Formal groups and lifts of the field of norms
Léo Poyeton



Let K be a finite extension of . The field of norms of a strictly APF extension K_∞/K is a local field of characteristic p equipped with an action of (K_∞/K). When can we lift this action to characteristic zero, along with a compatible Frobenius map? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of (ϕ,Γ)-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin-Tate group and that the power series giving the lift of the action of the Galois group of K_∞/K are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.


11S15; 11S20; 11S25; 11S31; 11S82; 13F25




	Energy Storage Management via Deep Q-Networks 



    
	Ahmed S. Zamzam,
	Bo Yang, and
	Nicholas D. Sidiropoulos
	The work of A.S. Zamzam and N.D. Sidiropoulos was supported in part by NSF grant 1525194. 
		A. S. Zamzam and B. Yang are with the ECE Dept., Univ. of Minnesota, Minneapolis, MN 55455, USA. N. D. Sidiropoulos is with the Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, VA 22904.
		E-mails: ahmedz@umn.edu, yang4173@umn.edu, nikos@virginia.edu.
	

    December 30, 2023
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§ INTRODUCTION
 
Let K be a finite extension of , and let K_∞/K be a totally ramified Galois extension which is “strictly arithmetically profinite” in the sense of Wintenberger <cit.>, so that we can attach to K_∞/K its field of norms X_K(K_∞). This field of norms is a field of characteristic p, isomorphic to k_K((π_K)) where k_K denotes the residue field of K and where π_K is a uniformizer of X_K(K_∞), and is naturally equipped with an action of Γ_K=(K_∞/K). Let E be a finite extension of  such that k_E ⊃ k_K. In this article, we consider the following slightly generalized question of Berger <cit.>: when can we lift the action of Γ_K on k_K((π_K)) to the p-adic completion of 𝒪_E[[T]][1/T], which is a complete ring of characteristic 0 that lifts an unramified extension of X_K(K_∞), along with a compatible 𝒪_E-linear power of the Frobenius map ϕ_d ? When it is possible to do so, we say, following Berger's definition, that the action of Γ_K is liftable. 

In the case where k_E = k_K and where we have an 𝒪_E-linear compatible Frobenius map ϕ_q, Fontaine's construction of (ϕ,Γ)-modules applies, and we get an equivalence of categories between (ϕ_q,Γ_K)-modules on _K, the p-adic completion of 𝒪_E[[T]][1/T], and 𝒪_E-linear representations of _K. Such a lift is possible when K_∞/K is the cyclotomic extension, or more generally when K_∞/K is generated by the torsion points of a formal relative Lubin-Tate group, relative to a subextension  E/F of K.

Note that, if the action of Γ_K is liftable, we then have power series F_g(T) and P(T) in _K that commute under the composition law, where P is a noninvertible series and the F_g are invertible. If we make the same assumption as in <cit.>, that is assuming that P(T) is actually a power series in _K^+:=𝒪_E[[T]], we shall call that case a lift of finite height and in that case, one can show that, up to a variable change, we have P(T) ∈ T·𝒪_E[[T]] and the series F_g(T) also belong to T·𝒪_E[[T]] (see <cit.> for the first claim and <cit.> and the part concerning this proposition in <cit.> for the second claim), and Berger proved that K_∞/K is necessary abelian.

The main result of this article is the following:


If the action of Γ_K is liftable, if E is a finite Galois extension of  containing K and if the 𝒪_E-linear Frobenius on _K is of finite height, then there exists a finite extension L of , contained in E_∞=E · K_∞, a subfield F of L and a relative Lubin-Tate group S, relative to the extension F^unr∩ L of F, such that if L_∞^S is the extension of L generated by the torsion points of S, then K_∞⊂ L_∞^S and L_∞^S/K_∞ is a finite extension.


In particular, this theorem puts a clear limit on which Galois extensions one can use to build a finite height one dimensional (ϕ,Γ_K)-modules theory: these have to be the extensions obtained when taking the invariants under a finite group of a relative Lubin-Tate extension.  

In that finite height case, we recover the situation of non archimedean dynamical systems studied by Lubin in <cit.>, that is families of elements of T·𝒪_E[[T]] who commute one to another for the composition law. The main ingredients of the proof of theorem <ref> are p-adic Hodge theory and some of the tools developed in <cit.> by Lubin to study these non archimedean dynamical systems.

In <cit.>, Lubin noticed that “experimental evidence seems to suggest that for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background”. Various results have been obtained in that direction (see for instance <cit.>, <cit.>, <cit.>, <cit.>, <cit.>) and our theorem <ref> shows that there is indeed a formal group that accounts for this. The last part of this article is dedicated to explicit a link between the power series by which Γ_K acts on _K and some endomorphims of the same relative Lubin-Tate formal group. 

As this article is dedicated to the question of lifts of finite height of the lifts of fields of norms, it also seems relevant to consider the following question:



Let K_∞/K be a strictly APF extension which is Galois, and for which the action of Γ_K is liftable of finite height. For which finite extensions L of K is the action of ((L· K_∞)/L) liftable of finite height ?


In the cyclotomic case, remarks from Wach <cit.> and Herr <cit.> show that the only extensions L of  answering this question are the ones for which L(μ_p^∞)/(μ_p^∞) is unramified. 

First, we prove the following, which is a partial converse of theorem <ref> and shows that a straightforward generalization of the remark of Wach and Herr does not hold for every extension for which the action is liftable:



Let K be a finite extension of  and let K_∞/K be the extension generated by the torsion points of a relative Lubin-Tate group, relative to a subfield F of K. Let L be an extension of K, contained in K_∞ such that K_∞/L is finite and Galois, of degree prime to p. Then there exists a finite extension E of K and a ϕ-iterate extension E_∞/E such that L=E_∞. 


In particular, the action of (L/E) and (K_∞/E) is liftable of finite height, but K_∞/L is totally ramified.

Then, we prove how one can relate the power series appearing from the Γ_K action of _K with endomorphisms of a formal group:



If K_∞/K is a strictly APF Galois extension such that the action of Γ_K=(K_∞/K) is liftable, with power series P(T) and {F_g(T)}∈_K, then the power series F_g are twists of semi-conjugates over a finite extension of  to endomorphisms of a relative Lubin-Tate group, relative to a subfield F of a Galois extension E of  which contains K and such that (E · K_∞)/E is totally ramified. Moreover, there is an isogeny from [α] to P.


In the meantime, some of the methods used to prove this theorem allow us to give an answer to question <ref> for relative Lubin-Tate extensions that is basically the same as in the cyclotomic case:



Let K_∞/K be a relative Lubin-Tate extension, relative to a subfield F of K. Then the action of Γ_K is liftable of finite height, and the only extensions L of K for which there is a finite height lift of (L · K_∞/L) are the ones for which L · K_∞/K_∞ is unramified.





 §.§ Acknowledgements
  
Most of the results presented here are part of the author's Ph.D. thesis. Without Laurent Berger's help, encouragement and comments, none of this work would have been possible. The author would also like to thank Gabriel Dospinescu for many useful discussions.




§ LIFTING THE FIELD OF NORMS

Let K be a finite extension of , and let K_∞ be an infinite totally ramified Galois extension of K which is “strictly arithmetically profinite” in the sense of <cit.>. This is a technical condition about the ramification of the extension K_∞/K. Note that, if Γ_K=(K_∞/K) is a p-adic Lie group, it follows from the main result of <cit.> that K_∞/K is strictly arithmetically profinite.  

We can apply to the extension K_∞/K the “field of norms” construction of Fontaine-Wintenberger <cit.> which is the following : let ℰ_K_∞/K be the set of finite extensions of K contained in K_∞, and let X_K(K_∞) be the set of sequences (x_E)_E ∈ℰ_K_∞/K such that N_E_2/E_1(x_E_2) = x_E_1 whenever E_1 ⊂ E_2. By <cit.>, X_K(K_∞) can be endowed with the structure of a valued field, and we have the following theorem, which is <cit.>:

The field X_K(K_∞) is a local field of characteristic p whose residue field is k_K.
 
In particular, if π_K denotes a uniformizer of X_K(K_∞) we have X_K(K_∞) = k_K((π_K)). By construction of X_K(K_∞), it is naturally endowed with an action of Γ_K. It is also endowed with the Frobenius map ϕ_q : x ↦ x^q which commutes with the action of Γ_K.

Now let E be a finite extension of  with residue field k_E=k_K, let ϖ_E be a uniformizer of E and let _K denote the ϖ_E-adic completion of 𝒪_E[[T]][1/T]. The notation _K is used for compatibility with the action of _K, but be careful that it is actually dependent on E, even though E does not appear in this notation. This ring _K is a ϖ_E-Cohen ring for X_K(K_∞)=k_K((π_K)). Following the definition of <cit.>, we will say that the action of Γ_K is liftable if there exists such a field E and power series {F_g(T)}_g ∈Γ_K and P(T) in _K such that:


  * F_g(π_K) = g(π_K) and P(π_K) = π_K^q ;

  * F_g ∘ P = P ∘ F_g and F_g ∘ F_h = F_hg for all g,h ∈Γ_K.


The main question we are interested in is trying to understand for which extensions K_∞/K the action of Γ_K is liftable. One reason for asking this is that in that case we get a (ϕ,Γ)-module theory a la Fontaine to study 𝒪_E-representations of _K, replacing the cyclotomic extension in the theory of Fontaine by the extension K_∞/K. In particular, if the action of Γ_K is liftable, then there is an equivalence of categories between étale (ϕ_q,Γ_K)-modules on _K and 𝒪_E-linear representations of _K (see <cit.>).

If L is a finite extension of K, let L_∞:=L· K_∞ and Γ_L=(L_∞/K). By <cit.>, every finite separable extension of X_K(K_∞) is of the form X_K(L_∞) for some L finite over K. To such an extension X_K(L_∞)/X_K(K_∞), there exists by Hensel lemma a unique étale extension of ϖ_E-rings _L/_K such that _L is a Cohen ring for X_K(L_∞).

