# The Hilbert symbol in the Hodge standard conjecture

Giuseppe Ancona and Adriano Marmora  IRMA, Université de Strasbourg et CNRS, 7
rue René-Descartes, 67084 Strasbourg, France<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study the Hodge standard conjecture for varieties over finite fields
admitting a CM lifting, such as abelian varieties or products of K3 surfaces.
For those varieties we show that the signature predicted by the conjecture
holds true modulo four. This amounts to determining the discriminant and the
Hilbert symbol of the intersection product. The first is obtained by
$\ell$-adic arguments whereas the second needs a careful computation in
$p$-adic Hodge theory.

This research was partly supported by the grant ANR–18–CE40–0017 of Agence
National de la Recherche.

###### Contents

  1. 1 Introduction
  2. 2 Conventions
  3. 3 Main statements
  4. 4 From archimedian to $p$-adic
  5. 5 Reduction to CM-simple objects
  6. 6 CM-quadratic forms over $p$-adic fields
  7. 7 Reduction to tensor generators
  8. 8 Characterization of $p$-adic periods
  9. 9 Lubin–Tate periods and local reciprocity
  10. 10 Proof of the main theorem
  11. A Lubin–Tate filtered $\varphi$-modules

##  1\. Introduction

The standard conjecture of Hodge type predicts the signature of the
intersection product of algebraic classes on a smooth projective variety. In
this paper we study the discriminant and the Hilbert symbol of the
intersection product and we show that they coincide with those predicted by
the conjecture for varieties over finite fields which admit a CM lifting, in
particular for abelian varieties and for products of K3 surfaces. This can be
reformulated by saying that the expected signature holds true modulo four.

This conjecture was formulated by Grothendieck in the sixties and it is
modelled on positivity results such as the Hodge index theorem, the
Hodge–Riemann relations and the positivity of the Rosati involution. For a
panorama on the history, the original motivation and potential applications of
this conjecture see [MR2115000, MR4199442, IIK]. This conjecture did not
progress until the 21st century. Then, Milne showed that the classical Hodge
conjecture for complex abelian varieties would imply the Hodge standard
conjecture for abelian varieties in positive characteristic [MR1906596] and
Ito studied the behaviour of this conjecture under blow-ups [MR2125735].

Very recently, some progress have been made. The first author proved the
conjecture for motives of rank two using $p$-adic Hodge theory [MR4199442,
Theorem 8.1]. This result implies the conjecture for abelian fourfolds
[MR4199442, Theorem 1.3] and for some other abelian varieties
[Koshi_HSC_prime]. On the other hand, Koshikawa–Ito–Ito proved the conjecture
for the square of a K3 surface, using the Kuga–Satake construction and
ultimately relying on the positivity of the Rosati involution [IIK].

The purpose of this article is to push the $p$-adic methods initiated in
[MR4199442] to CM motives of higher rank and to describe completely the
Hilbert symbol of the intersection product. To be more precise, let us recall
the formulation of the standard conjecture of Hodge type.

######  Definition 1.1.

Let $X$ be a smooth, projective and geometrically connected variety of
dimension $d$ and let $L$ be a hyperplane section of $X$. We denote by
$\mathcal{Z}^{n}_{\mathrm{num}}(X)_{\mathbb{Q}}$ the finite dimensional
$\mathbb{Q}$-vector space of $\mathbb{Q}$-algebraic cycles on $X$ of
codimension $n$ modulo numerical equivalence. For $n\leq d/2$ we define the
space of primitive cycles as

$\mathcal{Z}^{n,\mathrm{prim}}_{\mathrm{num}}(X,L)_{\mathbb{Q}}\coloneqq\\{\alpha\in\mathcal{Z}^{n}_{\mathrm{num}}(X)_{\mathbb{Q}},\hskip
5.69046pt\alpha\cdot L^{{d-2n+1}}=0\hskip 5.69046pt\textrm{in}\hskip
5.69046pt\mathcal{Z}^{d-n+1}_{\mathrm{num}}(X)_{\mathbb{Q}}\\}$

and we define the pairing
$q_{n,X,L}:\mathcal{Z}^{n,\mathrm{prim}}_{\mathrm{num}}(X,L)_{\mathbb{Q}}\times\mathcal{Z}^{n,\mathrm{prim}}_{\mathrm{num}}(X,L)_{\mathbb{Q}}\rightarrow\mathbb{Q}$
via the intersection product

$\alpha,\beta\mapsto(-1)^{n}\alpha\cdot\beta\cdot L^{d-2n}.$

######  Conjecture 1.2.

(Hodge standard conjecture.) Let $X$ be a smooth, projective and geometrically
connected variety of dimension $d$ and let $L$ be a hyperplane section of $X$.
Then for all $n\leq d/2$, the quadratic form $q_{n,X,L}$ is positive definite.

Our main result is the following.

######  Theorem 1.3.

  1. (1)

Let $(X,L)$ be an abelian variety over a finite field together with an
hyperplane section induced by a CM lifting. Let $n\leq\dim(X)/2$ be any
integer and $(s_{+},s_{-})$ be the signature of $q_{n,X,L}$. Then $s_{-}$ is
divisible by four.

  2. (2)

Let $X=S_{1}\times\ldots\times S_{m}$ be the product of K3 surfaces over
finite fields, $L_{i}$ be an hyperplane section induced by a CM lifting of
$S_{i}$, cf. [MR4241794], and let $L=\oplus_{i}p^{*}_{i}L_{i}$ be the induced
hyperplane section on $X$. Let $n\leq\dim(X)/2=m$ be any integer and
$(s_{+},s_{-})$ be the signature of $q_{n,X,L}$. Then $s_{-}$ is divisible by
four.

The above theorem follows from a general result on motives endowed with
quadratic forms (see Theorem 3.2 for the precise definitions).

######  Theorem 1.4.

Let $M$ be a CM motive in mixed characteristic with coefficients in
$\mathbb{Q}$. Suppose that its special fiber $M_{p}$ is supersingular and that
$M$ is endowed with a CM quadratic form $q$ whose Betti realization is a
polarisation of the underlying Hodge structure. Let $q_{Z}$ be the restriction
of $q$ to all algebraic classes of $M_{p}$. Write $(s_{+},s_{-})$ for the
signature of $q_{Z}$, then $s_{-}$ is divisible by four.

Let us give a sketch of the proof of Theorem 1.4. For a quadratic form over
$\mathbb{Q}$, the condition $4|s_{-}$ is equivalent to the fact that the
discriminant is positive and that the Hilbert symbol at infinity is positive.
Both conditions can be checked by studying the quadratic form in the non-
archimedean places, thanks to the classical product formula on Hilbert
symbols. In particular, Theorem 1.4 boils down to the study of the quadratic
form $q_{Z}$ after its $\ell$-adic and crystalline realization. The
$\ell$-adic computation is obtained from general results on $\ell$-adic
cohomology, and it is sufficient to prove that the discriminant is positive,
as in [MR4199442]. Instead, the computation of the Hilbert symbol needs to
control the crystalline contribution. Our arguments rely heavily on $p$-adic
Hodge theory. Let us give the mainlines below.

We begin by extending the coefficients of the motive $M$ from $\mathbb{Q}$ to
$\mathbb{Q}_{p}$. Precisely, let $F$ be the CM number field acting on $M$, the
motive $M\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ is endowed with an action of
$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$. Such an action decomposes the motive
into a sum of direct factors and we study them separately. Thus, we reduce to
the situation where the motive $M$ has an action of a finite extension of
$\mathbb{Q}_{p}$, which we will denote again by $F$, endowed with a non-
trivial involution $*\colon F\rightarrow F$. The crystalline and $p$-adic
étale realisations of $M$ are endowed by hypothesis with a non-degenerate
quadratic form. Faltings’s crystalline comparison theorem [MR1463696] allows
to compare these realisations via a matrix of periods laying in Fontaine’s
ring $\mathrm{B}_{\mathrm{cris}}$ of $p$-adic periods. By using the CM action
we characterise this matrix by a single invertible period
$\lambda\in\mathrm{B}_{\mathrm{cris},F}^{\times}$, where
$\mathrm{B}_{\mathrm{cris},F}$ is the smallest subring of
$\mathrm{B}_{\mathrm{dR}}$ containing $\mathrm{B}_{\mathrm{cris}}$ and $F$.
This period is unique up to a scalar in $F^{\times}$ and it controls
completely the arithmetic of our problem. Namely it produces a “re-
normalisation factor” $\lambda\lambda^{*}$ belonging to $F_{0}^{\times}$,
where $F_{0}$ denotes the subfield of $F$ of elements fixed by $*$. Thanks to
a theorem of Milnor on CM quadratic forms on $p$-adic fields [MR249519], we
show that our problem is equivalent to control precisely whether the re-
normalisation factor is a norm of an element of $F^{\times}$. We show that
this problem is multiplicative with respect to the tensor product of motives:
hence, we are reduced to treat the case of small Hodge weights. For those
motives we need to show that the re-normalisation $\lambda\lambda^{*}$ factor
is not a norm.

This is the crucial point of the article. Even if the definition of the period
$\lambda$, based on the crystalline comparison theorem, is not explicit, this
period satisfies a strong uniqueness property with respect to the behaviour
under the Frobenius and the de Rham filtration. Such a uniqueness follows from
the fundamental exact sequences in $p$-adic Hodge theory. Therefore, we can
describe $\lambda$ through any other periods with similar behaviour. For this
purpose, we use Lubin–Tate periods of Colmez [MR1956055] to get some control
of $\lambda$. It turns out that such a control is precisely what we need to
apply a theorem of Dwork [MR98079] in order to compute the image of the re-
normalisation factor $\lambda\lambda^{*}$ through the local reciprocity map
$F^{\times}_{0}\rightarrow\mathrm{Gal}({F}/{F_{0}})$. As the kernel of the
reciprocity map is the group of norms $N_{F/F_{0}}(F^{\times})$, we obtain
that $\lambda\lambda^{*}$ is not a norm.

######  Remark 1.5.

The main result of [MR4199442] is an instance of Theorem 1.4, namely the case
where the motive $M$ has rank two. In that particular case the field $F$ is a
quadratic extension of $\mathbb{Q}_{p}$. As there are finitely many such
extensions, it was possible in [MR4199442]*Section 12 to compute the period
$\lambda$ case by case. Moreover, those periods could be described in terms of
Lubin–Tate periods and elements algebraic over $F$.

In higher rank, $F$ is a quadratic extension of $F_{0}$, whose degree over
$\mathbb{Q}_{p}$ is equal to the half of the rank of the motive, with a priori
no restriction on the ramification of $F_{0}/\mathbb{Q}_{p}$ nor of $F/F_{0}$.
Hence an analysis case by case is impossible. Moreover, the period $\lambda$
is computed in terms of Lubin–Tate periods and a non-explicit element in
$\widehat{F}^{\mathrm{nr}}$, the completion of the maximal unramified
extension of $F$. In general such an element is transcendental over $F$, see
Remark 9.6.

### Organisation of the paper

We begin by setting up notation and conventions in Section 2. In Section 3 we
give the precise statement of Theorem 1.4 and explain how to deduce Theorem
1.3 from it. In Section 4 we recall classical results on quadratic forms and
use them, combined with general theorems on $\ell$-adic cohomology, to
translate Theorem 1.4 into a $p$-adic question. Section 5 presents the first
reduction step. We extend the coefficients of the motive from $\mathbb{Q}$ to
$\mathbb{Q}_{p}$ and decomposes it into a sum of direct factors. The goal of
the section is to show that it is enough to study each factor separately.

In the rest of the paper we fix one of these factors. It is endowed with the
action of a finite extension $F$ of $\mathbb{Q}_{p}$. Section 6 recalls
classical results on CM quadratic forms over the $p$-adic numbers. At the end
of the section we define the re-normalisation factor. This is used in Section
7 where we study the behaviour of the problem under tensor product. This
allows to reduce the problem to motives whose crystalline realization has
small Hodge weights. Section 8 gives a characterization of the $p$-adic
periods of those motives. This allows in Section 9 to describe them explicitly
by using Lubin–Tate periods and to apply Dwork’s theorem computing the local
reciprocity.

In Section 10 we put all the ingredients together and give the proof of the
main result. This section can also be red first in order to have a global
picture of the strategy.

We finish with an appendix concerning the filtered $\varphi$-modules attached
to a Lubin–Tate character of a $p$-adic field. We added it to keep the paper
as self contained as possible, by giving an alternative construction of
Lubin–Tate periods of [MR1956055].

### Acknowledgements

We thank Eva Bayer for her explanations on CM quadratic forms and pointing out
the reference [MR249519]. We thank Lionel Fourquaux and Stefano Morra for
discussions on Lubin–Tate formal groups and their filtered $\varphi$-modules.
We thank Emiliano Ambrosi for useful comments on a preliminary version of the
text. Some years ago we had some illuminating conversations on $p$-adic
periods with our colleague Jean-Pierre Wintenberger. We dedicate this work to
his memory.

##  2\. Conventions

Throughout the paper we will work with the following notation and conventions.

###  2.1. $p$-adic fields.

Let $p$ be a prime number. In this text, by $p$-adic field we mean a finite
extension of the field of $p$-adic numbers $\mathbb{Q}_{p}$. For such a field
$L$, we denote by $\mathcal{O}_{L}$ its ring of integers and by $k_{L}$ the
residue field. The degree of $k_{L}$ over $\mathbb{F}_{p}$ is called the
residual degree of $L/\mathbb{Q}_{p}$ and it will be denoted by $f_{L}$. We
denote111In literature $L_{a}$ is often denoted with subscript $0$ instead of
$a$, but we reserve the notation $L_{0}$ for another subfield (namely the
field of fixed points by an involution, see Section 6). The subscript “$a$”
stands for “absolute unramified” as $L_{a}$ is also called the absolute
unramified subextension of $L$. by $L_{a}$ the maximal subfield of $L$
unramified over $\mathbb{Q}_{p}$. It is equal to the field of fractions of the
ring of Witt vectors $W(k)$. The degree $e_{L}\coloneqq[L:L_{a}]$ is called
the ramification index of $L/\mathbb{Q}_{p}$, or the _absolute_ ramification
index of $L$. When there is no risk of confusion, we may drop the subscript
$L$ in the notation.

