us_2
  
 

where the bottom map ψ_M is defined to be the natural projection _2 → M_2 composed with ψ_∘ s_1. Tensoring the above diagram with , we see that (ψ_M) agrees with T_S(ψ_). If we let g_i: _i →_i denote the map _i →_i ⊗_, φ_ S, we obtain a similar diagram 

 scale=1,center
  [scale=1]    
      _1 rψ_ dg_1     _2dg_2

      _1 rψ_     _2 
  
 

which shows that T_^r(ψ_) will agree with T_S(ψ_). 

To complete the proof of fully-faithfulness, we need to justify the uniqueness of ψ_M. This follows from the established rational theory. Such a morphism ψ_M: M_1 → M_2 induces a morphism ψ_M[p^-1]: M_1[p^-1] → M_2[p^-1] which is known to be unique as <cit.> establishes an isomorphism M_i[p^-1] → D_^∨(T_(M_i) ⊗_Z_pℚ_p) and the induced morphism on D_^∨ is known to be unique by the original theory of Brinon in <cit.>. 

Thus we have established:

The functor 
    : MF^ff_∇, [0, r](R) →Rep_ℤ_p, [0, r]^cris(G_R)
 given by 
    (M)  (^r(A_cris(R) ⊗_R M))^φ_r=1
 is fully faithful. 


We can also establish fully-faithfulness for small base rings via base change. 

We also have the analogous result for the Tate Algebra: 

The functor 
    : MF^ff_∇, [0, r](T) →Rep_ℤ_p, [0, r]^cris(G_T)
 given by 
    (M)  (^r(A_cris(T) ⊗_T M))^φ_r=1
 is fully faithful. 


    Let M_1 and M_2 be Fontaine-Laffaille modules over T so that there exists a morphism 
    ψ: (M_1) →(M_2)
 of G_T representations. Write  for M ⊗_T . We get an induced map (_1) →(_2), for which there exists a unique morhphism _1 →_2  as  is fully faithful with the base ring R. We also have a unique induced map D_1 = M_1[p^-1] → M_2[p^-1] = D_2, setting up the following diagram:  
    scale=1,center
  [scale=1]    
      _1 rφ    _2 

      M_1 u    M_2 u
  
 
where the vertical maps are injective. But the image of M under φ is contained in _2 ∩ M_2[p^-1] = M_2, completing the proof of fully faithfulness. 











Essential Surjectivity of T_ over Power Series Rings
In this section we aim to prove the following two theorems, working over the base ring R. Let I_0 ⊂ S denote the kernel of the projection S → R given by u ↦ 0.



Let  be a Breuil module which arises from a Kisin module which arises from a Λ, an object in Rep_ℤ_p^ [0,r](G_R).  Set 
    M /I_0≅/u⊗_R, φ_ R.
 Then M carries the data of a Fontaine-Laffaille module and satisfies M ⊗_R S ≅ as in Definition <ref> We call it the “Fontaine-Laffaille module associated to Λ."



Let M be the Fontaine-Laffaille module associated to Λ. Then 
    T_cris(M) ≅Λ.
 


Combined, these theorems complete the proof of essential surjectivity of T_cris. For the remainder of Section 3, we will assume  satisfies the hypotheses of the above theorem.


Fontaile-Laffaille data on M
We will now define the Fontaine-Laffaille data on M and show that M is indeed a Fontaine-Laffaille module with our key tool being Theorem <ref>. As  modules, we have from Theorem <ref> and Proposition <ref> that M_g  M ⊗_R R_g and M_0  M ⊗_R W(k) are isomorphic, as modules, to Fontaile-Laffaille modules arising from ℤ_p stable lattices in crystalline representations, so they come equipped with the data of a classical Fontaine-Laffaille module as in Definition <ref>. We do not yet know, though, whether this data, particularly the filtration data, is compatible with the base change maps. Showing this will be a major focus of this section.
The Filtration and φ_i structure on M

We first develop the filtration and φ_i structure on M ℳ/I_0ℳ≅/u⊗_R, φ R. The key is to show that the following Lemma from <cit.> descends to the integral situation:



Consider the projection q: → M induced by the φ-compatible projection S → R, u ↦ 0. Then q admits a unique φ-compatible section s: M[p^-1] →[p^-1]. Furthermore, 1 ⊗ s: S[p^-1] ⊗_R[p^-1] M[p^-1] →[p^-1] is an isomorphism.