We have the following results, which are theorems 1.3 and 1.4 of <cit.>:



If the action of Γ_K on X_K(K_∞) is liftable and if L/K is a finite extension, then the action of Γ_L on X_K(L_∞) is liftable.



Let F_∞⊂ K_∞ be a Galois subextension of K_∞/K such that K_∞/F_∞ is finite, and let Γ_F:=(F_∞/K). If the action of Γ_K on X_K(K_∞) is liftable then the action of Γ_F on X_K(F_∞) is liftable.


In <cit.>, Berger showed the following:


If the action of Γ_K is liftable with P(T) ∈𝒪_E[[T]], then there is an injective character η: Γ_K →𝒪_E^× whose conjugates by (K,) are all de Rham with weights in _≥ 0.

In particular, if the action of Γ_K is liftable with P(T) ∈𝒪_E[[T]], then K_∞/K is abelian. We also know some cases where the action of Γ_K is liftable, namely in the cyclotomic case, which is the situation studied by Fontaine in <cit.>, and more generally in the Lubin-Tate case (<cit.> for the remark that it is liftable in characteristic zero and <cit.> for the construction of the related (ϕ,Γ)-modules).

We can actually generalize the notion of “liftable action”: let E be a finite extension of  with residue field k_E ⊃ k_K and let d be a power of q. We say that the action of Γ_K is liftable if there exists power series {F_g(T)}_g ∈Γ_K and P(T) in _K such that:


  * F_g(T) ∈ k_K((T)) and P(T) ∈ k_K((T)),

  * F_g(π_K) = g(π_K) and P(π_K) = π_K^d,

  * F_g ∘ P = P ∘ F_g and F_g ∘ F_h = F_hg for all g,h ∈Γ_K.


In this more general setting, there is no reason anymore to get a theory of (ϕ,Γ)-modules a priori.



§ EMBEDDINGS INTO RINGS OF PERIODS

In this section, we explain how to view the different rings we talked about in the previous section as subrings of some of Fontaine's rings of periods, and we give some key results about those rings and embeddings that will be used later on. We also recall some results of Cais and Davis about their “canonical Cohen rings for norm fields” <cit.> that we shall need later on.

Let K/ be a finite extension and let K_∞/K be a strictly APF extension. It will be convenient later on to see X_K(K_∞) as a subring of Fontaine's ring  which we define as follows: since K_∞/K is strictly APF, there exists by <cit.> a constant c = c(K_∞/K) > 0 such that for all F ⊂ F' finite subextensions of K_∞/K, and for all x ∈𝒪_F', we have 

    v_K(N_F'/F(x)/x^[F':F]-1) ≥ c.

We can always assume that c ≤ v_K(p)/(p-1). If F is a subfield of , let 𝔞_F^c be the set of elements x of F such that v_K(x) ≥ c, and let := _x ↦ x^d𝒪_/𝔞_^c. By §2.1 and §4.2 of <cit.>, there is a canonical _K-equivariant embedding ι_K : A_K(K_∞) ↪, where A_K(K_∞) is the ring of integers of X_K(K_∞). We can extend this embedding into a _K-equivariant embedding X_K(K_∞) ↪ where  is the fraction field of , and we note _K its image.

It will also be convenient to have the following interpretation for : 

    Ẽ^+ = _x → x^q𝒪__p = {(x^(0),x^(1),…) ∈𝒪__p^ : (x^(n+1))^q=x^(n)}.

To see that both definitions coincide, we refer to <cit.>.

Note that, even though _K depends on K_∞ rather than on K, it is still sensitive to K:


Let K' be a finite extension of K contained in K_∞. Let K_1 (resp. K_1') be the maximal tamely ramified extension of K_∞/K (resp. K_∞/K'). Then as subfields of , _K' is a purely inseparable extension of _K of degree [K_1':K_1]. In particular, _K_1=_K.


See <cit.>.


Assume now that K_∞/K is Galois and that the action of Γ_K is liftable to _K, a ϖ_E-Cohen ring of _K. Let = 𝒪_E ⊗_𝒪_E_0W() and endow it with the 𝒪_E-linear Frobenius map ϕ_q and the 𝒪_E-linear action of _K coming from those on  (this is well defined since K ⊃ E_0). Let _K = ^(/K_∞). 



There is a _K-equivariant embedding _K ↪_K and compatible with ϕ_q that lifts the embedding X_K(K_∞) ↪_K=^(/K_∞).


See <cit.> or <cit.>.


To simplify the notations, we will still denote _K for its image by this embedding. As stated in the previous section, we have the following lemma which is a consequence of Hensel's lemma:



Let L be a finite extension of K. Then there exists a unique étale extension of ϖ_E-rings _L/_K in  that lifts _L/_K.


See for example the discussion before <cit.>.


Let u ∈_K be the image of T by the embedding given by proposition <ref>, so that ϕ_q(u) = P(u) and g(u) = F_g(u) for g ∈Γ_K. Assume that the lift is of finite height, that is P(T) ∈𝒪_E[[T]]. Then we have the following:



If P(T) ∈𝒪_E[[T]] then u ∈.


See <cit.>.



If _K ⊂ is a ϖ_E-Cohen ring of _K, we define _K^+:=_K ∩, where = 𝒪_E ⊗_𝒪_E_0W().


Even though the assumption that the lift is of finite height might seem reasonable, it is a really strong assumption, as it is not stable under base change. Consider for example the cyclotomic case over , so that we can take P(T) = (1+T)^p - 1 and F_g(T) = (1+T)^χ(g)-1, where χ is the cyclotomic character, and u = [ϵ]-1 ∈. We let _ be the p-adic completion of _p[[u]][1/u] and _^+ = [[u]]. Consider K a finite extension of , and let K_∞ = K(μ_p^∞), the cyclotomic extension of K. Let Γ_K=(K_∞/K), and let _K be the unique étale extension of _ corresponding to the extension K_∞/(μ_p^∞) by the field of norms theory and lemma <ref>. Lemma <ref> shows that, if there exists a finite height lift of K_∞/K, then _K^+ generates _K, in the sense that there exists v ∈_K^+ such that _K is the p-adic completion of 𝒪_E[[v]][1/v]. However, Wach's remark in <cit.> and Herr's remark in <cit.> show that the only finite extensions K of  for which the action of (K_∞/K) is liftable are the ones for which K_∞/(μ_p^∞) is unramified.

One can wonder if Wach's and Herr's remarks still hold in any case of a finite height lift:

Question
Let K_∞/K be a strictly APF extension which is Galois and such that the action of Γ_K = (K_∞/K) is liftable of finite height on _K. What are the finite extensions L of K such that _L^+ generates _L ? Are they only the extensions L of K for which L_∞/K_∞ is unramified ?


We finish this section with some results about the canonical Cohen rings for norm fields of Cais and Davis <cit.>. They defined a ring _K_∞/K^+, canonically attached to a strictly APF extension K_∞/K:



Let K be a finite extension of , let π be a uniformizer of 𝒪_K, let K_∞/K be a strictly APF extension, and let {K_m}_m ≥ 0 be the tower of elementary extensions of K_∞/K as in <cit.>. Define

    _K_∞/K^+:={(x_q^i)_i ∈ W_π(𝒪_K_∞) : x_q^j∈ W_π(𝒪_K_m)  whenever  q^j|[K_m:K_1] },

viewed as a subring of 𝒪_K ⊗_𝒪_K_0W() via <cit.>, and where the W_π are some generalized Witt vectors defined in <cit.>.




Let L_∞ be a finite extension of K_∞ with wild ramification degree a, and let b ∈ be such that a ≤ q^b. Then the map 

    _K_∞/K^+ →_L_∞/K^+

given by

    (x_q^i)_i ↦ (x'_q^j)_j,   x'_q^j+b:=x_q^j

is a well defined embedding.


The proof is the same as the first part of the proof of <cit.>.



Note that this ring _K_∞/K^+ is a priori not related to the ring _K.




§ RELATIVE LUBIN-TATE GROUPS


In this section, we quickly recall de Shalit's construction (see <cit.>) of a family of formal groups that generalize Lubin-Tate groups and the results we will need about them.

Let F be a finite extension of , with ring of integers 𝒪_F and residue field k_F of cardinal q. Let h ≥ 1 and let E be the unramified extension of F of degree h. Let ϕ_q:E → E be the F-linear Frobenius map that lifts [x ↦ x^q]. If f(T) = ∑_i ≥ 0f_iT^i ∈ E[[T]], let f^ϕ_q(T) = ∑_i ≥ 0ϕ_q(f_i)T^i.

For α∈𝒪_F such that v_F(α) = h, let ℱ_α^r be the set of power series f(T) ∈𝒪_E[[T]] such that f(T) = π T+ O(T^2) with N_E/F(π) = α and such that f(T) ≡ T^q 𝔪_E[[T]]. Then ℱ_α^r is non empty since N_E/F(E^×) = {x ∈ F^×, v_F(x) ∈ h·} and we have the following results:


If f(T) ∈ℱ_α^r then:


  * there is a unique formal group law S(X,Y) ∈𝒪_E[[X,Y]] such that S^ϕ_q∘ f = f ∘ S, and the isomorphism class of S depends only on α,

  * for all a ∈𝒪_F, there is a unique power series [a](T) ∈𝒪_E[[T]] such that [a](T) = aT + O(T^2) and [a](T) ∈End(S).

Let x_0 = 0 and for m ≥ 1, let x_m ∈ such that x_m = x_m-1 with x_1 ≠ 0. Let Λ_m = {x ∈ : [π^m](x)=0 } and Λ = ⋃Λ_m and let E_m = E(x_m) and E_∞^S = ⋃_m ≥ 1 E_m. We have:


  * E_m = E(Λ_m) and the fields E_m depend only on α and not on the choice of f(T) ∈ℱ_α^r.

  * The extension E_m/E is Galois with Galois group isomorphic to (𝒪_F/𝔪_F^m)^×.

  * If g ∈(/E), there is a unique χ_α(g) ∈𝒪_F^× such that for all λ∈Λ, g(λ) = [χ_α(g)]_f(λ). 