We chose an algebraic closure $\overline{\mathbb{Q}}_{p}$ of $\mathbb{Q}_{p}$
and we denote by $\overline{\mathbb{F}}_{p}$ its residue field. We put
$\widehat{\mathbb{Q}}_{p}^{\mathrm{nr}}\coloneqq\operatorname{Frac}W(\overline{\mathbb{F}}_{p})$.
It is a completion of the maximal unramified extension
$\mathbb{Q}_{p}^{\mathrm{nr}}$ of $\mathbb{Q}_{p}$ in
$\overline{\mathbb{Q}}_{p}$. The fields
$\widehat{\mathbb{Q}}_{p}^{\mathrm{nr}}$ and $\mathbb{Q}_{p}^{\mathrm{nr}}$
are endowed with an isomorphism $\varphi$, called the absolute Frobenius,
which is the unique map lifting the $p$-power map on the residue field
$\overline{\mathbb{F}}_{p}$. For any subextension $L$ of
$\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}$ we put
$G_{L}\coloneqq\mathrm{Gal}({\overline{\mathbb{Q}}_{p}}/{L})$.

###  2.2. Motives.

We will work with the category $\operatorname{\mathcal{M}ot}(S)_{\mathbb{Q}}$
of homological motives over a base $S$ with coefficients in $\mathbb{Q}$. For
generalities on these categories see [MR2115000]*§4. The base $S$ will be
$\mathbb{C}$, a finite field or the ring of integers of a $p$-adic field. In
general these categories depend on the chosen Weil cohomology, but we will use
only cohomologies for which the classical comparison theorems will ensure that
these categories are independent of the choice. See also the convention in
[AncFra] for a detailed discussion.

We will also use the quotient category of motives with respect to numerical
equivalence and denote it by $\operatorname{NUM}(S)_{\mathbb{Q}}$. We will use
it only with $S$ being a finite field.

###  2.3. Realisations

We will make use of the classical realization functors, namely the de Rham
realization $R_{\mathrm{dR}}$, the Betti realization $R_{B}$, the $\ell$-adic
realization $R_{\ell}$ and the crystalline realization $R_{\mathrm{cris}}.$ We
consider them with their enriched structures, as in [MR2115000, §7.1]. In
particular, the functor $R_{\mathrm{dR}}$ will land in the category of
filtered vector spaces, $R_{B}$ in the category of Hodge structures,
$R_{\ell}$ in the category of Galois representations and $R_{\mathrm{cris}}$
in the category of modules endowed with an action of an absolute Frobenius.

###  2.4. Unit object.

The unit object in any tensor category we will consider will be denoted by
${\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}$.

###  2.5. Filtered Modules

Let $M$ be a module over some ring and $(\mathrm{Fil}^{i}M)_{i\in\mathbb{Z}}$
a decreasing filtration, exhaustive and separated. We will use the following
conventions: for any subset $S\subset M$, we put

$v_{\mathrm{dR}}(S)\coloneqq\sup\\{i\in\mathbb{Z}|\,S\subset\mathrm{Fil}^{i}M\\}.$

For $m$ in $M$, we write $v_{\mathrm{dR}}(m)=v_{\mathrm{dR}}(\\{m\\})$. (The
subscript $\mathrm{dR}$ comes from the fact that the filtrations in this
article will be the de Rham filtration on some cohomology group or on some
ring of periods.)

###  2.6. $p$-adic Hodge theory.

We denote by $\mathrm{B}_{\mathrm{dR}}$ and $\mathrm{B}_{\mathrm{cris}}$ the
Fontaine’s rings of periods (associated with
$\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p}$). They were introduced in
[MR657238, MR1293971]. We gather below some properties of these rings that we
will use in this article. For a detailed account on these rings and on
$p$-adic Hodge theory we refer for example to [MR1922833, fontaine_ouyang].

  1. (1)

$\mathrm{B}_{\mathrm{dR}}$ is a complete discrete valuation field, its ring of
integer is denoted by $\mathrm{B}_{\mathrm{dR}}^{+}$ and the residual field
identifies to the $p$-adic completion $\mathbb{C}_{p}$ of
$\overline{\mathbb{Q}}_{p}$.

  2. (2)

$\mathrm{B}_{\mathrm{dR}}$ is filtered by the (fractional) powers of the
maximal ideal of $\mathrm{B}_{\mathrm{dR}}^{+}$, and we denote this filtration
by $(\mathrm{Fil}^{i}\mathrm{B}_{\mathrm{dR}})_{i\in\mathbb{Z}}$. The map
$v_{\mathrm{dR}}$ defined in Convention 2.5 on $\mathrm{B}_{\mathrm{dR}}$ is
its discrete valuation.

  3. (3)

$\mathrm{B}_{\mathrm{dR}}$ is endowed with an action of $G_{\mathbb{Q}_{p}}$.
We have canonical inclusions
$\overline{\mathbb{Q}}_{p}\subset\mathrm{B}_{\mathrm{dR}}$ and
$\widehat{\mathbb{Q}}_{p}^{\mathrm{nr}}\subset\mathrm{B}_{\mathrm{dR}}$
compatible with the action of $G_{\mathbb{Q}_{p}}$.

  4. (4)

$\mathrm{B}_{\mathrm{cris}}\subset\mathrm{B}_{\mathrm{dR}}$ is a
sub-$\widehat{\mathbb{Q}}_{p}^{\mathrm{nr}}$-algebra stable under the
$G_{\mathbb{Q}_{p}}$-action.

  5. (5)

We have an endomorphism

$\varphi_{\mathrm{cris}}\colon\mathrm{B}_{\mathrm{cris}}\rightarrow\mathrm{B}_{\mathrm{cris}},$

semilinear with respect to the absolute Frobenius $\varphi$ of
$\widehat{\mathbb{Q}}_{p}^{\mathrm{nr}}$. We call it the Frobenius of
$\mathrm{B}_{\mathrm{cris}}$ and denote it simply by $\varphi$.

  6. (6)

There exists an element $t\in\mathrm{B}_{\mathrm{cris}}$, such that

$\varphi(t)=pt\quad\text{ and }\quad
t\in\mathrm{Fil}^{1}\mathrm{B}_{\mathrm{dR}}\,\text{ (actually
}\mathrm{Fil}^{1}\mathrm{B}_{\mathrm{dR}}=t\mathrm{B}_{\mathrm{dR}}^{+}\text{).}$

For any $g$ in $G_{\mathbb{Q}_{p}}$, we have

$g(t)=\chi(g)t,$

where $\chi\colon G_{\mathbb{Q}_{p}}\rightarrow\mathbb{Z}_{p}^{\times}$
denotes the cyclotomic character. We set
$\mathbb{Q}_{p}(1)\coloneqq\mathbb{Q}_{p}\cdot
t\subset\mathrm{B}_{\mathrm{cris}}$.

Let $L\subset\overline{\mathbb{Q}}_{p}$ be a finite extension over
$\mathbb{Q}_{p}$. Keep the notation of Convention 2.1.

  1. (7)

We will denote by $\mathrm{B}_{\mathrm{cris},L}$ the smallest subring of
$\mathrm{B}_{\mathrm{dR}}$ containing $\mathrm{B}_{\mathrm{cris}}$ and $L$. We
recall that the natural map

$\mathrm{B}_{\mathrm{cris}}\otimes_{L_{a}}L\xrightarrow{\,\simeq\,}\mathrm{B}_{\mathrm{cris},L}\subset\mathrm{B}_{\mathrm{dR}}$

is an isomorphism.

  2. (8)

The Frobenius of $\mathrm{B}_{\mathrm{cris}}$ does not extend to
$\mathrm{B}_{\mathrm{cris},L}$ in general, nevertheless its power
$\varphi_{\mathrm{cris}}^{f}$ is $L_{a}$-linear, hence we can extend it to
$\mathrm{B}_{\mathrm{cris},L}$ by
$\varphi_{\mathrm{cris}}^{f}\otimes\mathrm{Id}_{L}$. We will denote it still
by $\varphi_{\mathrm{cris}}^{f}$ or $\varphi^{f}$.

##  3\. Main statements

In this section we state our main result (Theorem 3.2) and we then give some
geometric consequences. To put the result into perspective, we formulate a
conjecture (Conjecture 3.1) and discuss the role of the different hypothesis
(Remark 3.3).

######  Conjecture 3.1.

Let $K$ be a $p$-adic field, $\mathcal{O}_{K}$ its ring of integers and $k$
its residue field. Let $(M,q)$ be a motive in mixed characteristic

$M\in\operatorname{\mathcal{M}ot}(\mathcal{O}_{K})_{\mathbb{Q}}$

together with a map $q\colon\operatorname{Sym}^{2}M\rightarrow{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}.$ Write $\cdot_{/_{k}}$ for the
restriction functor to the category
$\operatorname{\mathcal{M}ot}(k)_{\mathbb{Q}}$ and $R_{B}$ for the Betti
realization induced by a fixed embedding $\sigma:K\hookrightarrow\mathbb{C}$.

Define two $\mathbb{Q}$-quadratic spaces $(V_{B},q_{B})$ and $(V_{Z},q_{Z})$
as

$(V_{B},q_{B})\coloneqq R_{B}(M,q),\hskip
14.22636ptV_{Z}\coloneqq\mathrm{Hom}_{\operatorname{NUM}(k)_{\mathbb{Q}}}\bigl{(}{\rm
1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt},M_{/_{k}}\bigr{)}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptq_{Z}(-)\coloneqq(q_{/_{k}}\circ\operatorname{Sym}^{2}(-)).$

We conjecture that if $q_{B}$ is a polarization of Hodge structures then
$q_{Z}$ is positive definite.

######  Theorem 3.2.

Let $M,q,q_{B}$ and $q_{Z}$ be as in Conjecture 3.1.

Let $F$ be a number field which acts on $M$ and which is endowed with a non
trivial involution $*$. Assume the following:

  1. (1)

The equality $\dim_{\mathbb{Q}}V_{Z}=\dim M$ holds.

  2. (2)

The equality $[F:\mathbb{Q}]=\dim M$ holds.

  3. (3)

For every $z$ in $F$ the adjoint of $z$ with respect to $q$ is $*(z)$.

  4. (4)

The pairing $q_{B}$ on $V_{B}$ is a polarization of Hodge structures.

Then the signature $(s_{+},s_{-})$ of the quadratic form $q_{Z}$ satisfies
$4|s_{-}$.

The proof of Theorem 3.2 is given is Section 10.

######  Remark 3.3.

Hypothesis (4) is the crucial and absolutely necessary. Indeed, in order to
conclude a positivity statement one has to assume some positivity property.
The action of $F$ makes the hypothesis (1) essentially automatic. Indeed if
$\dim V_{Z}\neq\dim M$ then $V_{Z}=0$ and the statement is trivial. See
[MR4199442, proof of Proposition 6.8] for details.

The statement should be true without the hypothesis (2) and (3), see
Conjecture 3.1. On the other hand, the standard conjecture of Hodge type can
be reduced to the case of varieties over finite fields [MR4199442, Proposition
3.16] and for such a variety the Tate conjecture predicts that its motive
should be an abelian motive. In particular the hypothesis of Theorem 3.2
should not be restrictive in the study of the standard conjecture of Hodge
type.

Finally, homological and numerical equivalence should always coincide hence
one would like to replace $V_{Z}$ with the space of cycles modulo homological
equivalence. The following proposition is the crucial reason for which one
needs to work with numerical equivalence: its analogue with homological
equivalence is out of reach.

######  Lemma 3.4.

Keep the hypothesis from Theorem 3.2. Then hypothesis $(1)$ is equivalent to
the existence of an isomorphism

$M_{/_{k}}\cong{\rm 1\rule{0.43057pt}{6.54443pt}\hskip 0.86108pt}^{\oplus\dim
M}.$

In particular $V_{B}$ and all classical realizations of $M$ have weight zero
and if $E$ is the field of coefficients of a given realization $R$, then

$R(M_{/_{k}})=V_{Z}\otimes_{\mathbb{Q}}E,\hskip 14.22636pt\textrm{and}\hskip
14.22636ptR(q_{/_{k}})=q_{Z}\otimes_{\mathbb{Q}}E$

###### Proof..

This follows from the fact that numerical motives form a semisimple category
[MR1150598]. ∎

Let us now conclude the section with some applications of Theorem 3.2.

######  Corollary 3.5.

Let $A$ be an abelian variety over a finite field, $L$ be an hyperplane
section induced by a CM-lifting and $n\leq\dim(A)/2$ be a positive integer.
Let $(s_{+},s_{-})$ be the signature of the quadratic form $q_{A,L,n}$ from
Definition 1.1, then $s_{-}$ is divisible by four.

###### Proof..

We argue as in [MR4199442, §8] and use the complex multiplication to decompose
the motive of $A$ in an orthogonal sum of smaller motives. Among these
factors, the ones we need to study are the so called exotic. They all fit in
the hypothesis of Theorem 3.2. ∎

######  Remark 3.6.

It is known that the standard conjecture of Hodge type can be reduced to the
case of finite fields and that moreover its validity is actually independent
of the choice of $L$, see [MR4199442, §3] for details. Hence both hypothesis
in the above corollary should be thought as reasonable.

######  Corollary 3.7.

Let $X=S_{1}\times\ldots\times S_{m}$ be the product of K3 surfaces over
finite fields, $L_{i}$ be an hyperplane section induced by a CM lifting of
$S_{i}$, cf. [MR4241794], and let $L=\oplus_{i}p^{*}_{i}L_{i}$ be the induced
hyperplane section on $X$. Let $n\leq\dim(X)/2=m$ be any integer and
$(s_{+},s_{-})$ be the signature of the quadratic form $q_{X,L,n}$ from
Definition 1.1, then $s_{-}$ is divisible by four.

###### Proof..

Similar to the case of abelian varieties. ∎

######  Remark 3.8.

There are probably other varieties to which Theorem 3.2 can be applied. As
already mentioned in Remark 3.3, all varieties over finite fields should have
a CM lifting. Some other examples for which this is known are cubic Fermat
hypersurfaces [MR552586]. In order to apply Theorem 3.2 to them one has to
check that the decomposition induced by the CM action is orthogonal with
respect to the quadratic form $q_{X,L,n}$. Similarly, one might try to study
Kummer varieties or the Hilbert scheme of points on a K3.

##  4\. From archimedian to $p$-adic

We recall here some classical facts on quadratic forms. They will allow us to
reduce Theorem 3.2 to a $p$-adic statement. In what follows $\mathbb{Q}_{\nu}$
denotes the completion of $\mathbb{Q}$ at the place $\nu$. Recall that at
every place one defines $\varepsilon_{\nu}(q)\in\\{\pm 1\\},$ the Hilbert
symbol of $q$ at $\nu$, cf. [MR0498338, Ch. IV, §2.1 and §2.4].