We first prove the following:



The projection q as in Lemma <ref> admits a unique φ-compatible section 
    s: M →.



To attain this section we base change to the Shilov point obtaining the following commutative diagram: 

scale=0.85,center
  [scale=1]
    M [hook]r [hook]d     M[1/p] rs [hook]d     [1/p] [hook]d 

    M_g r     M_g[1/p] [hook]rs ⊗R_g     _g[1/p] 

  


Note that s ⊗ R_g determines a section M_g[1/p] →_g[1/p]. By <cit.> this section is unique and it takes M_g, and thus M, into _g. Tracing M through the top of the diagram, we see that the image of M in _g[1/p] is contained in [1/p]. 

From Lemma <ref>, we obtain:

    _g ∩[1/p]    = ⊗(S_g ∩ S[1/p]) 
       = ⊗ S.


We can verify S_g ∩ S[1/p] = S term by term. If we take x in the intersection, we can write it as 
    x = ∑E(u)^i/i!∑ a_iju^j
 with a_ij∈ W(k_g). We can also write it as 
    x = ∑E(u)^i/i!∑ b_ij u^j
 with b_ij∈ W(k)[1/p]. Comparing term-by-term we can see that a_ij = b_ij. 

 



The map 1 ⊗ s: S ⊗_R M → is an isomorphism.

Since S ⊗_R M and  are finite, free S-modules of the same rank, it suffices to show that 1 ⊗ s is surjective. Moreover, S is a local ring, so by Nakayama's Lemma it suffices to show that 1 ⊗ s is surjective after reducing modulo the maximal ideal of S which is the ideal generated by p, t_1, ⋯, t_d, u, and the divided powers E(u)^i/i!. After reducing modulo the maximal ideal, 1 ⊗ s becomes the identity map which is obviously surjective. 



We now define the filtration and φ_i structure on M. 

Realizing M as a submodule of  via the section s, we can define 
    ^i(M) ^i∩ M.
 


We set φ_M, i = φ_, i|_M.

Note that the filtration on  induces a filtration ^i(M) on M for which the composite ^i(M) →^i-1(M)  M is pφ_M, i. By the corresponding properties on ^i  (see Definition <ref>), it is easy to see that ^0M = M and ^r+1 M = {0}.  

Thus M has the structure of an object in MF_big^f(R). We rely on Proposition <ref> to show that we have an object of MF^ff(R) in the following section.



M ⊗_R R_g ≅ M_g as Fontaine-Laffaille modules


We need to show that the filtration data on the tensor product M 
⊗_R R_g is compatible with the filtration on M_g. The difficulty in doing this is establishing a base change theorem for the Breuil module . The ring S_g is very large, so treating  as an R-module and tensoring with R_g over R will not get us back _g: we would need to p-adically complete ⊗_R R_g. The fact S is non-Noetherian makes p-adic completion difficult to work with. 

Instead, we can work over the ring S/^i(S) which is Noetherian when i < p. This will allow us to decompose ^i(_g) and show that ^i(M_g) is contained in the nicer piece.



Let i be a nonnegative integer less than p. Then we can decompose ^i(_g) as follows: 
    ^i(_g) = ^i ⊗_R R_g + ⊗_S^i(S_g).


Another way to express the definition of ^i is as a kernel in the exact sequence 

    0 →^i →→(S/^iS) ⊗_.1
 
It follows quickly from the definition of ^i that ⊗_S ^i(S) is contained in the kernel, so we get an exact sequence 
    0 →^i /(⊗_S ^i(S)) →/(⊗_S ^i(S)) →(S/^iS) ⊗_.