  * (E_∞^S/E) ≃𝒪_F^× and that isomorphism is given by g ↦χ_α(g).

  * E_∞^S ⊂ F^ab and E_∞^S is the subfield of F^ab cut out by ⟨α⟩⊂ F^× by local class field theory.



See <cit.> and <cit.>.


In order not to confuse relative Lubin-Tate characters with the classical ones, we will note if needed χ_π^F the classical Lubin-Tate character associated with the uniformizer π of F. If u ∈𝒪_E^×, we note μ_u^E the unramified character of _E that sends Frob_E to u.



With these notations, we have χ_α(g) = χ_ϖ^F(g)·μ_u^E(g) where α = ϖ^hu. In particular, the action of (/E) on the torsion points of S is given by g(x) = [χ_ϖ^F·μ_u^E(g)](x).


Every part of the proof is basically in <cit.> but this result is not actually stated in <cit.>. For the convenience of the reader, we give a full proof of this result.

Let π∈ E such that N_E/F(π) = α = ϖ^hu. We also write ϕ and ϕ' for _F and _E respectively. Let f(T) = ϖ T+T^q and f'(T) ∈F_α. By <cit.>, there is a unique formal group law S_f(X,Y) (resp. S_f'(X,Y)) on 𝒪_Q̂_̂p̂^̂ûn̂r̂ such that f ∘ S_f = S_f^ϕ∘ f (resp. f' ∘ S_f' = S_f'^ϕ'∘ f'). We write Λ and Λ' for the torsion points of S_f and S_f' respectively.

By the discussion after proposition 4.4 of <cit.> and <cit.>, there is a unique power series θ∈𝒪_Q̂_̂p̂^̂ûn̂r̂[[T]] such that θ(T) = ϵ T +O(T^2), where ϵ is a unit of 𝒪_Q̂_̂p̂^̂ûn̂r̂, and such that:


  * f' ∘θ = θ^ϕ∘ f,

  * S_f^θ = S_f',

  * [a]_f^θ = [a]_f' for all a ∈𝒪_F.

The power series θ is thus such that

    f' ∘θ = θ^ϕ∘ f.

Since θ induces an isomorphism F_f →F_f' (θ being invertible in 𝒪__p^unr since θ'(0) = ϵ is a unit), we get an isomorphism Λ→Λ' of 𝒪_F-modules, and we have

    [a]_f^θ = [a]_f'.

Moreover, we have

    θ^ϕ' = θ∘ [u]_f

by <cit.>. Now let x ∈Λ and g ∈(/E) such that g acts as ϕ'^k on _p^unr. Then 

    g(θ(x)) = θ^ϕ'^k∘ g(x)

i.e.

    [χ_π(g)]_f'(θ(x)) = θ∘ [u]_f^k∘[χ_ϖ(g)]_f(x)

and since this holds for every x ∈Λ,

    [χ_π(g)]_f'=[χ_ϖ(g)u^k]_f^θ = [χ_ϖ(g)u^k]_f'.

Since g acts on Λ', that is on the torsion points of S, via g(x) = [χ_π(g)]_f'(x), the previous equality tells us that g acts on the torsion points of S by [χ_ϖ(g)u^k]_f'=[χ_ϖ^F(g)μ_u^E(g)]_f'.
 




§ Φ-ITERATE EXTENSIONS


In this section, we give a notion of “ϕ-iterate extension” that generalizes the Lubin-Tate extensions and is a bit more general that the definitions of <cit.> and <cit.>. The notion of ϕ-iterate extension will prove useful afterwards.



Let u_0 = π be a uniformizer of 𝒪_K, let d be a power of q and let P(T) ∈𝒪_K[[T]] a power series such that P(0) = 0 and P(T) ≡ T^d π𝒪_K. Let u_n+1∈𝒪_K be a root of P(T) - u_n and let K_n =K(u_n) and L = ⋃ K_n.

We call extensions of this form ϕ-iterate.



This definition is a bit more general than the one of Berger in <cit.> since he asks thatP(T) is a polynomial of the form T^d+a_d-1T^d-1+⋯+a_1T with the a_i in 𝔪_k for 1 ≤ i ≤ d-1.

Our definition <ref> is almost the same as the one of Cais and Davis in <cit.>, the only difference is that they ask d to be equal to q in their definition.


 


  * If P(T) = T^q and π is a uniformizer of K,  the corresponding ϕ-iterate extension K_∞/K is a Kummer corresponding to π.

  * If S is a relative Lubin-Tate group, relative to an extension E/F and α∈ F as in our §1, then E_∞^S/E_1 is ϕ-iterate with ϕ(T) = [α](T).



Cais and Davis actually proved the following for their definition of ϕ-iterate extension, and the generalization to our definition is straightforward:


Let K_∞/K be a ϕ-iterate extension. Then the ring _K_∞/K^+ defined in <ref> is a sub-𝒪_K-algebra of 𝒪_K ⊗_W(k_K)W(), of the form _K_∞/K^+ = 𝒪_K[[u]], such that X_K(K_∞) = k_K((u)), and 𝒪_K[[u]] is endowed with the action of a Frobenius map ϕ_q and  an action of (K_∞/K) that commute one to another and lift the ones on k_K((u)). 


See <cit.>.


In their article <cit.>, Cais and Davis need to make the assumption that the ϕ-iterate extension they consider is strictly APF. The following theorem, that they proved afterwards with Lubin, shows that this assumption is actually a consequence of their (and our) definition of a ϕ-iterate extension:

A ϕ-iterate extension is strictly APF.


This is a direct consequence of the main theorem of <cit.>.
 

In what follows, we will only consider ϕ-iterate extensions K_∞/K that are Galois. In particular, in that case, theorem <ref> proves that for such extensions, the action of Γ_K = (K_∞/K) is liftable of finite height.



§ LIFTS OF FINITE HEIGHT


We now go back to the question of the strictly APF extensions K_∞/K which are Galois with Galois group Γ_K = (K_∞/K) and such that the action of Γ_K is liftable of finite height. We will show in this section that such an extension is generated, up to a finite extension, by the torsion points of a relative Lubin-Tate group.

Let us now recall the hypotheses we will work with:

Let K_∞/K be a strictly APF extension which is Galois, with Galois group Γ_K = (K_∞/K). We suppose that the action of Γ_K is liftable of finite height, so that there exists a finite extension E of  with residue field k_E ⊃ k_K, d a power of q and power series {F_g(T)}_g ∈Γ_K and P(T) in _K such that:


  * F_g(π_K) = g(π_K) and P(π_K) = π_K^d,

  * F_g ∘ P = P ∘ F_g and F_g ∘ F_h = F_hg for all g,h ∈Γ_K,

  * P(T) ∈𝒪_E[[T]].


Recall that there exists a _K-equivariant embedding ι_K: X_K(K_∞) →𝐄̃_K defined by <cit.>. Let W_E(·) = 𝒪_E⊗_𝒪_E_0W(·) be the ϖ_E-Witt vectors. Let 𝐀̃=W_E(𝐄̃), endowed with the 𝒪_E-linear Frobenius map φ_d and with the 𝒪_E-linear action of _K coming from those on 𝐄̃. Let 𝐀̃_K = 𝐀̃^H_K so that 𝐀̃_K = W_E(𝐄̃_K). We also define =W_E().

Recall that by proposition <ref> there exists a Γ_K-equivariant embedding _K →_K, compatible with ϕ_q, that lifts the embedding ι_K : X_K(K_∞) →_K. Let u be the image of T by the embedding _K →_K so that ϕ_d(u)=P(u) and u lifts ι_K(π_K). Recall that, by lemma <ref>, this element u belongs to .

Let us recall some results of section 4 of <cit.>, which allow us to improve on the regularity of the power series P(T) and F_g(T) for g ∈Γ_K:


For all g ∈Γ_K, F_g(0) = 0.


See <cit.> and the erratum <cit.>.


Since we have F_g(0)=0, we can define a character η : Γ_K →𝒪_E^× by η(g)=F_g'(0).



If P(T) ∈𝒪_E[[T]], then there exists a ∈𝔪_E such that, if T'=T-a and u'=u-a, then ϕ_d(u') = Q(u') with Q(T') ∈ T' ·𝒪_E[[T']].


This is the same proof as in <cit.>.


In the following, we consider that we have made such a change of variable and we keep the previous notations, so that P(0)=0 and P(u)=ϕ_d(u). 



We have P'(0) ≠ 0.


This is <cit.>.




The character η : Γ_K →𝒪_E^× is injective.


This is a straightforward consequence of <cit.> which tells us that if Q'(0) ∈𝔪_K ∖{0}, then a power series f(T) ∈ T·𝒪_E[[T]] which commutes with Q is entirely determined by f'(0). This implies that F_g is determined by η(g). In particular, we have F_g(T)=T if and only if g(u)=u and thus if and only if g=𝕀.


Since we have P'(0) ≠ 0, we can define a logarithm attached to P as in <cit.>:


There exists a unique power series L_P(T) ∈ K[[T]], holomorphic on the open unit disk and such that:


  * L_P(T) = T+O(T^2) ;

  * L_P ∘ P(T) = P'(0)· L_P(T) ;

  * L_P ∘ F_g(T) = η(g)· L_P(T) for g ∈Γ_K.



See propositions 1.2, 2.2 and 1.3 of <cit.> which also show that

    L_P(T) = lim_n → +∞P^∘ n(T)/P'(0)^n=T·∏_n ≥ 0S(P^∘ n(T))/S(0),

where P(T) = T· S(T).


We will use this logarithm to construct crystalline periods for η as in <cit.>. First, we define _rig as the Fréchet completion of [1/p] as follows: every element of :=[1/p] can be written as ∑_k ≫ -∞π^k[x_k] where π is a uniformizer of 𝒪_E and {x_k}_k ∈ is a sequence of . Let e be the ramification index of E/, and for r ≥ 0, define a valuation V(·,r) on  by 

    V(x,r) = inf_k ∈(k/e+p-1/prv_(x_k))  if  x = ∑_k ≫ -∞π^k[x_k],

and let V(x) = inf_r ≥ 0V(x,r). Then _rig is defined as the completion of  for the valuation V. If u ∈ is the image of T by the embedding of proposition <ref> (and belongs to  by lemma <ref>), L_P(u) converges in _rig, and we have g(L_P(u))= η(g)· L_P(u) by the third item of <ref>. 