######  Proposition 4.1.

[MR0498338, Ch. IV, §2.4]. Let $q$ be a non-degenerate $\mathbb{Q}$-quadratic
form and let $(s_{+},s_{-})$ be its signature. Then the discriminant is
positive if and only if $2|s_{-}.$ In this case, $4|s_{-}$ if and only if
$\varepsilon_{\mathbb{R}}(q)=+1$.

######  Proposition 4.2.

[MR0498338, Ch. IV, §2.3, Theorem 7] Let $p$ be a prime number and $q_{1}$ and
$q_{2}$ two non-degenerate $\mathbb{Q}$-quadratic forms of same rank. Then

$q_{1}\otimes\mathbb{Q}_{p}\cong q_{2}\otimes\mathbb{Q}_{p}$

if and only if the discriminants of $q_{1}$ and $q_{2}$ coincide in
$\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}$ and

$\varepsilon_{p}(q_{1})=\varepsilon_{p}(q_{2}).$

######  Theorem 4.3.

[MR0498338, Ch. IV, §3.1] Let $q$ be non-degenerate $\mathbb{Q}$-quadratic
form. Then for all but a finite number of places $\nu$ the equality
$\varepsilon_{\nu}(q)=+1$ holds. Moreover, the following product formula
running on all places holds

$\prod_{\nu}\varepsilon_{\nu}(q)=+1.$

######  Corollary 4.4.

Let $q_{1}$ and $q_{2}$ be two non-degenerate $\mathbb{Q}$-quadratic forms and
let $p$ be a prime number. Suppose that, for all primes $\ell$ different from
$p$, we have

$q_{1}\otimes\mathbb{Q}_{\ell}\cong q_{2}\otimes\mathbb{Q}_{\ell}.$

Then the two quadratic forms $q_{1}$ and $q_{2}$ have the same discriminant.
Moreover, if this discriminant is positive, then $4|s_{-}(q_{1})$ if and only
if the equality

$\varepsilon_{p}(q_{1})=\varepsilon_{\mathbb{R}}(q_{2})\varepsilon_{p}(q_{2}).$

holds.

###### Proof.

The $\ell$-adic hypothesis implies in particular that the discriminants of
$q_{1}$ and $q_{2}$ coincide in
$\mathbb{Q}_{\ell}^{\times}/(\mathbb{Q}_{\ell}^{\times})^{2}$ for all
$\ell\neq p$. This implies that they coincide in
$\mathbb{Q}^{\times}/(\mathbb{Q}^{\times})^{2}$ by [MR1070716, Theorem 3 in
5.2].

Suppose from now on that the discriminant is positive. By Proposition 4.1,
$4|s_{-}(q_{1})$ if and only $\varepsilon_{\mathbb{R}}(q_{1})=+1.$

Now, Theorem 4.3 implies that
$\prod_{\nu}\varepsilon_{\nu}(q_{1})=1=\prod_{\nu}\varepsilon_{\nu}(q_{2}).$
Combining this with the $\ell$-adic isomorphisms we deduce

$\varepsilon_{\mathbb{R}}(q_{1})\varepsilon_{p}(q_{1})=\varepsilon_{\mathbb{R}}(q_{2})\varepsilon_{p}(q_{2}).$

This means that $\varepsilon_{\mathbb{R}}(q_{1})=+1$ if and only if
$\varepsilon_{p}(q_{1})=\varepsilon_{\mathbb{R}}(q_{2})\varepsilon_{p}(q_{2})$.
∎

######  Proposition 4.5.

Let us keep notation from Theorem 3.2. Let $p$ be the characteristic of $k$
and $h^{i}$ be the dimension of the $(i,-i)$-part of the Hodge structure
$V_{B}$. Define the positive integer

$s_{M}\coloneqq\sum_{i\geq 1,\,\text{odd}}h_{i}.$

Then the quadratic forms $q_{B}$ and $q_{Z}$ have the same discriminant, which
is positive. Moreover, Theorem 3.2 holds true if and only if the equality

$\varepsilon_{p}(q_{Z})/\varepsilon_{p}(q_{B})=(-1)^{s_{M}}.$

holds true.

###### Proof.

We want to apply Corollary 4.4 to $q_{1}=q_{Z}$ and $q_{2}=q_{B}$.

First of all, by Lemma 3.4, we have that
$q_{Z}\otimes\mathbb{Q}_{\ell}=R_{\ell}(q_{/_{k}})$. By Artin comparison
theorem, we have $q_{B}\otimes\mathbb{Q}_{\ell}=R_{\ell}(q_{/_{\mathbb{C}}})$.
Combining these equalities with smooth proper base change in $\ell$-adic
cohomology we deduce that

$q_{B}\otimes\mathbb{Q}_{\ell}=q_{Z}\otimes\mathbb{Q}_{\ell}.$

Now, by hypothesis $q_{B}$ is a polarization for the Hodge structure $V_{B}$
and recall that $V_{B}$ has weight zero (Lemma 3.4). In particular, the
Hodge–Riemann relations compute the signature $(s_{B,+},s_{B,-})$ of $q_{B}$
as

$s_{B,+}=\sum_{i\,\text{even}}h_{i}\hskip 14.22636pt\textrm{and}\hskip
14.22636pts_{B,-}=\sum_{i\,\text{odd}}h_{i}.$

Note that $s_{B,-}$ is even because of the Hodge symmetry. This implies
through Proposition 4.1 that the discriminant is positive and that the Hilbert
symbol of $q_{B}\otimes\mathbb{R}$ is $(-1)^{\sum_{i\geq 1,\text{odd}}h_{i}}$.
We can now conclude by applying Corollary 4.4 to $q_{1}=q_{Z}$ and
$q_{2}=q_{B}$. ∎

######  Remark 4.6.

We have ${\sum_{i\geq 1,\text{odd}}h_{i}}\equiv{\sum_{i\geq 0}ih_{i}}$ (mod
$2$), in particular the integer $s_{M}$ has the same parity as the minimum of
the Hodge polygon.

The role of the above proposition is to translate a positivity problem
(Theorem 3.2) into a $p$-adic problem. The advantage is that the latter has a
cohomological interpretation, as the following proposition shows.

######  Proposition 4.7.

Let us keep notation from Theorem 3.2. Then the following holds.

  1. (1)

The quadratic space $(V_{B},q_{B})\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ is
isomorphic to $(R_{p}(M),R_{p}(q))$.

  2. (2)

The Galois representation $R_{p}(M)$ is crystalline and the admissible
filtered-$\varphi$-module which corresponds to it is $R_{\mathrm{cris}}(M)$.

  3. (3)

The $\mathbb{Q}_{p}$-subspace of Frobenius invariant vectors in
$R_{\mathrm{cris}}(M)$ identifies with
$V_{Z}\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ and generates
$R_{\mathrm{cris}}(M)$.

  4. (4)

The quadratic form $q_{Z}\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ identifies with
the restriction of $R_{\mathrm{cris}}(q)$ to
$V_{Z}\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$.

###### Proof.

Point (1) comes from Artin’s comparison theorem. Point (2) comes from
Falting’s comparison theorem in $p$-adic Hodge theory [MR1463696], see also
[MR4199442]*Theorem 10.2. For part (3), notice that in general algebraic
classes are included in the Frobenius invariant ones, but for dimensional
reasons (hypothesis (1) in Theorem 3.2) this inclusion is an equality. Point
(4) follows from Lemma 3.4. ∎

##  5\. Reduction to CM-simple objects

In this section we study quadratic forms in the context of $p$-adic Hodge
theory. The setting we choose is inspired by the previous section and in
particular by Propositions 4.5 and 4.7.

######  Definition 5.1.

A supersingular pair $(V_{B,p},V_{Z,p})$ is the collection of the following
objects.

  1. (1)

A $\mathbb{Q}_{p}$-vector space $V_{B,p}$ endowed with the action of the
absolute Galois group of a $p$-adic field $K$ which makes the representation
crystalline.

  2. (2)

A $\mathbb{Q}_{p}$-vector space $V_{Z,p}$ which generates the filtered
$\varphi$-module corresponding to $V_{B,p}$ and on which the Frobenius acts
trivially.

An orthogonal supersingular datum $(V_{B,p},V_{Z,p},q_{B,p},q_{Z,p})$, or
simply an orthogonal supersingular pair $(q_{B,p},q_{Z,p})$ is the collection
of the following objects.

  1. (1)

A supersingular pair $(V_{B,p},V_{Z,p})$.

  2. (2)

A non-degenerate $\mathbb{Q}_{p}$-quadratic form $q_{B,p}$ on $V_{B,p}$ for
which the Galois action is orthogonal.

  3. (3)

The non-degenerate $\mathbb{Q}_{p}$-quadratic form $q_{Z,p}$ on $V_{Z,p}$
corresponding to $q_{B,p}$.

######  Lemma 5.2.

Let $(q_{B,p},q_{Z,p})$ be an orthogonal supersingular pair. Then the
discriminants of $q_{B,p}$ and $q_{Z,p}$ coincide.

###### Proof.

The quadratic form on $V_{Z,p}$ induces a quadratic form on the one
dimensional vector space $\operatorname{det}V_{Z,p}$. This form is
characterized by an element of
$\mathbb{Q}_{p}^{\times}/(\mathbb{Q}_{p}^{\times})^{2}$, which is the
discriminant of $q_{Z}$. (The same holds for $V_{B,p}$.) Note that the
Frobenius acts trivially on $\operatorname{det}V_{Z,p}$. Admissibility implies
that the filtration is trivial as well. This means that the comparison
isomorphism sends the $\mathbb{Q}_{p}$-structure $\operatorname{det}V_{Z,p}$
on the $\mathbb{Q}_{p}$-structure $\operatorname{det}V_{B,p}$. As the
comparison isomorphism respects the underlined quadratic forms the
discriminants coincide. ∎

######  Lemma 5.3.

Let $(V_{B,p},V_{Z,p},q_{B,p},q_{Z,p})$ be an orthogonal supersingular datum,
let $F$ be a number field and let $*$ be a non trivial field involution on
$F$. Then the following holds true.

  1. (1)

The ring $F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ is a product of $p$-adic fields

$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}=\prod_{i=1}^{n}F_{i}$

  2. (2)

The involution $*$ acts on this product as a composition of disjoint
transpositions, i.e. there is an integer $s$ such that, after changing the
numbering, one has

$*(F_{i})=F_{i+s}\hskip 5.69046pt\textrm{for}\hskip 5.69046pti\leq s,\hskip
5.69046pt*(F_{i})=F_{i-s}\hskip 5.69046pt\textrm{for}\hskip 5.69046pts<i\leq
2s\hskip 5.69046pt\textrm{and}\hskip 5.69046pt*\\!(F_{i})=F_{i}\hskip
5.69046pt\textrm{for}\hskip 5.69046pti>2s.$

Moreover the induced involution on $F_{i}$ for $i>2s$ is non trivial.

  3. (3)

Suppose that $F$ acts on $V_{B,p}$ and that this action commutes with the
Galois action. Then $F$ acts on $V_{Z,p}$ as well and the two actions are
compatible with respect to the comparison theorem. In particular the
decomposition of point (1) induces two decompositions

$V_{B,p}=\bigoplus_{i=1}^{n}V^{i}_{B,p}\ \hskip 14.22636pt\textrm{and}\hskip
14.22636ptV_{Z,p}=\bigoplus_{i=1}^{n}V^{i}_{Z,p}$

and for all $i$ the pair $(V^{i}_{B,p},V^{i}_{Z,p})$ is a supersingular pair.

  4. (4)

Suppose moreover that for all $z\in F$ the adjoint of $z$ with respect to
$q_{B,p}$ is $z^{*}$. Then the same holds true for $q_{Z,p}$. Moreover the
decompositions in $n-s$ factors

$V_{B,p}=\bigoplus^{\perp}_{s<i\leq 2s}(V^{i}_{B,p}\oplus
V^{i-s}_{B,p})\oplus^{\perp}\bigoplus^{\perp}_{i>2s}(V^{i}_{B,p})\hskip
14.22636pt\textrm{and}\hskip 14.22636ptV_{Z,p}=\bigoplus^{\perp}_{s<i\leq
2s}(V^{i}_{Z,p}\oplus
V^{i-s}_{Z,p})\oplus^{\perp}\bigoplus^{\perp}_{i>2s}(V^{i}_{Z,p})$

are orthogonal.

  5. (5)

Keep notation and hypothesis from point (4). For all $i>s$ denote by
$q^{i}_{B,p}$ the restriction of $q_{B,p}$ to each orthogonal factor of the
above decomposition and similarly for $q^{i}_{Z,p}$. Then the pair
$(q^{i}_{B,p},q^{i}_{Z,p})$ is an orthogonal supersingular pair.

  6. (6)

For $s<i\leq 2s$ consider the orthogonal supersingular pair
$(q^{i}_{B,p},q^{i}_{Z,p})$ defined in point (5). Then the two
$\mathbb{Q}_{p}$-quadratic forms $q^{i}_{B,p}$ and $q^{i}_{Z,p}$ are
isomorphic. More precisely, the two subspaces $V^{i}_{B,p}$ and
$V^{i-s}_{B,p}$ of $V^{i}_{B,p}\oplus V^{i-s}_{B,p}$ are maximal isotropic for
the quadratic form $q^{i}_{B,p}$ (and similarly for $q^{i}_{Z,p}$). Moreover
$q^{i}_{B,p}$ realizes the Galois representations $V^{i}_{B,p}$ and
$V^{i-s}_{B,p}$ as one the dual of the other.

###### Proof.

Point (1) comes from the fact that $F/\mathbb{Q}$ is a separable extension.

For point (2) consider
$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}=F_{1}\times\ldots\times F_{n}$. Its
spectrum is a disjoint union of $n$ closed points and $*$ acts on it as a
permutation of order two. Consider now $i>2s$ as in the statement and the
inclusion $F\hookrightarrow F_{i}$ induced from point (1). By construction the
involution on $F_{i}$ extends the one on $F$, hence it is non trivial.

For point (3), by the comparison theorem, as $F$ acts on $V_{B,p}$ seen as
crystalline Galois representation of $G_{K}$, then $F$ must act also on its
corresponding filtered $\varphi$-module. This means that the action of $F$
must commute with the Frobenius hence $F$ stabilizes the Frobenius invariant
part, i.e. $V_{Z,p}$.