 We can now tensor with R_g which maintains the exactness since R_g is a flat R-module. As a further conesequence of the flatness of R_g, the tensor product commutes with quotients and we obtain the exact sequence: 


    0 →^i ⊗_R R_g/⊗_S (^i(S) ⊗_R R_g)→/⊗_S (^i(S))⊗_R R_g →( (S/^iS) ⊗_)⊗_R R_g 2


We also have on the Shilov point the exact sequence 
    0 →^i_g →_g → (S_g/^iS_g) ⊗__g_g
 where the rightmost map restricts to the rightmost map of exact sequence (1) via the embedding ↪_g. We can also see that ⊗_S ^i S_g is in the kernel, so this gives an exact sequence 
    0 →^i_g/(⊗_S ^i S_g)→_g/(⊗_S ^i S_g)→ (S_g/^iS_g) ⊗__g_g.3
 

Now note that 
    _g/⊗_S ^iS_g≅⊗_S S_g/⊗_S ^iS_g≅⊗_S S_g/^iS_g
 since  is a finite, free S-module. Also, 
    /⊗_S ^iS⊗_R R_g≅⊗S/^i(S)⊗_R R_g.
 Since S/^iS is a finite, free R-module and S_g/^iS_g is a finite, free-R_g module of the same rank, the rightmost maps of (2) and (3) are the same with the same domain. Thus the kernels must be the same. 

Since ^iS ⊗_R R_g ⊂^i(S_g), we obtain that an element in ^i _g is determined by an element of ^i⊗_R R_g up to an element in ⊗_S ^iS_g, completing the proof of the lemma. 
 
 

We now want to use this decomposition to show that elements of ^i(M_g) cannot arise containing any ^i(S_g) ⊗_S components. Formally, we have the following lemma: 



    ^i(M_g) ⊂^i⊗_R R_g


Let x ∈^i(M_g), which we know to be a subset of ^i(_g). This means by Lemma <ref> we can write x = y + z with y ∈^i(S_g) ⊗_S and z ∈^i⊗_R R_g. Since  ^i(M_g) ⊂ M_g ≅ M ⊗_R R_g, we have that y = z-x ∈⊗_R R_g. We claim y ∈ (^i(S) ⊗_S ) ⊗_R R_g ⊂^i ⊗_R R_g. 

Let e_1, e_2, ⋯ , e_n be an S-basis of . Since y ∈^i(S_g) ⊗_S, we can write 
    y = ∑_m=1^n s_me_m
 with s_m ∈^i(S_g). Since y ∈⊗_R R_g, we can write 

    y    = ∑_j=1^ℓ[∑_m=1^n (∑_k=1^∞ a_k, m, jE(u)^k/k!)e_m ] ⊗ r_j 
       = ∑_m=1^n [∑_j=1^ℓ∑_k=1^∞((a_k, m, j⊗ r_j) E(u)^k/k!) ] e_m 
       = ∑_m=1^n [∑_k=1^∞(∑_j=1^ℓ  (a_k, m, j⊗ r_j)  )E(u)^k/k!]e_m


with a_k, m, j∈ R[u] and r_j ∈ R_g. Thus,  
    s_m = ∑_k=1^∞(∑_j=1^ℓ  (a_k, m, j⊗ r_j)  )E(u)^k/k!.
 Then since s_m ∈^i(S_g), we must have 
    (∑_j=1^ℓ  (a_k, m, j⊗ r_j)  ) = 0
 for k < i, and then it is clear that y ∈ ( ^i(S) ⊗_S ) ⊗_R R_g ⊂^i ⊗_R R_g and the lemma is proven.



 

^i(M_g) ≅^i(M) ⊗_R R_g. 

We know that ^i(M_g) ⊂ M ⊗_R R_g ≅ M_g and, by Lemma <ref>, ^i(M_g) ⊂^i ⊗_R R_g. By Lemma <ref>, we have:

    ^i(M_g)    ⊂(^i ⊗_R R_g) ∩( M ⊗_R R_g ) 
       = (^i ∩ M) ⊗_R R_g 
        = ^i(M) ⊗_R R_g.


The inclusion in the other direction is easy to establish.  Clearly ^i(M) ⊗_R R_g is inside M_g and ^i() ⊗_R R_g since ^i(M) is a subset of ^i() by definition, ^i(M) ⊂ M and R_g is a flat R-module. Thus we have:  

    ^i(M) ⊗_R R_g ⊂ (^i() ⊗_R R_g) ∩ M_g.