Let Σ be the set of embeddings of E in . If τ∈Σ, let n(τ) ∈ be such that τ = ϕ^n(τ) on k_K, and let u_τ=(τ⊗ϕ^n(τ))(u) ∈_τ(E). 

For F(T) = ∑_i ≥ 0f_iT^i ∈ E[[T]], let F^τ(T) = ∑_i ≥ 0τ(f_i)T^i ∈τ(E)[[T]]. We then have g(L_P^τ(u_τ))=τ(η(g))· L_P^τ(u_τ) in _τ(E),rig, with _τ(E),rig the Fréchet completion of _τ(E)[1/p]. We then get the following result, which is theorem 4.1 of <cit.> :


The character η: Γ→𝒪_E^× is de Rham, with nonnegative weights. 


See <cit.>.


In the case where K=E and K/ is Galois, one can actually show the following:



If K=E and K/ is Galois, then η: Γ→𝒪_K^× is crystalline with nonnegative weights.


The proof is the same as in <cit.>.


We recall an other useful result from <cit.>:


If f(T) ∈ T·𝒪_E[[T]] is such that f'(0) ∈𝔪_E ∖{0}, let Λ(f) be the set of the roots of all the iterates of f. If g(T) ∈ T·𝒪_E[[T]] is such that u'(0) ∈𝒪_E^× and is not a root of 1, let Λ(g) be the set of the fixed points of all the iterates of g. Then if f and g commute, we have Λ(f)=Λ(g).
 

See <cit.>.


We also get the following result (compare with <cit.>), where r_τ is the weight of η at τ for τ∈Σ:


If τ∈Σ, the following are equivalent:


  * r_τ≥ 1 ;

  * L_P^τ(u_τ) ∈Fil^1 ;

  * θ(u_τ) ∈ ;

  * θ(u_τ) ∈Λ(P^τ).



Since g(L_P^τ(u_τ))=τ(η(g))· L_P^τ(u_τ), the equivalence between (1) and (2) is straightforward. If L_P^τ(u_τ) ∈Fil^1, then L_P^τ(θ(u_τ))=0 and so θ(u_τ) ∈. It is clear that (4) implies (3). We now prove that (3) implies (4). Let x=θ(u_τ), so that g(x) = F_g^τ(x). If g is close enough to 1, then g(x) = x and so x ∈Λ(F_g^τ) = Λ(P^τ) by proposition <ref>. The only thing left to prove is that (4) implies (2). If there exists n ≥ 0 such that (P^τ)^∘ n(θ(u_τ))=0, then (P^τ)^∘ n(θ(u_τ)) ∈Fil^1 and so L_P^τ(u_τ) ∈Fil^1 by proposition <ref>.


Proposition <ref> is almost the same as <cit.>. However, in the ϕ-iterate setting, Berger had a canonical element u such that ϕ_d(u)=P(u) and θ(u) ∈. Here, we indeed have an element u such that ϕ_d(u) = P(u), but there is no reason to expect θ(u) ∈. This is the point of our next proposition:



There exists τ∈Σ such that θ(u_τ) ∈. Moreover, if v_n = θ∘ϕ_d^-n(u_τ), then τ(E)_∞ = ⋃_n ∈τ(E)(θ∘ϕ_d^-n(u_τ)).


The first thing to note is that there exists τ∈Σ such that r_τ≥ 1. Indeed, the character η has nonnegative weights so if such a τ did not exist, η would have all its weights equal to 0, so that the representation given by η would be -admissible and so η would be potentially unramified by Sen's theorem <cit.>. Since K_∞/K is totally ramified and infinite and η is injective, it can not be potentially unramified.

Let v_n = θ∘ϕ_d^-n(u_τ) and let K_n = K_1(v_n) where K_1 is the maximal tamely ramified extension of K_∞/K. We have v∈ and we write v = (v_n)_n ∈ for the components of v in =_x ↦ x^d𝒪__p/𝔞_^c. We have v_n = v_n in 𝒪_/𝔞_^c by <cit.>. In particular, the elements v_n belong to 𝒪_K_∞ mod 𝔞_K_∞^c and therefore belong to 𝒪_K̂_∞ and are invariant under the action of (/K_∞). Moreover, we have θ(u_τ) ∈ so that the v_n actually belong to K_∞. The v_n also satisfy the relations P^τ(v_n+1)=v_n and F_g^τ(v_n) = g(v_n) since ϕ_d(u_τ) = P^τ(u_τ) and g(u_τ) = F_g^τ(u_τ). Let us now prove that τ(E)_∞ = ⋃_n ≥ 1τ(E)(v_n). Let L = ⋃_n ≥ 1 K_n and let g ∈Γ_K be such that g_|L = 𝕀_L. Then g(v_n) = v_n for all n ∈, so that F_g^τ(v_n) = v_n for all n ≥ 0. Since F_g^τ(T)-T is a power series in 𝒪_E[[T]], it has only finitely many zeroes in the open unit disk. Since ||v_n|| → 1 and for all n ≥ 0, ||v_n|| < 1, this implies that F_g^τ(T) = T and therefore that g(u)=u, i.e. g =𝕀. This implies that τ(E)_∞ = ⋃_n ≥ 1τ(E)(v_n). 


Up to replacing E by a finite extension of E, which does not change the assumption that the action of Γ_K is liftable on 𝒪_E[[T]], one can assume that E/ is Galois (and thus in particular that τ(E)=E), that E contains K and that the extension E_∞:=E · K_∞ is totally ramified over E. In the following, we will assume that E satisfies those assumptions. Recall that our definition of  is actually dependent on E, so that we also have to change . We can always assume that d is a power of the cardinal of k_E (if not, we replace P by one of its iterate P^∘ m, which does not change the extension K_∞/K nor the extension E_∞/E, and we renumber the v_n and the extensions K_n and E_n to take this change into account, that is choosing v_n = θ∘ϕ_d^-n(u_τ) after having changed d). We then denote, for n ∈, E_n+1 :=E(v_n) (there might be some confusion since we also wrote E_1 for the maximal tamely ramified extension of E inside E_∞, but we won't use the maximal tamely ramified extension of E inside E_∞ in what follows). We also let Γ_E=(E_∞/E).



For all n ∈∪{+∞}, the extension E_n/E is Galois and we have an embedding (E_n/E) →(K_n/K) given by g ↦ g_|K_n. Moreover, the extension E_∞/E is strictly APF.


The first part of the lemma is a well knows result of Galois theory. The second part is contained in <cit.>.




There exists w ∈ such that 

    𝒪_E[[w]] = { x ∈, θ∘ϕ_d^-n(x) ∈𝒪_E_n for all  n ≥ 1 }.



Let R = { x ∈, θ∘ϕ_d^-n(x) ∈𝒪_E_n for all  n ≥ 1 }. It is an 𝒪_E-algebra, separated and complete for the ϖ_E-adic topology, where ϖ_E is a uniformizer of 𝒪_E. Moreover, if x ∈ R, its image in  belongs to _x ↦ x^d𝒪_E_n/𝔞_E_n^c. 

Proposition <ref> shows that u_τ∈ R, and so in particular we know that R/ϖ_ER contains k_E[[u_τ]]. We also know that _x ↦ x^d𝒪_E_n/𝔞_E_n^c ≃ k_E[[v]] for some v∈ by the theory of field of norms, and we consider an element w∈ R/ϖ_ER with minimal valuation within

    {x ∈ R/ϖ_ER, v_(x) > 0 }.

This set is nonempty since it contains u_τ, and so we can chose an element of minimal valuation within it since the valuation induced by the one on  on _x ↦ x^d𝒪_E_n/𝔞_E_n^c is discrete. Then R/ϖ_ER ≃ k_E[[w]], and so R = 𝒪_E[[w]] for w ∈ R lifting w, since R is separated and complete for the ϖ_E-adic topology.




The ring 𝒪_E[[w]] is stable by ϕ_d. Moreover, there exists a ∈𝔪_E such that if w'=w-a then there exists S(T) ∈ T·𝒪_E[[T]] such that S(w') = ϕ_d(w') and S(T) ≡ T^d 𝔪_E.


The set

    {x ∈, θ∘ϕ_d^-n(x) ∈𝒪_E_n for all  n ≥ 1 }

is clearly stable by ϕ_d and is equal to 𝒪_E[[w]] by proposition <ref>, so that ϕ_d(w) ∈𝒪_E[[w]] and so there exists Q ∈𝒪_E[[T]] such that Q(w) = ϕ_d(w). In particular, we have Q(w) = w^d and so Q(T) ≡ T^d 𝔪_E.

Now let R(T) = Q(T+a) with a ∈𝔪_E and let w'=w-a. Then ϕ_d(w')=ϕ_d(w-a)=Q(v)-a = R(w')-a and we let S(T) = R(T)-a so that ϕ_d(w')=S(w'). For S(0) to be 0, it suffices to find a ∈𝔪_E such that Q(a)=a. Such an a exists since we have Q(T) ≡ T^d 𝔪_E so that the Newton polygon of Q(T)-T starts with a segment of length 1 and of slope -v_p(Q(0)). 

Now, we have S(w')=ϕ_d(w') and so S(w') = w'^d, so that S(T) ≡ T^d 𝔪_E.


Lemma <ref> shows that one can choose w ∈{x ∈, θ∘ϕ_d^-n(x) ∈𝒪_E_n for all  n ≥ 1 } such that ϕ_d(w) = Q(w) with Q(T) ∈ T·𝒪_E[[T]], and we will assume in what follows that such a choice has been made.



The ring 𝒪_E[[w]] is stable under the action of Γ_E, and if g ∈Γ_E, there exists a power series H_g(T) ∈𝒪_E[[T]] such that g(w) = H_g(w). 