Let us now study point (4), let $q$ be one of the quadratic forms we want to
study and $b$ be its corresponding bilinear form. Let $p_{i}$ be the $i$-th
projector in the decomposition
$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}=F_{1}\times\ldots\times F_{n}$. By
hypothesis we have $b(p_{i}(-),p_{j}(-))=b((-),p_{i}^{*}p_{j}(-))$. But
$p_{i}^{*}p_{j}=0$ except if $p_{j}=p_{i}^{*}$.

Point (5) is automatic from the construction. For point (6) we have to study
the quadratic forms on the space $V^{i}_{B,p}\oplus V^{i-s}_{B,p}$ for
$s<i\leq 2s$ (and similarly for $V^{i}_{Z,p}\oplus V^{i-s}_{Z,p}$). Notice
that the proof of point (4) we gave, shows in particular that the two
subspaces $V^{i}_{B,p}$ and $V^{i-s}_{B,p}$ are maximal isotropic. Thus the
quadratic forms $q_{B,p}$ and $q_{Z,p}$ are both isomorphic to a sum of
hyperbolic planes. The duality statement comes from the fact that $q_{B,p}$ is
non-degenerate by hypothesis. ∎

######  Definition 5.4.

Let $M=(q_{B,p},q_{Z,p})$ be an orthogonal supersingular pair, let
$\varepsilon_{p}(q_{B,p})$ and $\varepsilon_{p}(q_{Z,p})$ be the Hilbert
symbols of the quadratic forms and define $s_{M}$ to be the minimum of the
Hodge polygon of the filtered $\varphi$-module underlying $M$. We say that $M$
is good if the following equality holds

(5.4.1) $\varepsilon_{p}(q_{Z,p})/\varepsilon_{p}(q_{B,p})=(-1)^{s_{M}}.$

######  Lemma 5.5.

A supersingular pair $M=(q_{B,p},q_{Z,p})$ is good if and only if one of the
following situation happens:

  1. (1)

The quadratic forms $q_{B,p}$ and $q_{Z,p}$ are isomorphic and $s_{M}$ is
even.

  2. (2)

The quadratic forms $q_{B,p}$ and $q_{Z,p}$ are not isomorphic and $s_{M}$ is
odd.

###### Proof.

The two quadratic spaces have the same discriminant by Lemma 5.2. Hence they
are isomorphic if and only if they have the same Hilbert symbol (Proposition
4.2). This translates the definition of good to the conditions (1) or (2) of
the statement. ∎

######  Proposition 5.6.

Keep notation as in Lemma 5.3. If for each $i>2s$ the pair
$(q^{i}_{B,p},q^{i}_{Z,p})$ is good then the pair $(q_{B,p},q_{Z,p})$ is good
as well.

###### Proof.

First of all, let us show that the pair $(q^{i}_{B,p},q^{i}_{Z,p})$ is good
also for $s<i\leq 2s$. In this case the two quadratic forms are isomorphic by
Lemma 5.3(6), hence
$\varepsilon_{p}(q^{i}_{Z,p})/\varepsilon_{p}(q^{i}_{B,p})=1.$ We need to show
that the height of the corresponding Hodge polygon is even.

Let $D_{Z,p}^{i}$ be the admissible filtered $\varphi$-module associated with
the factor $V^{i}_{Z,p}$. The height of the Newton polygon of each one of
those modules is zero by construction as they have a Frobenius invariant
basis. In particular, since they are admissible, the height of the Hodge
polygon is also zero. Moreover, by Lemma 5.3(6), $D_{Z,p}^{i}$ and
$D_{Z,p}^{i-s}$ are one the dual of the other. This duality together with
height zero implies that the two Hodge polygons are one the symmetric of the
other. In particular, they have the same minimum, hence the minimum of the
Hodge polygon of $D_{Z,p}^{i}\oplus D_{Z,p}^{i-s}$ is even.

Now, as the right hand side of the equation (5.4.1) is additive on direct
sums, let us show that the left hand side is additive as well, this will imply
the statement. Consider two orthogonal supersingular pairs $M_{1}$ and $M_{2}$
which are both partial sums of the factors in the decomposition of Lemma
5.3(4). Let $\delta_{B,1},\varepsilon_{B,1},\delta_{Z,1},\varepsilon_{Z,1}$ be
the discriminants and the Hilbert symbols of the two quadratic forms
associated $M_{1}$ and similarly let
$\delta_{B,2},\varepsilon_{B,2},\delta_{Z,2},\varepsilon_{Z,2}$ the invariants
for $M_{2}$. The Hilbert symbols of the sum $M_{1}\oplus M_{2}$ are
respectively

$\varepsilon_{B}(M_{1}\oplus
M_{2})=\varepsilon_{B,1}\cdot\varepsilon_{B,2}\cdot(\delta_{B,1},\delta_{B,2})$

and

$\varepsilon_{Z}(M_{1}\oplus
M_{2})=\varepsilon_{Z,1}\cdot\varepsilon_{Z,2}\cdot(\delta_{Z,1},\delta_{Z,2})$

because of the additivity of the total class.

If we write the quotient of the two equalities above the contribution of the
discriminants simplifies as $\delta_{B,1}=\delta_{Z,1}$ and
$\delta_{B,2}=\delta_{Z,2}$ by Lemma 5.2. ∎

##  6\. CM-quadratic forms over $p$-adic fields

From now on (except for Section 10) let $F$ be a finite extension of
$\mathbb{Q}_{p}$. We assume that $F$ is endowed with a non-trivial involution
$*\colon F\rightarrow F$, $\alpha\mapsto\alpha^{*}$. We collect here some
basic results on some quadratic forms endowed with an action of $F$ which we
call CM (Definition 6.1).

Throughout the section we fix a Galois closure $\tilde{F}\supset F$ of $F$,
and denote by $\Gamma$ the set of $\mathbb{Q}_{p}$-embeddings of $F$ in
$\tilde{F}$. The induced action of $*$ on $\Gamma$ will still be denoted by
$*$. We will denote by $F_{0}$ the subfield of $F$ where $*$ acts as the
identity.

Denote respectively by $\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}\colon
F_{0}\rightarrow\mathbb{Q}_{p}$ and $N_{F/F_{0}}\colon F\rightarrow F_{0}$ the
trace of the extension $F_{0}/\mathbb{Q}_{p}$ and the norm of $F/F_{0}$; the
extension $F/F_{0}$ has degree two and we have $N_{F/F_{0}}(x)=x^{*}x$ for
every $x\in F$.

######  Definition 6.1.

  1. (1)

A CM-space (with respect to $F$) is the datum of a $\mathbb{Q}_{p}$-vector
space $V$ and an action of $F$ on $V$ such that $V$ as an $F$-vector space has
dimension one.

  2. (2)

A CM-quadratic space (with respect to $F$) is the datum of a CM-space $V$ and
a non-degenerate quadratic form $q\colon V\rightarrow\mathbb{Q}_{p}$ on the
$\mathbb{Q}_{p}$-vector space $V$, such that for all $\alpha$ in $F$, the
adjoint with respect to $q$ of the multiplication by $\alpha$ is the
multiplication by $\alpha^{*}$.

For simplicity we may say that $q$ (or $V$) is a $CM$-quadratic form without
mentioning explicitly the other structures.

######  Proposition 6.2.

Let $V$ be a CM-quadratic space. Fix a nonzero vector $v$ in $V$ and consider
the induced identification $V\cong F$ as $F$-vector spaces. Under this
identification, there exists a unique $a\in F_{0}^{\times}$ such that for
every $x\in F$, we have

$q(x)=\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}(aN_{F/F_{0}}(x)).$

If one changes the choice of the vector $v$, the element $a\in F_{0}^{\times}$
is multiplied by some norm in $N_{F/F_{0}}(F^{\times})$. In particular the
class of $a$ in $F^{\times}_{0}/N_{F/F_{0}}(F^{\times})$ is well-defined and
depends only on $V$.

###### Proof..

Let $b\colon F\times F\rightarrow\mathbb{Q}_{p}$ the symmetric form associated
with $q$. Since $q$ is a CM-quadratic form, we have, for every $x\in F$,
$q(x)=b(x,x)=b(1,x^{*}x)=b(1,N_{F/F_{0}}(x))$. The restriction of $b(1,-)$ to
$F_{0}$ gives a $\mathbb{Q}_{p}$-linear form on $F_{0}$. Since
$(u,v)\mapsto\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}(uv)$ is a non-degenerate
bilinear form on $F_{0}$ we can conclude that there exists such an element
$a\in F_{0}$. It must be nonzero otherwise $q$ would be zero.

Let $w$ be a new nonzero vector in $V$ and $q^{\prime}(x)$ be the induced
quadratic form on $F$ under this new identification. There is a unique nonzero
$z\in F$ such that $zv=w$. This gives the relation $q^{\prime}(x)=q(zx)$.
Hence one deduces

$q^{\prime}(x)=\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}(aN_{F/F_{0}}(zx))=\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}((az^{*}z)\cdot
N_{F/F_{0}}(x)).\qed$

######  Definition 6.3.

Let $q$ be a CM-quadratic form. We define the gauge of $q$ as the element of
$F^{\times}/N_{F/F_{0}}(F^{\times})$ associated with $q$ through the above
proposition.

######  Theorem 6.4 (Milnor).

  1. (1)

The group $F^{\times}/N_{F/F_{0}}(F^{\times})$ has cardinality two.

  2. (2)

All CM-quadratic forms (with respect to the same $F$) have the same
discriminant.

  3. (3)

Two CM-quadratic forms are isomorphic if and only if they have the same gauge
in $F^{\times}/N_{F/F_{0}}(F^{\times})$.

  4. (4)

Up to isomorphism there are exactly two CM-quadratic spaces.

###### Proof..

The extension $F/F_{0}$ has degree $2$. The reciprocity isomorphism

$F^{\times}/N_{F/F_{0}}(F^{\times})\cong\mathrm{Gal}({F}/{F_{0}})$

of local class field theory proves (1).

Consider $F$ as $F_{0}$-vector space of dimension two and consider on it the
non-degenerate $F_{0}$-quadratic forms $\tilde{q}_{a}(x)=aN_{F/F_{0}}(x)$, for
$a\in F_{0}^{\times}$. By definition a CM-quadratic form $q$ of gauge $a$ is
isometric to $\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}\circ\,\tilde{q}_{a}$.
Now, notice that all quadratic forms $\tilde{q}_{a}$ have the same
discriminant in $F_{0}^{\times}/(F_{0}^{\times})^{2}$, independent of $a$,
since $\tilde{q}_{a}(x)=a^{2}\tilde{q}_{1}(x)$. (Actually it is the class of
the opposite of the discriminant of $F/F_{0}$.) By [MR249519, Lemma 2.2] this
implies (2).

For (3) we argue as above and consider the quadratic forms $\tilde{q}_{a}$. We
claim that $\tilde{q}_{a}\cong\tilde{q}_{b}$ if and only if $a$ and $b$ are in
the same class of $F^{\times}/N_{F/F_{0}}(F^{\times})$. This will give (3)
thank to [MR249519, Theorem 2.3]. The claim is elementary: if $a$ and $b$ are
in the same class, we can write $a/b=N_{F/F_{0}}(z)$ and the change of
variable $x\mapsto z\cdot x$ gives the isometry. Conversely, if the quadratic
forms are isomorphic then the images of $F^{\times}$ by them must be the same,
but those images are precisely $a\cdot N_{F/F_{0}}(F^{\times})$ and $b\cdot
N_{F/F_{0}}(F^{\times})$. Point (4) is the combination of (1) and (3). ∎

######  Proposition 6.5.

Let $V$ be a CM-space and consider $V\otimes_{\mathbb{Q}_{p}}F$. It is endowed
with two different actions of $F$, one on the left, induced by the action of
$F$ on $V$, and one on the right. To distinguish them we will write, for all
$w\in V\otimes_{\mathbb{Q}_{p}}F$ and $z\in F$, $z(w)$ for the left
multiplication and $w\cdot z$ for the right multiplication. The inclusion
$F\subset\tilde{F}$ gives a fixed inclusion $V\otimes_{\mathbb{Q}_{p}}F\subset
V\otimes_{\mathbb{Q}_{p}}\tilde{F}$ and, for all $\sigma\in\Gamma$, let us
denote again by $\sigma$ the induced embedding
$V\otimes_{\mathbb{Q}_{p}}F\subset V\otimes_{\mathbb{Q}_{p}}\tilde{F}.$ Then
the following hold.

  1. (1)

There exists a nonzero (eigen)vector $v\in V\otimes_{\mathbb{Q}_{p}}F$ such
that $z(v)=v\cdot z$. Such a $v$ is unique up to multiplication by an element
of $F^{\times}$.

  2. (2)

The (eigen)vectors $\\{\sigma(v)\\}_{\sigma\in\Gamma}$ form a basis of
$V\otimes_{\mathbb{Q}_{p}}\tilde{F}$ over $\tilde{F}$. We set
$L_{\sigma}\coloneqq\sigma(v)\cdot\tilde{F}$.

###### Proof..

Follows immediately from the classical description
$F\otimes_{\mathbb{Q}_{p}}\tilde{F}=\tilde{F}^{\Gamma}$ and the fact that the
dimension of $V$ over $F$ is one. ∎

######  Proposition 6.6.

Let $(V,q)$ be a CM-quadratic space and let $v\in V\otimes_{\mathbb{Q}_{p}}F$
be the eigenvector constructed in Proposition 6.5 above. Let $b$ be the
bilinear pairing induced by $q$. Then the following holds.

  1. (1)

One has $b(\sigma(v),\tau(v))=0$ except if $\sigma=\tau^{*}$.

  2. (2)

The number $2b(v,v^{*})\in F_{0}^{\times}$ is the gauge of $q$.

###### Proof..

For all $z\in F$ we have $b(\sigma(z(v)),\tau(v))=b(\sigma(v),\tau(z^{*}(v)))$
by definition of CM-quadratic forms. On the other hand we have the relations
$b(\sigma(z(v)),\tau(v))=b(\sigma(v),\tau(v))\cdot\sigma(z)$ and
$b(\sigma(v),\tau(z^{*}(v)))=b(\sigma(v),\tau(v))\cdot\tau^{*}(z)$, by
construction of $v$. These relations together give (1).