But ^i() ⊗_R R_g as a subset of _g is in the kernel of 1 ⊗φ: _g → S_g/^iS_g ⊗__g_g, so we have 
    ^i(M) ⊗_R R_g ⊂ (^i(_g)) ∩ M_g = ^i(M_g).






Now that we have shown the compatibility of the filtrations, M ⊗_R R_g has the full Fontaine-Laffaille module structure of M_g. We thus conclude:



M ⊗_R R_g is a Fontaine-Laffaille module over R_g, and M ⊗_R R_g ≅ M_g as Fontaine-Laffaille modules. 

We established in Theorem <ref> an isomorphism M ⊗_R R_g ≅ M_g compatible with Frobenius. We established in Theorem <ref> that ^i(M) ⊗_R R_g ≅^i(M_g). Since M_g carries the structure of a Fontaine-Laffaile module, we have completed the proof of this theorem.



M ⊗_R W(k) ≅ M_0 as Fontaine-Laffaille Modules

The strategy in this section is similar to the strategy in the above section where we analyze the Breuil module filtration to understand the Fontaine-Laffaille module filtration, but we have to deal with the additional concern that W(k) is not a flat R-module, which means analyzing various terms of the form  Tor_1^R( -, W(k)). These Tor terms aren't too concerning because the barrier to W(k) being a flat R-module is, effectively, t_i-torsion, which does not show up in most of our modules.  

 
 We have (for i < p):(a) _1^R(S/R, W(k)) = 0 (b) _1^R(/M, W(k)) = 0 (c) _1^R(S/^iS ⊗_, W(k)) = 0(d) Tor_1^R(/^i(), W(k)) = 0  
For (a), we compute Tor^R_1(S/R, W(k)) = 0 by recognizing that 
    0 → R → S → S/R → 0
 is a free resolution of S/R since S is a free R-module. When tensoring with W(k), it is clear that the first map remains injective, so indeed Tor^R_1(S/R, W(k)) = 0.  

For (b), by Lemma <ref>, we have that /M is a direct sum of finitely many copies of S/R, so Tor^R_1(/M, W(k)) = 0 by part (a). 

For part (c), we observe that S/^iS is (for i < p) a free R-module and hence a free W(k)-module. We also know  is a free -module, giving us our conclusion.

For Part (d), we have an exact sequence 
    0 →^i() ↪→ S/^i(S) ⊗_.
 Below in Lemma <ref>, we justify that the cokernel of this map 
 → S/^i(S) ⊗_ is a finite free R-module, hence a free W(k)-module.  

 Tensoring the induced sequence 
    0 →/^i() → S/^i(S) ⊗_
 with W(k) then applying Lemma <ref> and part (c) gives us the result. 


The following lemma completes the proof of (c):



Let f: → S/^iS ⊗_ be the map induced by 1 ⊗φ_:  = S ⊗_φ, → S ⊗_. Then coker(f) is a finite, free R-module. 

Write φ^* for ⊗_φ,. We have an exact sequence 
    0 →φ^* →→/φ^*→ 0
 Tensoring with S, we obtain 
    S ⊗_φ, → S ⊗_→ S ⊗_/φ^*→ 0
 where the first map is 1 ⊗φ_. Then we obtain the following diagram projecting to the  S/^i(S) level:

scale=0.85,center
  [scale=1]
    S ⊗_φ,  r1 ⊗φ_ drf     S ⊗_ r d     S ⊗_ /φ^*r d     0 

        S/^iS ⊗_ r(*) d    S/^iS ⊗_ /φ^*d    

        0     0    
  
 

from which we conclude that (*) is surjective and that coker(f) = S/^iS ⊗_/φ^*. By Lemma <ref> and the proof of Lemma <ref>(c) which justifies that _1^R(S/^iS, W(k)) = 0 and allows us to commute the base change with the quotient, coker(f) ⊗_R W(k) ≅ S_0/^iS_0 ⊗__0_0/φ^*_0. 