Since for all n ∈, E_n/E is Galois by lemma <ref> and since the action of _E commutes with ϕ and θ, the set

    {x ∈, θ∘ϕ_d^-n(x) ∈𝒪_E_n for all  n ≥ 1 }

is stable under the action of Γ_E, and by proposition <ref>, this set is equal to 𝒪_E[[w]]. In particular, if g ∈Γ_E, then g(w) ∈𝒪_E[[w]] and so there exists H_g(T) ∈𝒪_E[[T]] such that H_g(w) = g(w). 


The power series Q(T) and {H_g(T)} share the same regularity properties as the power series P and {F_g(T)}:

One has Q'(0) ≠ 0 and for all g ∈Γ_E, H_g(0) = 0.


The proof is the same as for proposition <ref> and lemma <ref>.


We now define κ(g) = H_g'(0), so that g ↦κ(g) defines a character κ: Γ_E →𝒪_E^×. The properties of regularity of the power series Q and {H_g(T)} allow us to prove that κ is injective: 



The character κ : Γ_E →𝒪_E^× is injective.


As in the proof of corollary <ref>, this follows from proposition 1.1 of <cit.>, which shows that if Q'(0) ∈𝔪_E ∖{0}, then a power series f(T) ∈ T·𝒪_E[[T]] which commutes with Q is determined by f'(0). Thus H_g is entirely determined by κ(g). It remains to prove that the action of g on w determines g, but if g ∈Γ_E acts trivially on w, then since u_τ∈𝒪_E[[w]], we get g(u_τ)=u_τ, and so g acts trivially on E_∞.


Since Q'(0) ≠ 0, we can also define a logarithm attached to Q as in proposition <ref>: 

There exists a unique power series L_Q(T) ∈ K[[T]], holomorphic on the open unit disk and such that:


  * L_Q(T) = T+O(T^2) ;

  * L_Q ∘ Q(T) = Q'(0)· L_Q(T) ;

  * L_Q ∘ H_g(T) = κ(g)· L_Q(T) for g ∈Γ_E.



We also define “conjugates” of the period w, which are the analogs of the periods u_τ: we define, for τ∈Σ=(E/), w_τ = (τ⊗ϕ^n(τ))(w) ∈, where n(τ) ∈ is such that τ =ϕ^n(τ) on k_E. Note that even though we fixed a particular element τ of Σ in proposition <ref>, we still use the notation τ for any element of Σ to avoid too much notations.

Just as in proposition <ref>, we have the following result:

The character κ is crystalline with nonnegative weights.


Again, this is the same proof as in <cit.>.


We also define for τ∈Σ, w_τ^n = θ∘ϕ_d^-n(w_τ) and w_τ^n,k = (Q^τ)^∘ k(w_n+k^p^n(τ)). These w_τ^n,k will allow us to give some approximations of the elements w_τ^n. 


One has θ∘ϕ_d^-n(w_τ) = lim_k → +∞w_τ^n,k.


This is the same proof as lemma 5.3 of <cit.>. 




For M > 0, there exists j ≥ 0 such that v_E(w_τ^n-w_τ^n,j) ≥ M for all n ≥ 1.


See <cit.>.


For τ∈Σ, let r_τ' be the weight of κ at τ. We then have:


If τ∈Σ, the following are equivalent:


  * r_τ' ≥ 1 ;

  * L_Q^τ(w_τ) ∈Fil^1 ;

  * θ(w_τ) ∈ ;

  * θ(w_τ) ∈Λ(Q^τ) ;

  * w_τ∈∪_j ≥ 0ϕ_d^-j(𝒪_E[[w]]).



The fact that the first four items are equivalent has already been proven in the proof of proposition <ref>. It thus remains to prove the equivalence with item (5). The proof is similar with the proof of <cit.>.

 If w_τ∈∪_j ≥ 0ϕ_d^-j(𝒪_E[[w]]), then since ϕ_d(w_τ) = Q^τ(w_τ) and since 𝒪_E[[w]] = { x ∈, θ∘ϕ^-n(x) ∈𝒪_E_n for all  n ≥ 1 } by <ref>, we get that θ(w_τ) ∈. 

Now let us assume that θ(w_τ) ∈. It suffices to prove that there exists j ≥ 0 such that w_τ^n ∈𝒪_E_n+j for all n by the characterisation of 𝒪_E[[w]] of proposition <ref>. Now let M be such that M ≥ 1+v_E((Q^τ)'(w_τ^n)) for all n ≥ 0. By lemma <ref>, there exists j ≥ 0 such that v_E(w_τ^n-w_τ^n,j) ≥ M for all n ≥ 1. Then since w_τ^n is a root of Q^τ(T)-w_τ^n-1, w_τ^n-w_τ^n,j is a root of Q^τ(w_τ^n,j+T)-w_τ^n-1. If w_τ^n-1∈𝒪_E_n+j-1, then R_n(T) = Q^τ(w_τ^n,j+T)-w_τ^n-1 belongs to 𝒪_E_n+j[T], and satisfies v_E(R_n(0)) ≥ M +v_E(R'(0)). This implies that R_n(T) has a unique root of slope v_E(R_n(0))-v_E(R_n'(0)) ≥ M by the theory of Newton polygons. This implies that this root belongs to E_n+j. In particular, since this root is w_τ^n-w_τ^n,j, we get that w_τ^n ∈𝒪_E_n+j, and so this finishes the proof by induction on n.



Proposition <ref> is exactly why we needed to work with 𝒪_E[[w]] instead of 𝒪_E[[u]]: we get item 5 in proposition <ref> (compare with proposition <ref>), which allows us to write the periods corresponding to the positive weights of κ as power series in some ϕ_d^-n(w). 


If τ satisfies those conditions, we can then write w_τ = f_τ(ϕ_d^-j_τ(w)) for some j_τ≥ 0 and some f_τ∈𝒪_E[[T]]. As in <cit.>, we get the following lemma:



One has f_τ(0)=0, f_τ'(0) ≠ 0, Q^τ∘ f_τ(T)=f_τ∘ Q(T), and H_g^τ∘ f_τ(T) = f_τ∘ H_g(T).


If w_τ=f_τ(ϕ_d^-j(w)), then Q^τ(w_τ)=Q^τ∘ f_τ(ϕ_d^-j(w)), and then ϕ_d(w_τ) = f_τ∘ϕ_d^-j(w)), so that Q^τ∘ f_τ(T)=f_τ∘ Q(T). For the same reason, we have H_g^τ∘ f_τ(T)=f_τ∘ H_g(T). Evaluating the relation Q^τ∘ f_τ(T) = f_τ∘ Q(T) at T=0 gives Q^τ(f_τ(0))=f_τ(0) and so f_τ(0) is a root of Q^τ(T) = T. 

Since Q^τ(0)=0 and since Q^τ(T) ≡ T^d 𝔪_E, we have by the theory of Newton polygons that either f_τ(0) is of valuation 0 or f_τ(0)=0. But the first case can not occur since θ∘ϕ_d^-n(w_τ) = f_τ(w_n+j) and so f_τ(0) ∈𝔪_E. Moreover, f_τ'(0) ≠ 0, since if we write f(T)=f_kT^k+O(T^k+1) with f_k ≠ 0, then since Q^τ∘ f_τ(T)=f_τ∘ Q(T), we get τ(Q'(0))f_k=f_kQ'(0)^k and so τ(Q'(0))=Q'(0)^k, thus k=1 since v_E(Q'(0)) > 0. 


Once again, we have a similar result as the one of Berger, and the following lemma is the analog of <cit.>.


Let G = {τ∈Σ such that  r_τ' ≥ 1 }. Then G is a subgroup of Σ and if F=E^G, then κ(g) ∈𝒪_F^×. Moreover, r_τ' is independent of τ∈ G.


Proposition <ref> shows that τ=𝕀 satisfies condition (3) of proposition <ref>, so that r_𝕀' ≥ 1. If σ,τ satisfy condition (5) of proposition <ref>, then one can write w_σ = f_σ(ϕ_d^-j_σ(w)) and w_τ=f_τ(ϕ_d^-j_τ(w)), so that w_στ = f_τ^σ∘ f_σ(ϕ_d^-j_τ-j_σ(w)), thus στ also satisfies condition (5). Since Σ is a finite group, this shows that G is a subgroup of Σ.

By lemma <ref>, Q^τ∘ f_τ=f_τ∘ Q(T), and so (Q^τ)^∘ n∘ f_τ=f_τ∘ Q^∘ n(T), so that 

    1/Q'(0)^n(Q^τ)^∘ n∘ f_τ(T) = 1/Q'(0)^nf_τ∘ Q^∘ n(T),

which implies passing to the limit that L_Q^τ∘ f_τ(T) = f_τ'(0)· L_Q(T). since we also have H_g^τ∘ f_τ(T) = f_τ∘ H_g(T), we get g(L_Q^τ∘ f_τ(v))=τ(η(g))·(L_Q^τ∘ f_τ(v)). This implies that τ(κ(g))= κ(g) since L_Q^τ∘ f_τ(v)=f_τ'(0)· L_Q(v). This holds for every τ∈ H, and so κ(g) ∈𝒪_F^×. Since κ(g) ∈𝒪_F^×, its weight r_τ' at τ depends only on τ_|F and so is independent of τ∈ H.


In order to finish the proof of theorem <ref>, we will need some local class field theory: for λ a uniformizer of E, let E_λ be the extension of E attached to λ by local class field theory. This extension is generated by the torsion points of a Lubin-Tate formal group defined over E and attached to λ, and we write χ_λ^K : (E_λ/E) →𝒪_E^× the corresponding Lubin-Tate character. Since E_∞/E is abelian and totally ramified, there exists λ a uniformizer of 𝒪_E such that E_∞⊂ E_λ.



One has κ = ∏_τ∈Στ(χ_λ^E)^r_τ'. 


This is the same proof as <cit.>.




There exists F ⊂ E and r ≥ 1 such that κ = N_E/F(χ_λ^E)^r.