For the second point, choose $\\{\sigma(v)\\}_{\sigma\in\Gamma}$ as basis. A
vector $w\in V\otimes_{\mathbb{Q}_{p}}\tilde{F}$ is actually in $V$ if and
only if there exists a scalar $z\in F$ such that the coordinates of $w$ with
respect to this basis are $\\{\sigma(z)\\}_{\sigma\in\Gamma}$. Let us now
compute the quadratic form on such a vector $w\in V$:

$q(w)=\sum_{\sigma,\tau\in\Gamma}b(\sigma(zv),\tau(zv))=\sum_{\sigma\in\Gamma}b(\sigma(zv),\sigma^{*}(zv))=\sum_{\sigma\in\Gamma_{0}}2\sigma(zz^{*})\sigma(b(v,v^{*})),$

where the second equality comes from part (1) and $\Gamma_{0}$ denotes the set
of $\mathbb{Q}_{p}$-embeddings of $F_{0}$ in $\tilde{F}$. In conclusion we
have

$q(w)=\operatorname{tr}_{F_{0}/\mathbb{Q}_{p}}(2b(v,v^{*})\cdot
N_{F/F_{0}}(z))$

which computes the gauge. ∎

######  Proposition 6.7.

Let $V_{1}$ and $V_{2}$ be two CM-spaces and let $B$ be an $F$-algebra.
Suppose that it is given an identification

$V_{1}\otimes_{\mathbb{Q}_{p}}B=V_{2}\otimes_{\mathbb{Q}_{p}}B$

which is compatible with the $F$-actions. Let $v_{1}$ and $v_{2}$ be the
vectors constructed in Proposition 6.5 associated respectively with $V_{1}$
and $V_{2}$. Then, there exists a unique invertible element $\lambda\in
B^{\times}$ such that

$v_{1}=\lambda v_{2}.$

Let now $\tilde{B}$ be an $\tilde{F}$-algebra and suppose that each
$\sigma\in\Gamma$ extends to a homomorphism
$\tilde{\sigma}:B\rightarrow\tilde{B}$. Then, for all $\sigma$, we have

$\sigma(v_{1})=\tilde{\sigma}(\lambda)\sigma(v_{2}).$

###### Proof..

Consider the $B$-module
$V_{1}\otimes_{\mathbb{Q}_{p}}B=V_{2}\otimes_{\mathbb{Q}_{p}}B$ and consider
on it the left action of $F$. By construction, $v_{1}$ and $v_{2}$ generate
the same eigenline with respect to this action, this gives the existence of
$\lambda$.

For the second part one applies $\tilde{\sigma}$ to the equality
$v_{1}=\lambda v_{2}$. As the $v_{i}$ live in
$V_{i}\otimes_{\mathbb{Q}_{p}}F\subset V_{i}\otimes_{\mathbb{Q}_{p}}B$ the
action on them coincides with $\sigma.$ ∎

######  Proposition 6.8.

Let $(V_{1},q_{1})$ and $(V_{2},q_{2})$ be two CM-quadratic forms and let $B$
be an $F$-algebra. Suppose that it is given an identification

$V_{1}\otimes_{\mathbb{Q}_{p}}B=V_{2}\otimes_{\mathbb{Q}_{p}}B$

which is compatible with the $F$-actions and with the quadratic forms.

Let $\lambda\in B^{\times}$ be the scalar constructed in Proposition 6.7 and
suppose that the involution $*$ of $F$ extends to an endomorphism $\tilde{*}$
of $B$ as $\mathbb{Q}_{p}$-algebra. Then $\lambda\cdot(\tilde{*}(\lambda))$
belongs to $F_{0}^{\times}$ and the two quadratic forms $q_{1}$ and $q_{2}$
are isomorphic if and only if

$\lambda\cdot(\tilde{*}(\lambda))\in N_{F/F_{0}}(F^{\times}).$

###### Proof..

Let $v_{i}$ be the vectors from Proposition 6.7. By Theorem 6.4, the two
quadratic forms are isomorphic if and only if they have the same gauge in
$F_{0}^{\times}/N_{F/F_{0}}(F^{\times})$. By Proposition 6.6(2), this means
that they are isomorphic if and only if the ratio
$2b(v_{1},v_{1}^{*})/2b(v_{2},v_{2}^{*})$, which belongs to $F_{0}^{\times}$,
is a norm of an element in $F$. On the other hand, by Proposition 6.7, this
ratio is equal to $\lambda\cdot(\tilde{*}(\lambda))$. ∎

##  7\. Reduction to tensor generators

We keep the same notation as in the previous section, in particular, $F$ is a
finite extension of $\mathbb{Q}_{p}$ endowed with a non-trivial involution
$*\colon F\rightarrow F$ and an embedding $\iota$ in a fixed Galois closure
$\tilde{F}\supset F$. Moreover $F_{0}\subset F$ is the subfield fixed by $*$
and $\Gamma$ is the set of $\mathbb{Q}_{p}$-embeddings of $F$ in $\tilde{F}$.
The maximal subfield of $\tilde{F}$ which is unramified over $\mathbb{Q}_{p}$
will be denoted by $\tilde{F_{a}}$, see Convention 2.1.

######  Definition 7.1.

A filtered-CM-space (with respect to $F$) is the datum of a CM-space $V$
(Definition 6.1) together with an integer $n_{\sigma}$ associated with each
eigenline $L_{\sigma}\subset V\otimes_{\mathbb{Q}_{p}}\tilde{F}$ of
Proposition 6.5(2). Such a module is called symmetric if
$n_{\sigma}=-n_{\sigma^{*}}.$

######  Lemma 7.2.

The datum of a filtered-CM-space $V$ is equivalent to the datum of a filtered
$\varphi$-module $D$ over $\tilde{F}$ such that:

  1. (1)

$\dim_{\tilde{F}_{a}}(D)=[F:\mathbb{Q}_{p}]$;

  2. (2)

$D$ is endowed with an action of $F$;

  3. (3)

$D$ has a basis of vectors fixed by the Frobenius $\varphi$.

The equivalence goes as follows. To $V$ one associates the module $D\coloneqq
V\otimes_{\mathbb{Q}_{p}}\tilde{F_{a}}$ endowed with the Frobenius which is
the identity on $V$ and extends semi-linearly. The filtration
$\mathrm{Fil}^{i}$ on $D_{\tilde{F}}\coloneqq
V\otimes_{\mathbb{Q}_{p}}\tilde{F}$ is the sum of the eigenlines $L_{\sigma}$
such that $n_{\sigma}\geq i.$ Conversely to $D$ one associates
$V=D^{\varphi=\mathrm{Id}}$ and $n_{\sigma}=v{{}_{\mathrm{dR}}}(L_{\sigma})$.

###### Proof..

By definition of filtered-$\varphi$-module each $\mathrm{Fil}^{i}$ is stable
under the action of $F$, hence it is the sum of eigenlines. The equivalence
follows. ∎

######  Lemma 7.3.

We define the tensor product of two filtered-CM-spaces $(V,n_{\sigma})$ and
$(V^{\prime},n^{\prime}_{\sigma})$ as

$(V\otimes_{F}V^{\prime},n_{\sigma}+n^{\prime}_{\sigma}).$

Under the equivalence of Lemma 7.2 this corresponds to taking the two
corresponding filtered-$\varphi$-modules $D$ and $D^{\prime}$ and to
considering inside the filtered-$\varphi$-module $D\otimes D^{\prime}$ the
sub-filtered-$\varphi$-module where the two $F$-actions coincide.

###### Proof..

Let $\\{L_{\sigma}\\}_{\sigma\in\Gamma}$ and
$\\{L^{\prime}_{\sigma}\\}_{\sigma\in\Gamma}$ be the eigenlines associated
with the actions of $F$ on $D$ and $D^{\prime}$. Then the action of
$F\otimes_{\mathbb{Q}_{p}}F$ on $D\otimes D^{\prime}$ has
$\\{L_{\sigma}\otimes L^{\prime}_{\tau}\\}_{\sigma,\tau\in\Gamma}$ as
eigenlines. By definition of the filtration on a tensor product one has
$v{{}_{\mathrm{dR}}}(L_{\sigma}\otimes
L^{\prime}_{\tau})=n_{\sigma}+n^{\prime}_{\tau}$. On the other hand the
submodule of $D\otimes D^{\prime}$ where the two $F$-actions coincide has
$L_{\sigma}\otimes L^{\prime}_{\sigma}$ as eigenline corresponding to
$\sigma\in\Gamma$, hence we do have that $n_{\sigma}+n^{\prime}_{\sigma}$ is
its de Rham valuation. ∎

######  Definition 7.4.

The fundamental filtered-CM-space (with respect to $(F,\iota)$) is the
(symmetric) filtered-CM-space whose associated integers are

$n_{\iota}=1,\hskip 14.22636ptn_{\iota^{*}}=-1,\hskip
14.22636pt\textrm{and}\hskip 14.22636ptn_{\sigma}=0,\hskip
14.22636pt\forall\sigma\neq\iota,\iota^{*}.$

######  Proposition 7.5.

If a filtered-CM-space is symmetric (Definition 7.1) then its corresponding
filtered-$\varphi$-module is admissible.

###### Proof..

Admissibility is stable under tensor product and direct factor, hence it is
enough to study a set of tensor generators of the symmetric
filtered-$\varphi$-modules. Choose as such a set the collection
$\\{V(\tau)\\}_{\tau\in\Gamma}$ where a given $V(\tau)$ is determined by the
choice of the following list of integers

$n_{\tau}=1,\hskip 14.22636ptn_{\tau^{*}}=-1,\hskip
14.22636pt\textrm{and}\hskip 14.22636ptn_{\sigma}=0,\hskip
14.22636pt\forall\sigma\neq\tau,\tau^{*}.$

On the other hand a given $V(\tau)$ is nothing but the fundamental filtered-
CM-space where the fixed embedding $F\subset\tilde{F}$ is chosen to be $\tau$
(instead of $\iota$). Hence it is enough to show admissibility for the
fundamental filtered-CM-space (for all pairs $(F,\iota)$). By [MR1779803] it
is enough to check the condition of weakly-admissibility.

Let $V$ be the fundamental filtered-CM-space and $D$ its corresponding
filtered-$\varphi$-module. The Newton polygon is constantly zero by
construction. Hence the only sub-filtered-$\varphi$-modules which might
contradict the condition of weakly-admissibility are those containing the
eigenline $L$ of de Rham valuation one. Let $N$ be such a module and let us
show that $N=D$, which will give the admissibility. It is not restrictive to
assume that $N$ is stable through the action of $F$, by
[MR1944572]*Proposition 3.1.1.5 (or by a short direct argument). Then the
inclusion $N^{\varphi=\mathrm{Id}}\subset D^{\varphi=\mathrm{Id}}$ is
$F$-equivariant, hence $N^{\varphi=\mathrm{Id}}=0$ or
$N^{\varphi=\mathrm{Id}}=D^{\varphi=\mathrm{Id}}$. On the other hand, as $D$
is generated by its $\varphi$-invariant part, the Newton slopes of $N$ are
also zero, cf. [MR563463, §(1.3) and Lemma 1.3.4]. Now, it is well-know that
for any $\varphi$-module $Q$ we have
$\dim_{\mathbb{Q}_{p}}(Q^{\varphi=\mathrm{Id}})\leq\dim_{\tilde{F}_{a}}(Q)$.
By construction
$\dim_{\mathbb{Q}_{p}}(D^{\varphi=\mathrm{Id}})=\dim_{\tilde{F}_{a}}(D)$ and
$N$ is not zero (it contains $L$); thus by arguing on the dimension over
$\mathbb{Q}_{p}$ in the exact sequence

$0\rightarrow N^{\varphi=\mathrm{Id}}\rightarrow
D^{\varphi=\mathrm{Id}}\rightarrow(D/N)^{\varphi=\mathrm{Id}},$

we get $N^{\varphi=\mathrm{Id}}=D^{\varphi=\mathrm{Id}}$. Therefore $N=D$. ∎

######  Definition 7.6.

Let $V$ be a symmetric filtered-CM-space, $D$ be the corresponding admissible
filtered $\varphi$-module (cf. Lemma 7.2 and Proposition 7.5). Let $W$ be the
crystalline representation of $G_{\tilde{F}}$ corresponding to $D$ via
Fontaine’s $p$-adic comparison theorem. Consider the identification

$V\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris}}=D\otimes_{\tilde{F}_{a}}\mathrm{B}_{\mathrm{cris}}=W\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris}},$

and the induced one by tensoring it with $F$ over $F_{a}$,

$V\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris},F}=W\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris},F},$

where $\mathrm{B}_{\mathrm{cris},F}$ is defined in Convention 2.6(7). Then we
define the period associated with $V$ as the invertible element
$\lambda_{V}\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ deduced from Proposition
6.7 applied to $V_{1}=V,V_{2}=W$ and $B=\mathrm{B}_{\mathrm{cris},F}$. We
recall that $\lambda_{V}$ is well-determined up to a constant in $F^{\times}$
since it is defined as the period such that

$v_{1}=\lambda_{V}v_{2},$

for a given choice of a basis $v_{1}\in V$ (resp. $v_{2}\in W$) of the
$F$-line in $V\otimes_{\mathbb{Q}_{p}}F$ (resp. $W\otimes_{\mathbb{Q}_{p}}F$)
where the two $F$-actions coincide, see Proposition 6.5.

We define _the fundamental period_ (with respect to $(F,\iota)$) as the period
associated with the fundamental filtered-CM-space (Definition 7.4).

The goal of the next two sections is to prove the following theorem.

######  Theorem 7.7.

Let $\lambda$ be the fundamental period defined in Definition 7.6 above. Then

$\lambda\lambda^{*}\notin N_{F/F_{0}}(F^{\times}).$

The proof of Theorem 7.7 will be given in 9.8.

######  Corollary 7.8.

Let $V$ be a symmetric filtered-CM-space and $\lambda_{V}$ be the
corresponding period (Definition 7.6). Then $\lambda_{V}\lambda^{*}_{V}\in
F_{0}^{\times}$ belongs to $N_{F/F_{0}}(F^{\times})$ if and only if the
positive integer

$\sum\limits_{\sigma,n_{\sigma}\geq 0}n_{\sigma}$

is even, where the sum is taken over all $\sigma\in\Gamma$ such that
$n_{\sigma}\geq 0$.

###### Proof..

Recall that the group $F_{0}^{\times}/N_{F/F_{0}}(F^{\times})$ has cardinality
two (Theorem 6.4). By construction the map $V\mapsto\lambda_{V}$ from
symmetric filtered-CM-spaces to $\mathrm{B}_{\mathrm{cris},F}^{\times}$ is
multiplicative on tensor products. In particular the statement is stable under
tensor product, hence it is enough to check it on tensor generators. Arguing
as in the beginning of the proof of Proposition 7.5, it is enough to show the
statement for the fundamental filtered-CM-space, which is Theorem 7.7 above. ∎

######  Remark 7.9.

By construction, the integer $\sum_{\sigma,n_{\sigma}\geq 0}n_{\sigma}$ is the
(opposite of the) minimum of the Hodge polygon of the filtered
$\varphi$-module associated with the symmetric filtered-CM-space $V$ of
Corollary 7.8 above.