By tensoring 0 →φ^*_0 →_0 →_0/φ^*→ 0 with S_0, we find that S_0/^iS_0 ⊗__0_0/φ^*_0 is the cokernel of f_0: _0 → S_0/^iS_0 ⊗_ induced by 1 ⊗φ__0. If i < p, by <cit.> we know that S_0/^iS_0 = W(k)[u]/E(u)^i and _0/φ^*_0 = ⊕_i W(k)[u]/E(u)^h_i where h_i are the Hodge-Tate weights from which it is clear that S_0/^iS_0 ⊗__0/φ^*  = coker(f_0) is a finite free W(k)-module of rank, say, m. 

Let e_1, ⋯ , e_m be a basis of coker(f_0) as a W(k)-module. By Nakayama's Lemma, we can lift it to e_1, ⋯ , e_m which generate coker(f) as an R-module. Analogously, we can see that coker(f) ⊗_R R_g ≅ S_g/^i(S_g) ⊗__g_g/φ^*_g which, by the same reasoning replacing W(k) with W(k_g), is a free W(k_g)-module of rank m. The rank is the same since  arises from a crystalline representation. A crystalline representation in the relative case is Hodge-Tate and will have the same Hodge-Tate weights on all of its fibers, so the decomposition of _0/φ^*_0 will use the same Hodge-Tate weights h_i as the analogous decomposition of _g/φ^*_g. Then the images of e_1, ⋯ , e_m in coker(f) ⊗_R R_g must generate S_g/^i(S_g) ⊗__g_g/φ^*_g. Therefore they are linearly independent over W(k_g) and, hence, over R, as well. Thus coker(f) is a finite, free R-module. 



On the Shilov point we had to be concerned that S_g is very large: it is not true that ⊗_R R_g = _g, we would need an additional p-adic completion which does not behave nicely over non-Noetherian rings. On the closed fiber we have a little more freedom since S_0 is smaller than S and this concern is not present on the closed fiber. We can thus obtain a stronger version of Lemma <ref> more easily: 
 
We have 
    ^i(_0) = ^i⊗_R W(k).



    We have the exact sequence 
    0 →^i→ (S/^iS) ⊗_→coker(f) → 0
 from the definition of ^i, and let f be the labelled map. Factoring through the kernel, we obtain the short exact sequence 
    0 →/^i→ (S/^iS) ⊗_→coker(f) → 0.
 We now base change to W(k), noting that coker(f) is a finite, free R-module by Lemma <ref> and the tensor produdct commutes with the quotients by Lemma <ref>. We thus obtain the exact sequence 
    0 →_0/(^i⊗_R W(k))  → (S_0/^iS_0) ⊗__0_0 →coker(f) ⊗_R W(k) → 0.
 As ^i_0 is exactly the kernel of _0 → (S_0/^iS_0) ⊗__0_0 inside of _0 and ^i⊗_R W(k) is contained in this kernel, the proof of the lemma is complete. 



Now that we can reduce the Breuil module filtration to the classical case, we will introduce a second filtration on M which behaves a little more nicely under base change. In the classical case, the two filtrations agree, which is what we will show in our power series ring setting.


Let 
    f_p: → M
 be the map determined by u ↦ p. Note this map sends E(u)^i/i! to 0. Define ^iM to be the image of ^i under f_p.




    ^iM = ^iM


We first show that ^iM ⊂^iM. Recall that we set 𝒟 = [p^-1] and D = M[p^-1] upon which we can utilize the rational theory. Set ^iD ^i∩ D. We have the exact sequence 
    0 →^i-1𝒟/(^pS)^i𝒟/(^pS)^iD → 0.
 The kernel of f_p: ^i →^iD is (^1S)^i-1 by <cit.>, and removing the ^pS part ensures that ^i-1 will give the entire kernel. Recalling from the rational theory that 
    ^i = ^i𝒟∩,
 we obtain the exact sequence 
    0 →^i-1/(^pS)→^i/(^pS)^iM → 0.
 Since ^iM ⊂^i∩ M ⊂^i∩ D, we have an injective map 
    ^iM ↪^iD ↪^i/(^pS)^iD.
 On the other hand, this agrees with the map 
    ^iM ↪^i/(^pS)^iM ↪^iD
 giving us that 
    ^iM ⊂^iM.