Let F be the field given by lemma <ref> and let r = r_τ' for τ∈ G, which does not depend on the choice of τ∈ G by the same lemma. Lemma <ref> also shows that G = (E/F), and so, combining these results with proposition <ref>, we get that η=N_E/F(χ_λ^E)^r.


These results and lemma <ref> allow us to prove the following:


If K is a finite extension of  and if K_∞/K is an infinite strictly APF Galois extension with Galois group Γ_K = (K_∞/K),  and such that there exists a finite extension E of  such that k_K ⊂ k_E, an integer d which is a power of the cardinal of k_K and power series {F_g(T)}_g ∈Γ_K and P(T) in _K such that:


  * for all g ∈Γ_K, F_g(T) ∈ k_K((T)) and P(T) ∈ k_K((T));

  * F_g(π_K)=g(π_K) and P(π_K) = π_K^d;

  * F_g ∘ P = P ∘ F_g and F_g ∘ F_h = F_hg for g,h ∈Γ_K;

then there exists a finite extension L of E, a subfield F of L and a relative Lubin-Tate group S, relative to the extension F^unr∩ L of F, such that if L_∞^S is the extension of L generated by the torsion points of S, then K_∞⊂ L_∞^S and L_∞^S/K_∞ is a finite extension.


Let L be a finite extension of E such that L contains K, L/ is Galois and L_∞/L is totally ramified. Let λ be a uniformizer of L such that L_∞⊂ L_λ, let F be the subfield of L given by theorem <ref> and let E'= F^unr∩ L and let π = N_L/E'(λ) and α = N_L/F(λ), so that π is a uniformizer of E' and α = N_E'/F(π). Let S be the relative Lubin-Tate group attached to α, and let L_∞^S be the extension of L generated by the torsion points of S. If g ∈(/L_∞^S), then N_L/F(χ_λ^L(g))=1 by lemma <ref>, and so by theorem <ref>, we get κ(g) = 1. In particular, we have L_∞⊂ L_∞^S. By theorem <ref> and some Galois theory, we get:


  * L_∞^S is the field cut out by {g ∈_L : N_L/F(χ_λ^L(g))=1 } ;

  * L_∞ is the field cut out by {g ∈_L : N_L/F(χ_λ^L(g))^r=1}.

so that L_∞^S/L_∞ is a finite Galois extension whose Galois group is isomorphic to {x ∈𝒪_F^×,x^r=1}. The conclusion comes from the fact that L_∞/K_∞ is finite.
 




§ FROM LUBIN-TATE EXTENSIONS TO Φ-ITERATE EXTENSIONS


In this section, we show a partial converse of theorem <ref>. Let K be a finite extension of  and let K_∞/K be the extension generated by the torsion points of a relative Lubin-Tate group, relative to a subfield F of K. We will prove that, if L is an extension of K, contained in K_∞ and such that K_∞/L is finite and Galois, of degree prime to p, then L/K is ϕ-iterate up to a finite level. In particular, using theorem <ref>, the action of L/K will be liftable of finite height. To do so, we use some results from <cit.>. These results will be used in the next section.

Note however that a full converse of theorem <ref> does not hold. A full converse result of theorem <ref> would be that when K/ is a finite extension and K_∞/K is the extension generated over K by the torsion points of a relative Lubin-Tate group S, relative to an extension E/F with E ⊂ K, then any extension L of K such that K_∞/L is finite and Galois would be ϕ-iterate. In particular, taking L=K_∞, the action of Γ_K=(K_∞/K) would be liftable of finite height thanks to theorem <ref>. But in the case where K_∞ is the cyclotomic extension of K and K is such that K_∞/(ζ_p^∞) is totally ramified, <cit.> and <cit.> show that such a lift is not possible.

In what follows, F is a finite extension of  and K is a finite unramified extension of F. Let q be the cardinal of k_K. Let S(X,Y) be a formal relative Lubin-Tate group, relative to the extension K/F, attached to α∈𝒪_F and K_∞ is the extension of K generated by the torsion points of S. We note Γ_K=(K_∞/K), χ_α : Γ_K →𝒪_F^× the relative Lubin-Tate character attached to α. Let L be an extension of K, contained in K_∞ such that K_∞/L is finite and Galois, and let W=(K_∞/L). Let d be the cardinal of W = (K_∞/L) so that χ_α(W) is the set of d-th roots of unity. To keep the notations simple, we will stille write W for its image by χ_α. Let M=K((T)).

For a ∈ K, let [a] = [a](T) ∈ M be the unique endomorphism of S over K whose derivative at 0 is a. Let P=[α].

The group W acts on M by (right) composition: (w,Q) ↦ w · Q = Q ∘ [w].

By Artin's theorem, M/M^W is Galois with Galois group isomorphic to W, and we define R:=N_M/M^W(T) = ∏_w ∈ W[w](T).
 


We have M^W = K((R)).


This is lemma 2.1 of <cit.> and it is a straightforward consequence of a more general theorem of Samuel <cit.>, which says that if 𝒪 is a local noetherian complete domain, and if G is a finite group of 𝒪-automorphisms of 𝒪[[T]], then the ring 𝒪[[T]]^G of G-invariants is the subring 𝒪[[f(T)]], where f = ∏_s ∈ Gs· T.




For all a in K, there exists a unique power series Γ_a such that

    Γ_a ∘ R = R ∘ [a] = N_M/M^W([a])

We then have the relation Γ_a∘Γ_b = Γ_ab and Γ_a is for a ∈𝒪_K a power series whose derivative at 0 is a^d.


The fact that, for all a ∈ K, there exists a unique power series Γ_a such that 

    Γ_a ∘ R = R ∘ [a] = N_M/M^W([a])

is a direct consequence of lemma <ref>. The relation Γ_a∘Γ_b = Γ_ab follows from the fact that [a] ∘ [b] = [ab].


The power series Γ_k for k ∈𝒪_K^× are entirely determined by their derivative at 0:


Let k,ℓ∈𝒪_K^×. We have Γ_k = Γ_ℓ if and only if Γ_k'(0) = Γ_ℓ'(0).


Let L_P be Lubin's logarithm attached to P as in proposition <ref>. Then 

    L_P(Γ_k ∘ R) = L_P(R ∘ [k]) = L_P(∏_w ∈ W[wk]) = (-1)^d-1k^d· L_P

by item 2 of proposition <ref>. In particular, Γ_k = Γ_ℓ if and only if ℓ^d=k^d, which are the derivatives at 0 of Γ_ℓ and Γ_k by proposition <ref>.


We will now consider the power series Γ_α and use it to construct a ϕ-iterate extension. To do so, we will need the following lemma:



The power series Γ_α satisfies the following:


  * Γ_α(0) = 0 and Γ_α'(0) = α^d ;

  * Γ_α(T) ≡ T^q 𝔪_K.



The first item has already been proven in proposition <ref>. In order to prove the second item, we use the relation Γ_α∘ R = R ∘ [α](T), which gives us by reducing mod π : Γ_α(R) = R(T^q) π and so Γ_α(R) = R(T)^q π. This implies that Γ_α = T^q π.




Let K be a finite extension of  and let K_∞/K be the extension generated by the torsion points of a relative Lubin-Tate group, relative to a subfield F of K. Let L be an extension of K, contained in K_∞ such that K_∞/L is finite and Galois, of degree prime to p. Then there exists a finite extension E of K and a ϕ-iterate extension E_∞/E such that L=E_∞. Moreover, we can write E_∞ = ⋃ E(u_n) and there exists a family {F_g(T)}_g ∈(E_∞/E) of power series such that g ∈(E_∞/E) acts on the u_n by g(u_n) = F_g(u_n) with F_g(0)=0, and g ↦ F_g'(0) defines a character η : (E_∞/E) →𝒪_F^× satisfying η(g) = χ_α(g)^d.


Let x_0 ∈𝔪_K, x_0 ≠ 0, such that [α](x_0) = 0 and let u_0 = S(x_0). Since Γ_α∘ R = R ∘ [α], we get Γ_α(u_0) = 0. 

We will now prove that u_0 is a uniformizer of E:= K(u_0). The discussion following <cit.> (it concerns the classical Lubin-Tate case but its generalization to relative Lubin-Tate groups is straightforward), for d prime to p, shows that u_0 = R(x_0) is a root of multiplicity d of Γ_α and that the other roots of Γ_α other than 0 are also of multiplicity d. In particular, we have that [K(u_0):K] ≤q-1/d. Since R(x_0) = u_0 and one can write R(T) as R(T)=(-1)^d-1T^d(1+R_1(T)) with R_1 invertible in 𝒪_K[[T]] (see the proof of <cit.>). This implies that [K(x_0):K(u_0)] ≤ d. Since x_0 generates the first level of the relative Lubin-Tate extension K_1 over K, we have [K(x_0):K] = q-1. It follows that the previous inequalities on the degree of the extensions are equalities, and so [K(x_0):K(u_0)] = d. Moreover, since K(x_0)/K is totally ramified, so is K(x_0)/K(u_0). Since R(T)=(-1)^d-1T^d(1+R_1(T)) with R_1 invertible in 𝒪_K[[T]], we get that v_K(x_0)(u_0)=d and so u_0 is a uniformizer of K(u_0).

Let us now define a sequence (x_n) such that [α](x_n+1)=x_n, and let u_n=R(x_n). Then 

    Γ_α(u_n) = Γ_α∘ R(x_n) = R ∘ [α](x_n) = R(x_n-1) = u_n-1.


Since the power series Γ_α satisfies conditions of definition <ref> and since u_0 is a uniformizer of L, the extension L_∞:=⋃ L_n where L_n=L(u_n) is ϕ-iterate. Since u_n=R(x_n) we have E_∞⊂ K_∞ and since K_∞ is an abelian extension of K, so is E_∞. 