##  8\. Characterization of $p$-adic periods

In Definition 7.6 we defined a period $\lambda$ in
$\mathrm{B}_{\mathrm{cris},F}$, which we called the fundamental period. The
goal of this section is to give some properties of $\lambda$ as element of
$\mathrm{B}_{\mathrm{cris},F}$ which are enough to characterize it up to a
constant in $F^{\times}$, see Propositions 8.7 and 8.8.

We keep the same notation as in Sections 6 and 7, in particular $F$ is a
finite extension of $\mathbb{Q}_{p}$ with ramification index $e$ and residual
degree $f$. Recall that $F$ is endowed with an involution $*\colon
F\rightarrow F$, whose subfield of fixed points is denoted by $F_{0}$, which
has not to be confused with the maximal unramified subfield of $F$, denoted by
$F_{a}$. The degree of $F$ over $F_{0}$ is $2$, so
$ef=2[F_{0}:\mathbb{Q}_{p}]$.

We start this section with some preliminary constructions and lemmas,
certainly known to experts, that we recall here for the convenience of the
reader.

### 8.1.

Let us choose an embedding of $F$ in $\mathrm{B}_{\mathrm{dR}}$ and let be
$\tilde{F}$ the Galois closure of $F$ in $\mathrm{B}_{\mathrm{dR}}$; when
needed we will denote by $\iota\colon F\hookrightarrow\tilde{F}$ this
embedding. As in Sections 6 and 7 we denote by $\Gamma$ the set (pointed by
$\iota$) of $\mathbb{Q}_{p}$-embeddings of $F$ in $\tilde{F}$ ; we will denote
still by $*$ the embedding $\iota\circ*\in\Gamma$. Since
$F_{a}/\mathbb{Q}_{p}$ is a cyclic unramified extension, every $\sigma$ in
$\Gamma$ stabilizes $F_{a}$ and its restriction to $F_{a}$ is a power of the
absolute Frobenius $\varphi$ of $F_{a}$: we will set
$\sigma_{|F_{a}}=\varphi^{\nu(\sigma)}$, for a unique integer
$0\leq\nu(\sigma)\leq f-1$. By construction, we have $\nu(\iota)=0$. We put
$\nu\coloneqq\nu(*)$, it is either $0$ or $f/2$ whether $F/F_{0}$ is ramified
or unramified.

### 8.2.

We keep the notation of 8.1. We denote by $\mathrm{B}_{\mathrm{cris},F}$ the
smallest subring of $\mathrm{B}_{\mathrm{dR}}$ containing
$\mathrm{B}_{\mathrm{cris}}$ and $F$, cf. Convention 2.6(7); the ring
$\mathrm{B}_{\mathrm{cris},F}$ identifies to the image of the natural map
$\mathrm{B}_{\mathrm{cris}}\otimes_{F_{a}}F\rightarrow\mathrm{B}_{\mathrm{dR}}$.
Similarly, we will use $\mathrm{B}_{\mathrm{cris},\tilde{F}}$, which is
isomorphic to $\mathrm{B}_{\mathrm{cris}}\otimes_{\tilde{F}_{a}}\tilde{F}$,
where $\tilde{F}_{a}$ is the absolute unramified subfield of $\tilde{F}$.

We recall that the ring $\mathrm{B}_{\mathrm{cris},F}$ is endowed with a
$F$-linear endomorphism
$\varphi^{f}\colon\mathrm{B}_{\mathrm{cris},F}\rightarrow\mathrm{B}_{\mathrm{cris},F}$,
defined as
$\varphi_{\mathrm{cris}}^{f}\otimes_{\mathrm{Id}_{F_{a}}}\mathrm{Id}_{F}$,
where $\varphi_{\mathrm{cris}}$ is the Frobenius of
$\mathrm{B}_{\mathrm{cris}}$, cf. Convention 2.6(8).

Furthermore, every $\sigma$ in $\Gamma$ extends to an homomorphism, denoted by
$\sigma_{\mathrm{cris}}\colon\mathrm{B}_{\mathrm{cris},F}\rightarrow\mathrm{B}_{\mathrm{cris},\tilde{F}}$,
defined as

(8.2.1)
$\sigma_{\mathrm{cris}}\colon\mathrm{B}_{\mathrm{cris},F}=\mathrm{B}_{\mathrm{cris}}\otimes_{F_{a}}F\xrightarrow{\varphi_{\mathrm{cris}}^{\nu(\sigma)}\otimes\sigma}\mathrm{B}_{\mathrm{cris}}\otimes_{\tilde{F}_{a}}\tilde{F}=\mathrm{B}_{\mathrm{cris},\tilde{F}},$

where the product $\varphi_{\mathrm{cris}}^{\nu(\sigma)}\otimes\sigma$ is
taken over the inclusion $F_{a}\hookrightarrow{\tilde{F}_{a}}$, _cf._
[MR0274237, A.II, §3, no3, p.53]. Note that $\iota_{\mathrm{cris}}$ coincide
with the inclusion
$\mathrm{B}_{\mathrm{cris},F}\subset\mathrm{B}_{\mathrm{cris},\tilde{F}}$ as
subrings of $\mathrm{B}_{\mathrm{dR}}$. Moreover, although we will not need it
in following, it is not difficult to show that $\sigma_{\mathrm{cris}}$ is
injective, by using the fact that the natural maps
$\mathrm{B}_{\mathrm{cris}}\otimes_{F_{a}}F\rightarrow\mathrm{B}_{\mathrm{dR}}$,
$\mathrm{B}_{\mathrm{cris}}\otimes_{\tilde{F}_{a}}\tilde{F}\rightarrow\mathrm{B}_{\mathrm{dR}}$
and the Frobenius $\varphi_{\mathrm{cris}}$ are.

Finally, the involution $*$ of $F$ extends also to an endomorphism
$*_{\mathrm{cris}}\colon\mathrm{B}_{\mathrm{cris},F}\rightarrow\mathrm{B}_{\mathrm{cris},F}$
by putting
$*_{\mathrm{cris}}\coloneqq\varphi_{\mathrm{cris}}^{\nu}\otimes_{\varphi^{\nu}}*$.
Beware that in general it is not an involution of
$\mathrm{B}_{\mathrm{cris},F}$: precisely, as it follows from Lemma 8.3 below,
if $F/F_{0}$ is unramified, then
$*_{\mathrm{cris}}^{2}=\varphi_{\mathrm{cris}}^{f}$; if $F/F_{0}$ is ramified,
then $*_{\mathrm{cris}}^{2}=\mathrm{Id}_{\mathrm{B}_{\mathrm{cris},F}}$.

######  Lemma 8.3.

We keep the notation of 8.1 and 8.2.

  1. (1)

For all $\sigma$ in $\Gamma$, we have
$\sigma_{\mathrm{cris}}\circ*_{\mathrm{cris}}=(\sigma\circ*)_{\mathrm{cris}}$
except if $F/F_{0}$ is unramified and $\nu(\sigma)\geq f/2$, as in that case
$\sigma_{\mathrm{cris}}\circ*_{\mathrm{cris}}=(\sigma\circ*)_{\mathrm{cris}}\circ\varphi_{\mathrm{cris}}^{f}$.

  2. (2)

$*_{\mathrm{cris}}\circ\varphi_{\mathrm{cris}}^{f}=\varphi_{\mathrm{cris}}^{f}\circ*_{\mathrm{cris}}.$

###### Proof..

The property (2) follows directly by definition:

$*_{\mathrm{cris}}\circ\varphi_{\mathrm{cris}}^{f}=(\varphi_{\mathrm{cris}}^{\nu}\otimes*)\circ(\varphi_{\mathrm{cris}}^{f}\otimes\mathrm{Id}_{F})=\varphi_{\mathrm{cris}}^{\nu+f}\otimes*=(\varphi_{\mathrm{cris}}^{f}\otimes\mathrm{Id}_{F})\circ(\varphi_{\mathrm{cris}}^{\nu}\otimes*)=\varphi_{\mathrm{cris}}^{f}\circ*_{\mathrm{cris}}.$

Let us prove (1). We have

$\sigma_{\mathrm{cris}}\circ*_{\mathrm{cris}}=(\varphi_{\mathrm{cris}}^{\nu(\sigma)}\otimes_{\varphi^{\nu(\sigma)}}\sigma)\circ(\varphi_{\mathrm{cris}}^{\nu}\otimes_{\varphi^{\nu}}*)=\varphi_{\mathrm{cris}}^{\nu(\sigma)+\nu}\otimes_{\varphi^{\nu(\sigma)+\nu}}(\sigma\circ*).$

If $F/F_{0}$ is ramified then $\nu=0$ and $\nu(\sigma\circ*)=\nu(\sigma)$,
_cf._ 8.1, and the statement is clear. If $F/F_{0}$ is unramified then
$\nu=f/2$: if $\nu(\sigma)<f/2$, the statement is also clear. Finally, if
$F/F_{0}$ is unramified and $\nu(\sigma)\geq f/2$, we have
$\nu(\sigma)+f/2=f+\nu(\sigma\circ*)$, so

$\varphi_{\mathrm{cris}}^{\nu(\sigma)+{f}/{2}}\otimes_{\varphi^{\nu(\sigma)+{f}/{2}}}(\sigma\circ*)=\varphi_{\mathrm{cris}}^{\nu(\sigma\circ*)+f}\otimes_{\varphi^{\nu(\sigma\circ*)+f}}(\sigma\circ*)=(\sigma\circ*)_{\mathrm{cris}}\circ\varphi_{\mathrm{cris}}^{f}.\qed$

######  Convention 8.4.

When there is no risk of confusion we will write abusively $\sigma$ (resp.
$*$, resp. $\varphi$) instead of $\sigma_{\mathrm{cris}}$ (resp.
$*_{\mathrm{cris}}$, resp. $\varphi_{\mathrm{cris}}$); also, for any
$\lambda\in\mathrm{B}_{\mathrm{cris},F}$, we will sometimes write
$\lambda^{*}$ (resp. $\lambda^{*_{\mathrm{cris}}}$) instead of $*(\lambda)$
(resp. ${*_{\mathrm{cris}}}(\lambda)$).

######  Lemma 8.5 (Colmez).

We keep the notation of 8.1, 8.2 and Convention 8.4. Let
$\mu\in\mathrm{B}_{\mathrm{cris},F}$ be an element such that:

  1. (1)

for all $\sigma\in\Gamma$, $v_{\mathrm{dR}}(\sigma(\mu))=0$;

  2. (2)

$\varphi^{f}(\mu)=\mu$.

Then $\mu$ belongs to $F^{\times}\subset\mathrm{B}_{\mathrm{cris},F}$.

###### Proof..

This follows from the fundamental exact sequence [MR1956055]*Lemma 9.25
(SEF3E). We briefly recall the argument. By construction
$\mathrm{B}_{\mathrm{cris}}$ is the localization
$\mathrm{A}_{\mathrm{cris}}[{t}^{-1}]$ of Fontaine’s ring
$\mathrm{A}_{\mathrm{cris}}$, where $t$ is the period of $\mathbb{Q}_{p}(1)$,
see Convention 2.6(6). Thus there exists some integer $n\geq 0$, such that
$\mu
t^{n}\in\mathrm{B}_{\mathrm{cris},F}^{+}\coloneqq\mathrm{A}_{\mathrm{cris}}\otimes_{W(k_{F})}F$.
By evaluating the sequence of [MR1956055, Lemma 9.16] in $\mathbb{C}_{p}$ and
by multiplying with $t^{n-1}$ we get a short exact sequence of
$\mathbb{Q}_{p}$-vector spaces

(8.5.1) $0\rightarrow F\cdot
t^{n}\rightarrow(B_{F})^{\varphi^{f}=p^{nf}}\xrightarrow{\,\Theta_{F}\,}\bigoplus_{\Gamma}\mathbb{C}_{p}\rightarrow
0,$

where $B_{F}\subset\mathrm{B}_{\mathrm{dR}}^{+}$ is some ring of periods
containing $\mathrm{B}_{\mathrm{cris},F}^{+}$, cf. [MR1956055]*§8.5222The ring
$B_{F}$ is the ring $\mathrm{B}_{\mathrm{max},F}^{+}$ in the notation of
loc.cit.; and the map $\Theta_{F}$ is given by

$\Theta_{F}\colon
x\longmapsto\Bigl{(}\theta\bigl{(}\sigma\bigl{(}{x}{t^{-n+1}}\bigr{)}\bigr{)}\Bigr{)}_{\sigma\in\Gamma}\,,$

where $\theta\colon\mathrm{B}_{\mathrm{dR}}^{+}\rightarrow\mathbb{C}_{p}$
denotes the reduction map and $\sigma\colon
B_{F}\rightarrow\mathrm{B}_{\mathrm{dR}}^{+}$ extends the homomorphism
$\sigma_{\mathrm{cris}}$ of (8.2.1). By the hypothesis (2) of the statement,
the element $x=\mu t^{n}$ belongs to
$(\mathrm{B}_{\mathrm{cris},F}^{+})^{\varphi^{f}=p^{nf}}$; we get
$\Theta_{F}(x)=(\theta(\sigma(\mu t)))_{\sigma\in\Gamma}$. By the hypothesis
(1) of the statement we have $v_{\mathrm{dR}}(\sigma(\mu))=0$ and since
$\sigma{{}_{\mathrm{cris}}}(t)=\varphi_{\mathrm{cris}}^{\nu(\sigma)}(t)=p^{\nu(\sigma)}t$,
we get $\Theta_{F}(x)=0$. Hence $\mu\in F$; moreover, $\mu$ is nonzero
otherwise the hypothesis (1) above would fail. ∎

######  Definition 8.6.

An element $\beta$ in $\mathrm{B}_{\mathrm{cris},F}$ is called a _fundamental
period_ if it satisfies the following properties:

  1. a)

$v_{\mathrm{dR}}(\beta)=1$;

  2. b)

$v_{\mathrm{dR}}(\beta^{*})=-1$;

  3. c)

for all $\sigma\in\Gamma\backslash\\{\iota,*\\}$,
$v_{\mathrm{dR}}(\sigma(\beta))=0$;

  4. d)

$\varphi^{f}(\beta)=\beta$;

where $\sigma(\beta)$, $\varphi^{f}$, and
$\beta^{*}\in\mathrm{B}_{\mathrm{cris},\tilde{F}}\subset\mathrm{B}_{\mathrm{dR}}$
are defined in 8.2 and Convention 8.4.