Tensoring (1) with W(k) and applying Lemma <ref> we obtain the exact 
    ^i-1_0/(^pS_0)_0→^i_0/(^pS_0)_0^iM ⊗_R W(k) → 0.
 But we also know from the classical theory that ^i(M_0) is the image of ^i_0 under f_p, so we obtain that ^iM_0 = ^iM ⊗_R W(k) and hence ^iM ⊗_R W(k) is a finite, free W(k)-module. Let e_1, ⋯ , e_a be a basis and by Nakayama's Lemma we can lift it to a generating set e_1, ⋯, e_a of ^iM. Since ^iM is an image after u ↦ p, it is clear that we can safely mod out (1) by ^pS to obtain the exact 
    0 →^i-1/(^pS)→^i/(^pS)^iM → 0.
 Now we can safely base change to the Shilov point to obtain 
    0 →^i-1_g/(^pS_g)_g→^i_g/(^pS_g)_g^iM ⊗_R R_g → 0.
 the base change in the first two terms is as written due to Theorem <ref> and the classical functor from Fontaine-Laffaille modules to Breuil modules described in Definition <ref>. Applying the classical theory to the Shilov point, we know ^iM ⊗_R R_g = ^iM_g and hence is a finite, free R_g module and is of the same rank as ^iM_0. Note that the rank is the same again because of the Hodge-Tate property of crystalline representations. The places where the filtrations of M_0 and M_g increase in rank and the amount they increase in rank by are determined completely by the Hodge-Tate weights of the crystalline representation associated to M. These are the same on each fiber and, consequently, the ranks of ^iM_0 and ^iM_g are the same. 

Thus with our embedding ^iM ↪^iM ⊗_R R_g, we see that e_1, ⋯ , e_a is a basis of ^iM and hence ^iM is a finite, free R-module.  

Now we know that ^iM is a finite, free R-module contained in both ^iM_g and ^i We conclude that 
    ^iM    ⊂ (^iM ⊗_R R_g) ∩^i
       = ^iM_g ∩^i 
       = ^iM





 

^i(M_0) ≅^i(M) ⊗_R W(k). 

In the proof of the previous lemma we established that ^iM ⊗_R W(k) = ^iM_0. The result of the previous lemma is that ^iM = ^iM, completing the proof. 



This completes the proof of the following theorem:



M ⊗_R W(k) is a Fontaine-Laffaille module over W(k), and M ⊗_R W(k) ≅ M_0 as Fontaine-Laffaille modules. 



M is an object of MF_∇^ff
We now complete the proof of Theorem <ref>:

Having established Theorem <ref> and Theorem <ref>, we have shown Condition (1) of Proposition <ref>. Since M is a finite, free R-module, condition (2) holds, and thus M is a Fontaine-Laffaille module over R and belongs to the category MF^ff(R).  


 M is an object of the category MF^ff_∇. 

By Theorem <ref> and Theorem <ref>, we only need to show that M is stable under ∇_ and that it satisfies integral Griffiths Transversality. 

Using Lemma <ref> and Lemma <ref> we see that M = M[p^-1] ∩  as submodules of [p^-1]. We know M[p^-1] is stable under ∇_ (which restricts to ∇_ on M[p^-1]) and, by Proposition <ref>, so is , showing that M is stable under ∇_. 

Because ^i(M) = ^i() ∩ M, M is stable under ∇_, and ∇_ satisfies integral S-Griffiths transversality by lemma <ref> we conclude ∇_ satisfies Griffiths transversality, as needed. 



Proof of Essential Surjectivity
We now complete the proof that  is essentially surjective. Let Λ be an object of Rep_ℤ_p, [0,r]^(G_R). Let (, φ_, f) be the Kisin module with descent data associated to Λ guaranteed by Theorem <ref> (note Remark <ref>). Note also this theorem associates a Kisin module with descent data to an étale (φ_, )-module ℳ with T_(ℳ)(r) ≅Λ, and then T_^r() ≅Λ by Lemma <ref>. 

Let ≅⊗_φ,  S be the Breuil module associated with . We first prove the following stepping stone to Theorem <ref>.
 