Moreover, Γ_k(u_n) = Γ_k ∘ R(x_n) = R ∘ [k](x_n) = R(g(x_n)) if g ∈(K_∞/K) is such that χ_α(g) = k and R(g(x_n)) = g(R(x_n)) = g(u_n), and so Γ_k(u_n) = g(u_n) for χ_α(g) = k. Now let η be the character (K_∞/K) →𝒪_F^×, given by g ↦Γ_χ_α(g)'(0)=χ_α(g)^k. Since (K_∞/K) ≃𝒪_F^× via g ↦χ_α(g), this character is surjective onto {x^d : x ∈𝒪_K^×}, which is a subgroup of 𝒪_F^× with finite index d. If g ∈(K_∞/K) is such that η(g)=1, then χ_α(g)^d=1 and so Γ_χ_α(g)(T)=T since the power series Γ_k are determined by their derivative at 0 by lemma <ref>. In particular, Γ_χ_α(g)(u_n) = u_n for all n and so g ∈(K_∞/E_∞). If g ∈(K_∞/E_∞), then for all n, g(u_n) = u_n since E_∞ is generated by the u_n over K. But g(u_n) = g(R(x_n))=R(g(x_n))=Γ_χ_α(g)(u_n) and so Γ_χ_α(g)(T)-T is a power series with coefficients in 𝒪_K[[T]] which vanishes on an infinite subset of the unit disk of , and so Γ_χ_α(g)(T)=T and η(g)=1.


In particular, we deduce the following result which shows that in a general setting there exist extensions K_∞/K for which the lift of Γ_K is of finite height and for which there exists extensions L_∞/K_∞ that are not unramified but such that the action of Γ_L is liftable of finite height:

There exists a finite extension K of  and a ϕ-iterate extension K_∞/K and a finite extension E of K such that, if u denotes the image of T by the embedding of proposition <ref> for _K and if v denotes the image of T by the embedding of proposition <ref> for _E, then 𝒪_E[[u]] ≠𝒪_E[[v]].


Let K/ be a finite extension. Let E_∞/K be a relative Lubin-Tate extension attached to a relative Lubin-Tate group, relative to an extension K/F where K/F is unramified, and let W be a finite subgroup of 𝒪_K^× with cardinal prime to p (for example, one can take the set of (p-1)-nth roots of 1). Let K_∞/K be the invariants under W of E_∞. Then there exists a finite extension K' of K, contained in K_∞ such that K_∞/K' is ϕ-iterate. Moreover, E_∞/E_1 is ϕ-iterate. Let v be the element of  which is the image of T by the embedding given by proposition <ref> for the extension E_∞ and let u be the element of  which is the image of T by the embedding given by proposition <ref> for the extension K_∞. Then the rings 𝒪_E[[u]] and 𝒪_E[[v]] can not be equal since u is invariant under W but v is not.





§ FORMAL GROUPS, SEMI-CONJUGATION AND CONDENSATION

In this section, we show that the power series F_g and P given in the case of a finite height lift of the field of norms are related to endomorphisms of a formal group. 

Let us first recall the setting of non archimedean dynamical systems as studied in <cit.>. We say that a power series g(T) ∈ T·𝒪_K[[T]] is invertible if g'(0) ∈𝒪_K^× and noninvertible if g'(0) ∈𝔪_K. We say that g is stable if g'(0) is neither 0 nor a root of 1. The case studied by Lubin is the case where two stable power series f and u commute, with f noninvertible and u invertible. Following Lubin's statement that “experimental evidence seems to suggest that for an invertible series to commute with a noninvertible series, there must be a formal group somehow in the background”, it seems reasonable to expect that, given such power series f and u (at least when f is non zero modulo 𝔪_K), there exists a formal group S, a nonzero power series h and two endomorphisms f_S and u_S of S such that f ∘ h = h ∘ f_S and u ∘ h = h ∘ u_S. Following the definitions of <cit.>, we say in that case that f and f_S, and u and u_S, are semi-conjugate and that h is an isogeny from f_S to f.

One of the main examples of this situation is the condensation in the sense of Lubin (see <cit.>), of which the construction of the power series Γ_a in the previous section is a particular case. 

We will see how the power series F_g given in the case of a finite height lift of the field of norms can be seen as semi-conjugates of endomorphisms of a relative Lubin-Tate group. But first, we will prove our theorem <ref>. Let K_∞/K be a relative Lubin-Tate extension, relative to a subfield F of K and attached to α∈𝒪_F. Let u_0 = 0 and let (u_n)_n ∈ be a compatible sequence of roots of iterates of [α], that is such that [α](u_n+1)=u_n and u_1 ≠ 0. Let u = (u_0,⋯) ∈. By §9.2 of <cit.>, there exists u ∈, whose image in  is u, and such that ϕ_q(u) = [α](u) and g(u) = [χ_α(g)](u) for g ∈_K. Let _K be the p-adic completion of 𝒪_K[[u]][1/u] inside . This element u is exactly the image of T by the embedding of proposition <ref>, where P(T)=[α](T) and F_g(T) = [χ_α(g)](T). Note that we have _K^+=𝒪_K[[u]]. Now let E be a finite extension of K, and let E_∞= E · K_∞. By lemma <ref>, there exists a unique étale extension of ϖ_K-Cohen rings _E of _K lifting the extension _E/_K. Let v ∈ be such that _E^+ = 𝒪_K[[v]]. Since u belongs to _K^+ ⊂_E^+, we can write u = h(v) for some h ∈𝒪_K[[T]]. In particular, one has θ∘ϕ_q^-n(u) = h(θ∘ϕ_q^-n(v)), so that the elements v_n = θ∘ϕ_q^-n(v) belong to . Let E_n = E(v_n), and let w ∈𝒪_E ⊗ W() be such that 𝒪_E[[w]] = { x ∈𝒪_E ⊗ W() , θ∘ϕ_q^-n(x) ∈𝒪_E_n for all  n ≥ 1 }. Such a w exists since v belongs to that set, and we have 𝒪_E[[w]] ⊃𝒪_E[[v]] ⊃𝒪_E[[u]]. In particular, we can write u=f(w) with f ∈𝒪_E[[T]]. By lemma <ref> and lemma <ref>, one can assume that g(w) = H_g(w) with H_g(T) ∈ T·𝒪_E[[T]] and that ϕ_q(w) = Q(w) with Q ∈ T·𝒪_E[[T]]. Moreover, by lemma <ref>, κ(g)=H_g'(0) defines an injective character: Γ_E →𝒪_E^×, where Γ_E=(E_∞/E).


One has f(0)=0, f'(0)≠ 0 and κ = χ_α. 


We have g(u) = F_g(u) = F_g ∘ f(w) and g(u) = g(f(w)) = f(g(w))=f ∘ H_g(w), so that F_g ∘ f = f ∘ H_g. We know that we can write F_g(T)=χ_α· T+O(T^2) and H_g(T) = κ(g)· T+O(T^2). Write f(T) = α_kT^k+O(T^k+1), with α_k ≠ 0. The equality F_g ∘ f = f ∘ H_g gives us that α_kχ_α(g)T^k=α_kκ(g)T^k, so that χ_α(g)=κ(g)^k, and since χ_α : Γ_E →𝒪_F^× is injective, this implies that f(0)=0. 

It remains to prove that k=1 in order to prove the lemma. By proposition <ref>, if λ is a uniformizer of 𝒪_E such that E_∞⊂ E_λ, then there exists a subfield F' of E and an integer r ≥ 1 such that κ = N_E/F'(χ_λ^E)^r. Since χ_α = κ^k, this implies that F ⊂ F'. However, Γ_E is isomorphic to an open compact subgroup of 𝒪_F^× by χ_α (and the embedding Γ_E ↪Γ_K) and to an open compact subgroup of 𝒪_F'^× by κ, which are analytic groups of respective dimension [F:] and [F':]. Since F ⊂ F', this implies that F=F'. Since we have χ_α = N_E/F(χ_λ^E), this implies that κ = χ_α and k=r=1, so that f'(0) ≠ 0.


Following Li's terminology, f is an isogeny of order 1 from Q to [α], and we can apply his results of <cit.>.



There exists n ≥ 0 such that 𝒪_E[[w]] = ϕ_q^-n(𝒪_E[[u]]).


By <cit.>, there exists f̌∈𝒪_E[[T]] and n ∈ such that f̌∘ f = Q^∘ n. Evaluating at w gives us f̌(u) = ϕ_q^-n(w), so that w ∈ϕ_q^-n(𝒪_E[[u]]).



We have _K^+ = _E^+. 


One has _K^+ ⊂_E^+ ⊂ϕ_q^-n(_K^+) by the previous proposition. In particular, one can write v=h(ϕ_q^-n(u)) with h ∈𝒪_E[[T]]. This implies that v belongs to a purely inseparable extension of k_E[[u]]. However, by definition of v and the theory of field of norms, v belongs to a separable extension of k_E[[u]], so that v∈ k_E[[u]]. This implies that _K^+/ϖ_E_K^+ = _E^+/ϖ_E_E^+ and therefore that _K^+ = _E^+.


In particular, this answers our question <ref> in the case of Lubin-Tate extensions: the only extensions of K_∞ for which there exists a finite height lift are the unramified ones. 

We will now show how to relate the power series given in the case of a finite height lift of the field of norms with endomorphisms of some relative Lubin-Tate group. To do so, we will use the “canonical Cohen ring” for norm fields constructed by Cais and Davis in <cit.>. Let K_∞/K be an infinite strictly APF extension which is Galois, and such that the action of Γ_K = (K_∞/K) is liftable on _K = 𝒪_E[[T]][1/T] with power series P(T) and {F_g(T)}_g ∈Γ_K. As in section <ref>, we can assume that E contains K, is Galois over  and is such that E_∞/E is totally ramified. By proposition <ref>, there exists u_τ∈ such that ϕ_q(u_τ) = P^τ(u_τ), g(u_τ) = F_g^τ(u_τ) and the elements v_n:=θ∘ϕ_q^-n(u_τ) belong to . Let κ be the character Γ_E →𝒪_E^× given by κ(g)=(F_g^τ)'(0). By theorem <ref>, there exists a subfield F of E and a relative Lubin-Tate group S, relative to the extension F^unr∩ E and attached to an element α of F, such that if E_∞^S denotes the extension of E generated by the torsion points of S, then E_∞⊂ E_∞^S and the extension E_∞^S/E_∞ is finite. Moreover, there exists r ≥ 1 such that κ = χ_α^r. Up to replacing E by a finite extension of E, we can also assume that E_∞^S=E_∞ and we do so in what follows. Let w_0 = 0 and let (w_n)_n ∈ be a compatible sequence of roots of iterates of [α], that is such that [α](w_n+1)=w_n and w_1 ≠ 0. Let w = (w_0,⋯) ∈. By §9.2 of <cit.>, there exists w ∈, whose image in  is w, and such that ϕ_q(w) = [α](w) and g(w) = [χ_α(g)](w) for g ∈_K. We now prove a link between the ring that we called _E^+ and the canonical Cohen ring _E_∞/E^+ of Cais and Davis:



There exists k ≥ 0 such that _E_∞/E^+ = ϕ_q^-k(𝒪_E[[w]]).