We denote by $\mathcal{P}\subset\mathrm{B}_{\mathrm{cris},F}$ the set of
fundamental periods.

######  Proposition 8.7.

The fundamental period $\lambda\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ of
Definition 7.6 is a fundamental period according to Definition 8.6.

###### Proof..

Let us put ourselves in the setting of Definition 7.6 and Proposition 6.7. We
have $v_{1}\in V_{F}\coloneqq V\otimes_{\mathbb{Q}_{p}}F$, where $V$ is the
fundamental filtered-CM-space, and $v_{2}\in W_{F}\coloneqq
W\otimes_{\mathbb{Q}_{p}}F$, where $W$ is the Galois representation associated
with the admissible filtered $\varphi$-module $D\coloneqq
V\otimes_{\mathbb{Q}_{p}}\tilde{F}_{a}$.

By applying $\sigma$ to the equality $v_{1}=\lambda v_{2}$, we get the
relation $\sigma(v_{1})=\sigma(\lambda)\sigma(v_{2})$ in

$V\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris},\tilde{F}}=W\otimes_{\mathbb{Q}_{p}}\mathrm{B}_{\mathrm{cris},\tilde{F}}\,.$

Now, by construction we know that:

  * -

for all $\sigma\in\Gamma$, $v_{\mathrm{dR}}(\sigma(v_{2}))=0$;

  * -

$v_{\mathrm{dR}}(v_{1})=1$ and $v_{\mathrm{dR}}(v_{1}^{*})=-1$;

  * -

for all $\sigma\in\Gamma\backslash\\{\iota,*\\}$,
$v_{\mathrm{dR}}(\sigma(v_{1}))=0$.

Therefore, we get:

  * -

for $\sigma=\iota$, $v_{1}=\lambda v_{2}$, so $v_{\mathrm{dR}}(\lambda)=1$;

  * -

for $\sigma=*$, $v_{1}^{*}=\lambda^{*}v_{2}^{*}$, so
$v_{\mathrm{dR}}(\lambda^{*})=-1$;

  * -

for $\sigma\in\Gamma\backslash\\{\iota,*\\}$,
$\sigma(v_{1})=\sigma(\lambda)\sigma(v_{2})$, so
$v_{\mathrm{dR}}(\sigma(\lambda))=0$.

Finally, by applying $\varphi^{f}$ to $v_{1}=\lambda v_{2}$, since by
construction both $v_{1}$ and $v_{2}$ are fixed by $\varphi^{f}$, we get

$v_{1}=\varphi_{D}^{f}(v_{1})=\varphi^{f}(\lambda)\varphi^{f}(v_{2})=\varphi^{f}(\lambda)v_{2},$

hence $\varphi^{f}(\lambda)=\lambda$. ∎

######  Proposition 8.8 (Uniqueness of fundamental periods).

Let $\lambda\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ be as in Proposition 8.7
and $\mathcal{P}$ the set of fundamental periods of Definition 8.6, then

$\mathcal{P}=F^{\times}\cdot\lambda.$

In particular all fundamental period are invertible.

###### Proof..

Since $\lambda$ is invertible in $\mathrm{B}_{\mathrm{cris},F}$, we can
consider the subset $\lambda^{-1}\mathcal{P}$ of
$\mathrm{B}_{\mathrm{cris},F}$. We have to prove
$\lambda^{-1}\mathcal{P}=F^{\times}$, which is exactly the statement of Lemma
8.5.∎

######  Proposition 8.9.

Let $\lambda\in\mathrm{B}_{\mathrm{cris},F}$ be a fundamental period. Then
$\lambda\lambda^{*}\in F_{0}^{\times}.$

######  Remark 8.10.

For the fundamental period $\lambda$ defined in Definition 7.6, the statement
of Proposition 8.9 follows directly from Proposition 6.8. We give below an
alternative proof based on the $p$-adic properties of the period which does
not use quadratic forms.

###### Proof..

In this proof in order to avoid any possible confusion we will write
$\lambda^{*_{\mathrm{cris}}}$ instead of $\lambda^{*}$. First, let us show
that the period
$\lambda\lambda^{*_{\mathrm{cris}}}\in\mathrm{B}_{\mathrm{cris},F}$ satisfies
the hypothesis of Lemma 8.5. For any $\sigma$ in $\Gamma$ we have

$\sigma_{\mathrm{cris}}(\lambda\lambda^{*_{\mathrm{cris}}})=\sigma_{\mathrm{cris}}(\lambda)\cdot(\sigma_{\mathrm{cris}}\circ{*_{\mathrm{cris}}})(\lambda)$

and by Lemma 8.3(1),
$(\sigma_{\mathrm{cris}}\circ{*_{\mathrm{cris}}})(\lambda)$ is equal either to
$(\sigma\circ{*})_{\mathrm{cris}}(\lambda)$ or to
$(\sigma\circ{*})_{\mathrm{cris}}(\varphi_{\mathrm{cris}}^{f}(\lambda))$.
Since $\varphi_{\mathrm{cris}}^{f}(\lambda)=\lambda$ by property d of
Definition 8.6, we get

$\sigma_{\mathrm{cris}}(\lambda\lambda^{*_{\mathrm{cris}}})=\sigma_{\mathrm{cris}}(\lambda)\cdot(\sigma\circ{*})_{\mathrm{cris}}(\lambda).$

Applying $v_{\mathrm{dR}}$ and considering the properties a, b and c of
Definition 8.6 for $\lambda$, we get that $\lambda\lambda^{*_{\mathrm{cris}}}$
verifies the property (1) of Lemma 8.5. Let us check the property (2) of Lemma
8.5:

$\varphi_{\mathrm{cris}}^{f}(\lambda\lambda^{*_{\mathrm{cris}}})=\varphi_{\mathrm{cris}}^{f}(\lambda)\varphi_{\mathrm{cris}}^{f}(\lambda^{*_{\mathrm{cris}}})=\lambda\lambda^{*_{\mathrm{cris}}},$

because $\varphi_{\mathrm{cris}}^{f}$ and $*_{\mathrm{cris}}$ commute, _cf._
Lemma 8.3(2), and $\varphi_{\mathrm{cris}}^{f}(\lambda)=\lambda$. Therefore,
by Lemma 8.5, the period
$\lambda\lambda^{*_{\mathrm{cris}}}\in\mathrm{B}_{\mathrm{cris},F}$ belongs to
$F\subset\mathrm{B}_{\mathrm{cris},F}$ . Applying $*$ we get

$(\lambda\lambda^{*_{\mathrm{cris}}})^{*}=\lambda^{*_{\mathrm{cris}}}\lambda^{*_{\mathrm{cris}}^{2}}=\lambda^{*_{\mathrm{cris}}}\varphi_{\mathrm{cris}}^{2\nu}(\lambda)=\lambda^{*_{\mathrm{cris}}}\lambda,$

where $\nu=0$ or $f/2$, _cf._ 8.1 and 8.2; and we used again
$\varphi_{\mathrm{cris}}^{f}(\lambda)=\lambda$. Being stable under $*$, the
element $\lambda\lambda^{*_{\mathrm{cris}}}\in F$ belongs actually to $F_{0}$.
It is different from zero since $\mathrm{B}_{\mathrm{cris},F}$ is an integral
domain, and neither $\lambda$ nor $\lambda^{*_{\mathrm{cris}}}$ can be zero by
the properties a and b of Definition 8.6. ∎

##  9\. Lubin–Tate periods and local reciprocity

The goal of this section is the proof of Theorem 7.7, which is given in 9.8.
Before that we need to rely the fundamental period $\lambda$ of Definition 7.6
with Lubin–Tate periods, _cf._ Proposition 9.5 and Corollary 9.7.

We keep the notation of Section 8 and specifically we refer to 8.1, 8.2 and
Convention 8.4. In particular, recall that $F$ is a $p$-adic field endowed
with non-trivial involution $*$ and we denoted by $F_{0}$ the subfield of $F$
of points fixed by $*$, whereas we denoted by $F_{a}$ the absolute unramified
subfield of $F$.

######  Definition 9.1.

Let $\pi$ be a uniformizer of $F$. We say that $\alpha$ in
$\mathrm{B}_{\mathrm{cris},F}$ is a Lubin–Tate period (relative to $\pi\in F$)
if:

  1. (1)

$v_{\mathrm{dR}}(\alpha)=1$;

  2. (2)

for all $\sigma\in\Gamma\backslash\\{\iota\\}$,
$v_{\mathrm{dR}}(\sigma(\alpha))=0$;

  3. (3)

$\varphi^{f}(\alpha)=\pi\alpha$;

  4. (4)

$\alpha$ is invertible in $\mathrm{B}_{\mathrm{cris},F}$.

######  Theorem 9.2 (Colmez).

There exist Lubin–Tate periods relative to any choice of a uniformizer $\pi$
of $F$.

######  Remark 9.3.

  1. (1)

Lubin–Tate periods are constructed in [MR1956055, §9.3-§9.5]333Precisely, cf.
loc.cit. Proposition 9.10 and Lemma 9.18. Note that these periods are
constructed in $B_{\mathrm{max},F}$, but actually they belong to
$\mathrm{B}_{\mathrm{cris},F}$, since, by Definition 9.1(3) above, they belong
to $\varphi^{f}(B_{\mathrm{max},F})\subset\mathrm{B}_{\mathrm{cris},F}$., via
a direct computation in periods rings using Lubin–Tate formal groups
[MR172878], see also [MR1247996]*§2 [fourquaux_these]*§3.6. Therefore the
action of $G_{F}$ on $\alpha\in\mathrm{B}_{\mathrm{cris},F}$ is given
precisely by multiplication with the Lubin–Tate character. We will not need
this property, so we did not mention it in Definition 9.1 above.

  2. (2)

To keep this article more self-contained, we recall in Appendix A another
construction of Lubin–Tate periods, relying on the theorem “weakly admissible
$\Rightarrow$ admissible”, _cf._ [MR1779803], applied to a well-chosen
filtered $\varphi$-module over $F$, see Definition A.4.

### 9.4.

Let us denote by $\widehat{F}^{\mathrm{nr}}$ (resp.
$\widehat{F}_{a}^{\mathrm{nr}}$) the completion of maximal unramified
extension of $F$ (resp. $F_{a}$) and by
$\mathrm{Gal}({\widehat{F}^{\mathrm{nr}}}/{F})$ the group of continuous
$F$-automorphisms of $\widehat{F}^{\mathrm{nr}}$. We have
$\widehat{F}^{\mathrm{nr}}=\widehat{F}_{a}^{\mathrm{nr}}\otimes_{F_{a}}F$,
that we embed canonically in $\mathrm{B}_{\mathrm{cris},F}$. By definition,
the endomorphism $\varphi_{\mathrm{cris}}^{f}$ of
$\mathrm{B}_{\mathrm{cris},F}$, restricted to $\widehat{F}^{\mathrm{nr}}$, is
the arithmetic Frobenius $\varphi^{f}$ in
$\mathrm{Gal}({\widehat{F}^{\mathrm{nr}}}/{F})$, _i.e._ the unique isomorphism
of $\widehat{F}^{\mathrm{nr}}$ lifting the map $x\mapsto x^{q}$, $q=p^{f}$, on
the residue field.

######  Proposition 9.5 (Explicit construction of the fundamental period).

Assume that $\alpha\in\mathrm{B}_{\mathrm{cris},F}$ is a Lubin–Tate period,
for some uniformizer $\pi\in F$. Then there exists
$c\in(\widehat{F}^{\mathrm{nr}})^{\times}$ such that
$c\frac{\alpha}{\alpha^{*}}$ is a fundamental period, see Definition 8.6.
Moreover $c$ can be chosen as follow:

  * -

if $F/F_{0}$ is unramified, and $\pi$ belongs to $F_{0}$, then $c=1$;

  * -

if $F/F_{0}$ is (totally) ramified, then $c$ verifies

(9.5.1) $\varphi^{f}(c)=\frac{\pi^{*}}{\pi}c,$

where $\varphi^{f}\in\mathrm{Gal}({\widehat{F}^{\mathrm{nr}}}/{F})$ is the
arithmetic Frobenius, see 9.4 above.

###### Proof..

The Lubin–Tate period $\alpha$ is invertible in
$\mathrm{B}_{\mathrm{cris},F}$, so is ${*_{\mathrm{cris}}}(\alpha)$. Consider
the period

$\frac{\alpha}{\alpha^{*}}\coloneqq{\alpha}\cdot{{*_{\mathrm{cris}}}(\alpha)}^{-1}\in\mathrm{B}_{\mathrm{cris},F}.$

First let us show that $\frac{\alpha}{\alpha^{*}}$ satisfies the properties a,
b and c of Definition 8.6; then, we will multiply it by a well chosen constant
$c$ in $\widehat{F}^{\mathrm{nr}}$ in order to get the property d of 8.6.
Since, for every $\sigma$ in $\Gamma$,
$\sigma_{\mathrm{cris}}(\widehat{F}^{\mathrm{nr}})=\widehat{F}^{\mathrm{nr}}\subset\mathrm{B}_{\mathrm{cris},\tilde{F}}\cap\mathrm{B}_{\mathrm{dR}}^{+}$,
normalizing the period by a nonzero constant in $\widehat{F}^{\mathrm{nr}}$
will not change the first three properties.

As $\alpha$ is a Lubin–Tate period, by using Definition 9.1, we get the
following.

  1. a)

$v_{\mathrm{dR}}\bigl{(}\tfrac{\alpha}{\alpha^{*}}\bigr{)}=v_{\mathrm{dR}}(\alpha)-v_{\mathrm{dR}}(*_{\mathrm{cris}}({\alpha}))=1-0=1.$

  2. b)

$v_{\mathrm{dR}}\bigl{(}*_{\mathrm{cris}}\bigl{(}\tfrac{\alpha}{\alpha^{*}}\bigr{)}\bigr{)}=v_{\mathrm{dR}}(*_{\mathrm{cris}}({\alpha}))-v_{\mathrm{dR}}(*_{\mathrm{cris}}^{2}({\alpha}))=-v_{\mathrm{dR}}(*_{\mathrm{cris}}^{2}({\alpha})).$

Since $*_{\mathrm{cris}}^{2}$ is either
$\mathrm{Id}_{\mathrm{B}_{\mathrm{cris},F}}$ or $\varphi_{\mathrm{cris}}^{f}$,
cf. 8.2, and $\varphi_{\mathrm{cris}}^{f}(\alpha)=\pi\alpha$, we get
$-v_{\mathrm{dR}}(*_{\mathrm{cris}}^{2}({\alpha}))=-1$.