 The natural injection 
    T_^r() → T_S() ^r(⊗_S )^φ_r=1
 induced by the embedding → is an isomorphism of G_R̃_∞ representations.  

We prove the lemma by reducing to the Shilov point where we argue as in <cit.>. First note that using Lemma <ref>, 
    Λ≅ T_(ℳ)(r) ≅ T^r_() → (^rφ^*⊗_)^φ_r = 1≅ (𝒪_ℰ, g^ur⊗_𝒪_ℰ, gℳ_g)^φ = 1(r) ≅Λ
 is an isomorphism of G_R̃_g,∞ representations. 

Then we obtain the diagram: 

 scale=1,center
  [scale=1]    
      T_^r() r d≅     T_S() d

      (^rφ^*⊗_ )^φ_r = 1 r≅     ^r((R_g) ⊗_S _g)^φ_r=1
  
 
where the bottom map is an isomorphism by <cit.> and  represents the usual  from the classical theory. We do not know whether the right vertical map is an injection since the natural map (R) →(R_g) requires a p-adic completion. We can, however, extend the diagram to the right because if M is the Fontaine-Laffaille module associated to , then the unique section of Lemma <ref> and the discussion of Section <ref> induces an isomorphism T_S() ≅(M), and we get a similar isomorphism on the Shilov point.  This gives us the diagram: 

 scale=1,center
  [scale=1]    
      T_^r() r d≅     T_S() d r≅     (M) d≅

      (^rφ^*⊗_ )^φ_r = 1 r≅     ^r(_g ⊗_S  (R_g) )^φ_r=1 r≅     (M_g)
  
 
where the rightmost vertical arrow is an isomorphism by <cit.>. Since T_^r() → T_S() is an injection, it must then be an isomorphism, completing the proof of the lemma.  



The last diagram in the proof of this lemma then establishes that (M) ≅Λ as G_R_∞ representations. We now are able to prove Theorem <ref> by showing this is actually an isomorphism of G_R representations which comes from the fact that the above maps are compatible with the relevant Galois actions. 

The isomorphism T_S() →(M) is determined by the unique ∇-compatible section M →, and it is clear that the Galois action on (M) determined by ∇ on M is compatible with the Galois action on T_S() determined by ∇ on , confirming that this map is actually an isomorphism of G_R representations. 

To see that the map T_^r() → T_S() is compatible with the G_R action, <cit.> confirms that there is a unique descent datum f on  such that f_S = id_S ⊗_φ,  f. As the descent data is used to construct the full G_R action in the same way as described in Section <ref> and Section <ref> and this is the same as the Galois action defined via the ∇ structure, the Galois action established on T_^r() is compatible with the Galois action on T_S() and thus the map T_^r() → T_S() which was previously shown to be an isomorphism of G_R_∞ representations is actually an isomorphism of G_R representations, completing the proof of Theorem <ref>. 

From the diagram in the above proof, we have also shown that T_cris(M) ≅ T_^r() ≅Λ as G_R representations, which completes the proof of essential surjectivity over R. 



Essential Surjectivity of  over small base rings
We will now prove theorems analaogous to those in the last section but in more generality over small base rings. While the results are analogous, the proof is easier because we are able to leverage the theory from the previous section to define a φ_i structure via the rational theory, which would be difficult to do for the power series ring. Throughout this section, fix T a small base ring. Write = _T = T u and S = S_T. Let I_0 ⊂ S denote the kernel of the projection S → T given by u ↦ 0.
Statement of Main Theorems


Let  be a Breuil module which arises from a Kisin module which arises from a Λ, an object in Rep_ℤ_p^ [0,r](G_T).  Set 
    M /I_0≅/u⊗_T, φ_ T.
 Then M carries the data of a Fontaine-Laffaille module and satisfies M ⊗_R S ≅ as in Definition <ref> We call it the “Fontaine-Laffaille module associated to Λ."


Let M be the Fontaine-Laffaille module associated to Λ. Then 
    T_cris(M) ≅Λ.
 