Let E'= F^unr∩ E and let F_∞^S be the extension of F generated over E' by the torsion points of S. For n≥ 1, let F_n:=E'(w_n), so that F_∞^S=⋃_n ≥ 1F_n. Let F_0 = E'.

The computation of the ramifications subgroups of the Galois group of a relative Lubin-Tate extension (see for example <cit.>) shows that the sequence {F_n}_n ≥ 0 is the tower of elementary extensions attached to F_∞^S/E' defined in <cit.>. Following definition <ref>, this implies that the element w ∈ belongs to the ring _F_∞^S/E'^+. By definition <ref>, the image of _F_∞^S/E'^+ modulo a uniformizer is contained inside _F_∞^S/E'^+. Moreover, since w is a uniformizer of the ring _F_∞^S/E'^+ and since _F_∞^S/E'^+ is an 𝒪_E'-algebra which is separated and complete for the p-adic topology, this implies that _F_∞^S/E'^+=𝒪_E'[[w]].

Let w' be such that _E_∞/E^+ = 𝒪_E[[w']]. By proposition <ref>, we have w ∈𝒪_E[[w']], and the same proof as the one of proposition <ref> shows that 𝒪_E[[w']] = ϕ_q^-n(𝒪_E[[w]]). 


The following proposition allows us to see some power of the Frobenius of u_τ inside the ring _E_∞/E^+:


There exists i ≥ 0 such that ϕ_q^i(u_τ) ∈_E_∞/E^+.

The strategy to prove this proposition is basically the same as the one proposed by Cais and Davis in <cit.> to show that there was a lift of a uniformizer to _L/K^+ for ϕ-iterate extensions. Since most of the arguments are the same, we will only explain the points where we have to adapt the proof. The idea is to relate in some way the fields E(v_n) and the tower {E^(m)} of elementary extensions extensions attached to the APF extension E_∞/E.



There exists n_0 ≥ 0 such that for all n ≥ n_0, the extension E(v_n+1)/E(v_n) is totally ramified of degree q and such that v_E(v_n)(v_n) is independent of n ≥ n_0.
 

We have P^τ(v_n+1)=v_n, with P^τ∈ T·𝒪_E[[T]] of Weierstrass degree q. Since v_n = u_n  p^c, we know that v_E(v_n) → 0 as n → +∞. The fact that for n big enough, E(v_n+1)/E(v_n) is totally ramified of degree q then follows from the theory of Newton polygons. 
Now let n_0 ≥ 0 be such that, for all n ≥ n_0, E(v_n+1)/E(v_n) is totally ramified of degree q and such that v_p(v_n) < c. We then have v_p(v_n+1) = v_p(u_n+1) = 1/qv_p(u_n) = 1/qv_p(v_n), so that, since E(v_n+1)/E(v_n) is totally ramified of degree q, v_E(v_n)(v_n) is independent of n ≥ n_0, and is equal to v_E_v_n_0(v_n_0).


Let π_n be a uniformizer of 𝒪_E(v_n) such that E(v_n) = E(π_n) (this is possible since E(v_n)/E is totally ramified).


There exist positive constants A and B such that, for all n ≥ 1, for all σ∈(E(v_n)/E(v_n-1), 

    Aq^n ≤ v_E(v_n)(σ(π_n)-π_n) ≤ Bq^n.



Using the same arguments as in the proof of <cit.>, we can prove that there exist positive constants A' and B' such that, for all n ≥ 1 and for all σ∈(E(v_n)/E(v_n-1)):

    A'q^n ≤ord_v_n(σ(v_n)-v_n) ≤ B'q^n.

Let n_0 be such as in lemma <ref>. Since v_E(v_n)(v_n) = k_0 and since π_n is a uniformizer of 𝒪_E(v_n), one can write

    v_n = ∑_k ≥ k_0a_kπ_n^k,

where the a_k can be written as Teichmüller representatives of elements of k_E (in particular, σ(a_k)=a_k for all k and v_E(a_k)=0 if a_k ≠ 0), and with a_k_0≠ 0.
This gives us that

    σ(v_n)-v_n = ∑_k ≥ k_0a_k(σ(v_n)^k-v_n^k)

so that

    σ(v_n)-v_n = (σ(π_n)-π_n)∑_k ≥ k_0a_k(∑_m=0^kσ(π_n)^mπ_n^k-m).

In particular, we get that

    ord_π_n(σ(v_n)-v_n) = ord_π_n(σ(π_n)-π_n)+k_0,

so that ord_π_n(σ(π_n)-π_n)+k_0 = k_0ord_v_n(σ(v_n)-v_n). In particular, by letting B=k_0(B'+1), we get that ord_π_n(σ(π_n)-π_n) ≤ Bq^n. There also exists n_1 ≥ 0 such that q^n_1 > A', so that k_0(A'-1/q^n) > 0 for n ≥ n_1, and if we let 

    A:=min( 1/q^nmin_n ≤ n_1, σ∈(E(v_n)/E(v_n-1)(ord_π_n(σ(π_n)-π_n)),k_0(A'-1/q^n_1)),

then we have q^nA ≤ord_π_n(σ(π_n)-π_n) for all n ≥ 1 and σ∈(E(v_n)/E(v_n-1)).




Let σ∈(E(v_n)/E). Then

    v_E(v_n)(σ(π_n)-π_n) ≤ Bq^n,

where B is the same as in lemma <ref>.


The same proof as the one of <cit.> shows that for all σ∈(E(v_n)/E),

    ord_v_n(σ(v_n)-v_n) ≤ B'q^n,

with B' the same constant as the one in the proof of lemma <ref>. Using the fact that ord_π_n(σ(π_n)-π_n)+k_0 = k_0ord_v_n(σ(v_n)-v_n), we get that ord_π_n(σ(π_n)-π_n) ≤ Bq^n with B=k_0(B'+1), which is the same constant B as in lemma <ref>.



Lemma <ref> and lemma <ref> allow us to apply lemmas 7.5, 7.6, corollary 7.7, proposition 7.9 and lemmas 7.10 and 7.11 of <cit.> (note that we do not need lemma 7.4 of ibid. since all our extensions are already Galois) in order to obtain the equivalent of corollary 7.12 of ibid., that is that there exist i, i_0 ≥ 0 such that if q^m|[E^(i+j):E^(i)], then π_i_0+m belongs to E^(i+j) and a fortiori v_i_0+m∈ E^(i+j). In particular, if i_1 is such that [E^(i):E]|q^i_1, then ϕ_q^i_1(u_τ) ∈_E_∞/E^+.


Lemma <ref> and proposition <ref> now allow us to relate the power series F_g^τ and P^τ to the power series [α](T) and [χ_α(g)](T):


If K_∞/K is an infinite strictly APF Galois extension such that the action of Γ_K = (K_∞/K) is liftable of finite height with power series P(T) and {F_g(T)}_g ∈Γ_K, then the power series F_g are twists of semi-conjugates over a finite extension of  of endomorphisms of a relative Lubin-Tate group S, relative to a subfield F of a Galois extension E of  and attached to an element α of F, and such that if E_∞^S denotes the extension of E generated by the torsion points of S, then K_∞⊂ E_∞^S and E_∞/K_∞ is finite. Moreover, there is an isogeny from [α] to P^τ.  


Lemma <ref> and proposition <ref> show that there exists i ≥ 0 and h(T) ∈𝒪_E[[T]] such that u_τ = h(ϕ_q^-i(w)). Let w̃=ϕ_q^-i(w), so that u_τ = h(w̃). For g ∈_E, we have g(u_τ) = F_g^τ(u_τ), and ϕ_q(u_τ) = P^τ(u_τ). We also have g(w) = [χ_α(g)](w) and ϕ_q(w)=[α](w) so that g(w̃) = [χ_α(g)](w̃) and ϕ_q(w̃)=[α](w̃). This gives us that F_g^τ∘ h = h ∘ [χ_α(g)] and that P^τ∘ h = h ∘ [α], which is what we wanted to prove.



The fact that the power series F_g are twists of semi-conjugates of the relative Lubin-Tate group seems necessary, as there is no reason to expect that the character g ↦ F_g'(0) has weight ≥ 1 at 𝕀. For example, let K_∞/K be a (classical) Lubin-Tate extension, attached to a uniformizer π of 𝒪_K, where K/ is totally ramified and Galois, and let u ∈ = 𝒪_K ⊗_W() be the element constructed in the Lubin-Tate setting at the beginning of this section, so that [χ_π(g)](u) = g(u) for g ∈_K. For any τ≠𝕀 in (K/), the element u_τ = (τ⊗𝕀)(u) satisfies F_g^τ(u_τ) = g(u_τ), P^τ(u_τ) = ϕ(u_τ) and u_τ = u, so that the action of Γ_K is liftable with power series [π]^τ and {[χ_π(g)]^τ}_g ∈Γ_K, yet the weight of the character η_τ : g ↦ ([χ_π(g)]^τ)'(0) at 𝕀, which is the weight of η_𝕀 : g ↦ ([χ_π(g)])'(0) at τ, is zero (see for example <cit.> or §3.2 of <cit.>).

However, in the case where K/ is unramified, the author expects that the power series F_g are actual semi-conjugates of the relative Lubin-Tate group.


amsalpha