 
Preliminaries
Similar to the previous section, our strategy will be to show the Fontaine-Laffaille data is compatible with the base change to ≅ R discussed in Section <ref> and then to apply Theorem <ref>. This time we will be able to rely on the rational theory combined with the theory we have established for a power series ring. 

Recall that associated to V Λ⊗_ℤ_p⊗ℚ_p is (V) (which we will denote with D in this section), a finite, projective T[p^-1] module equipped with a filtration structure ^iD, a Frobenius structure φ_D, and a ∇_D structure. We will transfer the data on D to M in a similar way as how in the last section we transferred the data on  to M. 

By Theorem <ref>, we can see that M ⊗_T and (Λ|_), the Fontaine-Laffaille over  associated to Λ|_ are isomorphic as modules (note that  is a local ring, and a projective module over a local ring is free). Write M for (Λ|_) through the remainder of this section. 

Consider the projection q: → M induced by the φ-compatible projection S → T, u ↦ 0. Then q admits a unique φ-compatible section s: M →. 


    The proof of Lemma <ref> works identically when the base ring is T. 



This section gives us an injective map M ↪↪[p^-1] and the projection u ↦ 0 takes  to D, and thus we have an injective map M ↪ D. 

The Filtration Structure
We first define the filtration structure on M via the filtration on D from the rational theory.

Set 
    ^iM ^iD ∩ M.


It remains to show this filtration behaves well under base change. 

    ^iM ⊗_T ≅^i

    Note that <cit.> describes the compatibility of D under base change. Since  is a Fontaine-Laffaille module we know  (Λ|_) ≅[p^-1] and ^i = ^i(Λ|_) ∩ M. Using Lemma <ref> we have 
    
    (M) ⊗_T    = (^i D ∩ M) ⊗_T 
       = (^iD ⊗_T ) ∩ (M ⊗_T ) 
       = ^i((Λ|_)) ∩
       = ^i .




The φ_i structure
We now define the φ_i structure on M. Fix 𝔪 a maximal ideal of T and again write  For M ⊗ As  is a Fontaine-Laffaille module, there is a φ_i structure on  which we will denote φ_i. Via our φ-compatible embedding M ↪M, we can define a φ_i structure on M as 
    φ_i φ_i|_^iM.
 Let D[p^-1] = (Λ|_) Note that φ_i = φ_D/p^i, and the compatibility of  with base change in <cit.> gives us that φ_D/p^i acts the same on elements of ^iM ⊂ D and sends them into D. Thus φ_i sends ^iM into 
    D ∩M = M ⊗_T T[p^-1] ∩ M ⊗_T  = M
 by Lemma <ref>. Since the φ_i satisfies φ_i-1 = pφ_i, the same holds true for our new φ_i structure. 

The ∇ Structure
As D comes equipped with a ∇_D structure, set 
    ∇_M ∇_D|_M.
 The proof that M is stable under ∇_M is similar to the proof that M is stable under the φ_i structure. 

We can embed M into M which is stable under its ∇_ structure, and we can also embed M into D which is stable under its ∇_D structure. As D ∩ = M as above, we see that M is stable under ∇_M.

 is essentially surjective over T.
Combining the last few subsections, we have equipped M with the data of an object in MF_big, ∇(T) which is compatible with each base change to , completing the proof of Theorem <ref> by utilizing Theorem <ref>. It only remains to show that 

    (M) ≅Λ.
    
We define a map (M) →() via the functors to Galois representations from Kisin modules. As (M) = T_S(), we obtain an injective map  
    T_^r()  = (^r⊗(T))^φ_r=1↪ (^r ⊗(T))^φ_r = 1 = (M)
 and analogously for _^r(⊗__T_R) →(). We obtain the diagram     

 scale=1,center
  [scale=1]    
      (M) r      () 

      T_^r() r≅ u    _^r(⊗__T _)u≅
  
 
as we also have a map (M) →() since (M) ⊂(D) and () ⊂(D) and (D) = (D). 

The right vertical arrow in the diagram above is an isomorphism by the proof of Lemma <ref> and the bottom horizontal arrow is an isomorphism by Remark <ref> The left vertical arrow is an injection. Thus the injective (M) →() is forced to be an isomorphism, which quickly completes the proof.
    