  3. c)

For every $\sigma$ in $\Gamma\backslash\\{\iota,*\\}$,

$v_{\mathrm{dR}}\bigl{(}\sigma_{\mathrm{cris}}\bigl{(}\tfrac{\alpha}{\alpha^{*}}\bigr{)}\bigr{)}=v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}({\alpha}))-v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(*_{\mathrm{cris}}({\alpha})))=-v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(*_{\mathrm{cris}}({\alpha}))),$

as $\sigma\not=\iota$. By using Lemma 8.3(1), the period
$\sigma_{\mathrm{cris}}(*_{\mathrm{cris}}({\alpha}))$ is either
$(\sigma\circ*)_{\mathrm{cris}}({\alpha})$ or
$(\sigma\circ*)_{\mathrm{cris}}(\varphi_{\mathrm{circ}}^{f}({\alpha}))=(\sigma\circ*)_{\mathrm{cris}}(\pi\alpha)=\sigma(\pi^{*})(\sigma\circ*)_{\mathrm{cris}}(\alpha)$.
As
$\sigma(\pi^{*})\in\tilde{F}\subset\mathrm{B}_{\mathrm{cris},\tilde{F}}\cap\mathrm{B}_{\mathrm{dR}}^{+}$
and $\sigma\circ*\not=\iota$, we have, in both cases,
$v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(*_{\mathrm{cris}}({\alpha})))=0$ by
Definition 9.1(3).

Finally we need to check the property d of Definition 8.6. As ${\pi^{*}}$ and
${\pi}$ have the same valuation (one), we have
${\pi^{*}}{\pi}^{-1}\in\mathcal{O}_{F}^{\times}$. In particular the equation

$\frac{\varphi^{f}(c)}{c}=\frac{\pi^{*}}{\pi}$

has a solution $c\in(\widehat{F}^{\mathrm{nr}})^{\times}$, see for example
[MR1697859, Chap. V, §2, Lemma (2.1)]. Note that if $F/F_{0}$ is unramified
and $\pi$ is in $F_{0}$, we can take $c=1$. We finish by computing

$\varphi_{\mathrm{cris}}^{f}\left(c\frac{\alpha}{\alpha^{*_{\mathrm{cris}}}}\right)=\varphi^{f}(c)\frac{\varphi_{\mathrm{cris}}^{f}(\alpha)}{\varphi_{\mathrm{cris}}^{f}(*_{\mathrm{cris}}(\alpha))}=c\frac{\pi^{*}}{\pi}\frac{\varphi_{\mathrm{cris}}^{f}(\alpha)}{*_{\mathrm{cris}}(\varphi_{\mathrm{cris}}^{f}(\alpha))}=c\frac{\pi^{*}}{\pi}\frac{\pi\alpha}{(\pi\alpha)^{*_{\mathrm{cris}}}}=c\frac{\alpha}{\alpha^{*_{\mathrm{cris}}}},$

where we used $\varphi_{\mathrm{cris}}^{f}(\alpha)=\pi\alpha$ and Lemma
8.3(2). ∎

######  Remark 9.6.

If $F/F_{0}$ is tamely ramified, we can choose a uniformizer $\pi\in F$, such
that $\pi^{*}=-\pi$ and the constant $c$ can be chosen in a quadratic
unramified extension of $F$, such that $\varphi^{f}(c)=-c$. In general, the
constant $c$ is algebraic over $F$ if and only if its orbit under
$\mathrm{Gal}({\widehat{F}^{\mathrm{nr}}}/{F})$ is finite. Since $\varphi^{f}$
is a topological generator, the relation (9.5.1) implies that $c$ is algebraic
if and only if $\pi^{*}\pi^{-1}$ is a root of unity. When the break of the
ramification filtration of $\mathrm{Gal}({F}/{F_{0}})$ is large, it may happen
that for any choice of a uniformizer $\pi$ in $F$, $\pi^{*}\pi^{-1}$ is never
a root of unity (e.g. $F=\mathbb{Q}_{2}(\zeta_{8})$ and
$F_{0}=\mathbb{Q}_{2}(\zeta_{4})$, where $\zeta_{8}$ is a primitive 8th-root
of unity and $\zeta_{8}^{2}=\zeta_{4}$).

######  Corollary 9.7.

Let $\lambda$ be a fundamental period in $\mathrm{B}_{\mathrm{cris},F}$, then

$\lambda\in F^{\times}\cdot c\frac{\alpha}{\alpha^{*}},$

with $c$ and $\alpha$ as in Proposition 9.5.

###### Proof..

As $c\not=0$ and $\frac{\alpha}{\alpha^{*}}$ is invertible in
$\mathrm{B}_{\mathrm{cris},F}$, this is an immediate consequence of
Propositions 8.8 and 9.5. ∎

###  9.8. Proof of Theorem 7.7

Let $\lambda\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ be the fundamental
period defined in Definition 7.6. By Proposition 8.7 it is a fundamental
period in the sense of Definition 8.6. By Proposition 8.9 (or by Proposition
6.8), the period $\lambda\lambda^{*}$ belongs to $F_{0}^{\times}$: we have to
prove that $\lambda\lambda^{*}$ does not belong to $N_{F/F_{0}}(F^{\times})$,
where $*=*_{\mathrm{cris}}$ is the endomorphism of
$\mathrm{B}_{\mathrm{cris},F}$ defined in 8.2.

By Corollary 9.7, we have $\lambda=bc\tfrac{\alpha}{\alpha^{*}}$, for some
$b\in F^{\times}$, a Lubin–Tate period
$\alpha\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ and
$c\in\widehat{F}^{\mathrm{nr}}$ as in Proposition 9.5. Considering that
$bb^{*}=N_{F/F_{0}}(b)\in F_{0}^{\times}$ is a norm we can suppose $b=1$.
Following Proposition 9.5, we need to treat separately two cases, depending on
whether the extension $F/F_{0}$ is unramified or ramified.

  1. (1)

Assume that the extension $F/F_{0}$ is unramified. We have
$*_{\mathrm{cris}}^{2}=\varphi^{f}$, _cf._ 8.2. Choose $\pi$ in $F_{0}$ so
that we can have $c=1$, _cf._ 9.5. We get

$\lambda\lambda^{*}=\frac{\alpha}{(\alpha^{*})}\frac{\alpha^{*}}{(\alpha^{*})^{*}}=\frac{\alpha}{(\alpha^{*})^{*}}=\frac{\alpha}{\varphi^{f}(\alpha)}=\frac{\alpha}{\pi\alpha}=\frac{1}{\pi}.$

Since $F/F_{0}$ is unramified every norm has even valuation in $F_{0}$, hence
$\lambda\lambda^{*}$ cannot be a norm.

  2. (2)

Assume that the extension $F/F_{0}$ is (totally) ramified. In particular, the
endomorphism $*_{\mathrm{cris}}$ of $\mathrm{B}_{\mathrm{cris},F}$ is an
involution, _cf._ 8.2. We get

$\lambda\lambda^{*}=c\frac{\alpha}{(\alpha^{*})}c^{*}\frac{\alpha^{*}}{(\alpha^{*})^{*}}=cc^{*}=N_{\widehat{F}^{\mathrm{nr}}/\widehat{F}_{0}^{\mathrm{nr}}}(c),$

where the last equality follows given that the restriction of
$*_{\mathrm{cris}}$ to
$\widehat{F}^{\mathrm{nr}}=\widehat{F}_{a}^{\mathrm{nr}}\otimes_{F_{a}}F\subset\mathrm{B}_{\mathrm{cris},F}$
is the unique non-trivial element in
$\mathrm{Gal}({\widehat{F}^{\mathrm{nr}}}/{\widehat{F}_{0}^{\mathrm{nr}}})$.
Let

$(-,F/F_{0})\colon F_{0}^{\times}\rightarrow\mathrm{Gal}({F}/{F_{0}})$

be the local reciprocity map. The equation
$\varphi^{f}(c)c^{-1}=\pi^{*}\pi^{-1}$ from (9.5.1) fits exactly the
hypothesis of a theorem of Dwork, _cf._ [MR0354618, Ch. XIII, §5, Cor. of Th.
2]; whence we get

$\bigl{(}N_{\widehat{F}^{\mathrm{nr}}/\widehat{F}_{0}^{\mathrm{nr}}}(c),F/F_{0}\bigr{)}=*^{-1}=*.$

Since $*\not=\mathrm{Id}_{F}$, that exactly means that
$N_{\widehat{F}^{\mathrm{nr}}/\widehat{F}_{0}^{\mathrm{nr}}}(c)\in
F_{0}^{\times}$ is not a norm, _i.e._ it does not belong to
$N_{F/F_{0}}(F^{\times})$, and that finishes the proof. $\square$

######  Remark 9.9.

The proof above, as all the statements of this section does not use that
$\lambda=\lambda_{V}$ is coming from a CM-quadratic space $V$, _cf._
Definition 6.1, but only the properties of Definition 8.6. (See also Remark
8.10).

##  10\. Proof of the main theorem

We now put all the ingredients together and prove Theorem 3.2.

### Proof of Theorem 3.2

By Proposition 4.5, the result is reduced to a $p$-adic statement. In
particular, following Proposition 4.7, it is enough to study the $p$-adic
étale realization and the Frobenius invariant part of the crystalline
realization of the motive $M$. By this same proposition, this pair of
quadratic spaces is an orthogonal supersingular pair, in the language of
Definition 5.1. We can reformulate Proposition 4.5 using Definition 5.4: we
have to show that our orthogonal superisingular pair is good (see also Remark
4.6).

These two quadratic spaces are endowed with the action of
$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$, where $F$ is the number field given by
hypothesis. By Proposition 5.6 we can use the action of
$F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$ to decompose the two quadratic spaces
and study them separately. More precisely it is enough to study the case where
the two quadratic spaces are endowed with an action of a $p$-adic field (which
is a factor of $F\otimes_{\mathbb{Q}}\mathbb{Q}_{p}$) and is stable by the
involution $*$ induced by $F$. By abuse of notation we will denote such a
$p$-adic field again by $F$.

We have now an orthogonal supersingular pair endowed with an action of a
$p$-adic field. Both quadratic space are now CM-quadratic spaces with respect
to $F$ in the sense of Definition 6.1. We can now apply Proposition 6.7 to our
two quadratic spaces and to the ring $B=\mathrm{B}_{\mathrm{cris},F}$. This
constructs a period $\lambda\in B^{\times}$, well defined up to multiplication
by an element of $F^{\times}$.

By Proposition 6.8 applied to the endomorphism $\tilde{*}=*_{\mathrm{cris}}$
defined in 8.2, the two orthogonal spaces are isomorphic if and only if
$\lambda\cdot(\tilde{*}(\lambda))$ lies in the group of norms
$N_{F/F_{0}}(F^{\times}).$ By Corollary 7.8, this norm condition is equivalent
to the fact that the minimum of the Hodge polygon of the underlying filtered
$\varphi$-module is an even number (see also Remark 7.9). Altogether we have
that the two orthogonal spaces are isomorphic if and only if the minimum of
the Hodge polygon is even. By Lemma 5.5 this means precisely that the pair is
good. $\square$

######  Remark 10.1.

Note that Corollary 7.8 is a consequence of Theorem 7.7 whose proof is the
object of Sections 8 and 9.

##  Appendix A Lubin–Tate filtered $\varphi$-modules

Lubin–Tate periods were constructed by Colmez in [MR1956055], via a direct
construction based on Lubin–Tate’s formal group law [MR172878]. The goal of
this appendix is to present to the reader an alternative construction of these
periods as self contained as possible. Beyond [MR1956055], there is a vast
literature for the $p$-adic representations associated with Lubin–Tate groups
and their periods, cf. for example [MR1247996, fourquaux_these, MR2565906,
MR4068300].

Let $\pi$ be a uniformizer of a $p$-adic field $F$. The plan is the following
:

  1. (1)

Describe concretely a filtered $\varphi$-module
$D=(D,\varphi_{\pi},\mathrm{Fil}^{\bullet}(D_{F}))$ over $F$ using only semi-
linear algebra data, see Definitions A.1 and A.4.

  2. (2)

Show that $D$ is weakly-admissible, see Proposition A.5; hence admissible by
[MR1779803].

  3. (3)

Show that a Lubin–Tate period as in Definition 9.1 appears as period
$\alpha=\alpha_{\pi}$ of $D$, see (A.8.1) and Proposition A.11.

  4. (4)

Rely $\alpha$ to Colmez’ original construction, hence $D$ to the Galois
representation given by the Lubin–Tate character, see Proposition A.13.

In this appendix we do not assume anymore that $F$ is endowed with a non-
trivial involution.

Keep the notation of Conventions 2.1 and 2.6. In particular recall that we
denoted by $F_{a}$ the absolute unramified subfield of $F$, which has degree
$f$ over $\mathbb{Q}_{p}$. Let $E_{\pi}(x)\in F_{a}[x]$ be the minimal (monic)
polynomial of $\pi$ over $F_{a}$, which is an Eisenstein polynomial of degree
$e$.

######  Definition A.1 (The $\varphi$-module).

Let us define a filtered $\varphi$-module: we set

$D\coloneqq{(F_{a})}^{ef},$

as $F_{a}$-vector space and we endow it with the semi-linear Frobenius
$\varphi_{\pi}\colon D\rightarrow D$ given, in the canonical basis
$\mathscr{C}\coloneqq(\mathfrak{e}_{1},\ldots,\mathfrak{e}_{ef})$, by the bloc
matrix

(A.1.1) $A\coloneqq\left(\begin{array}[]{c|c}0&C\\\ \hline\cr
I_{e(f-1)}&0\end{array}\right)\in\mathrm{M}_{ef}(F_{a}),$

where $I_{e(f-1)}\in M_{e(f-1)}(F_{a})$ is the identity matrix, $0$ stands for
the zero (rectangular) blocs, and $C\in M_{e}(F_{a})$ is the companion matrix
of the minimal polynomial $E_{\pi}(x)$ of $\pi$.

######  Lemma A.2.

  1. (1)

The $\varphi$-module $(D,\varphi_{\pi})$ is endowed with an $F$-action, i.e. a
linear action of $F$ on the $F_{a}$-vector space $D$ which commutes with
$\varphi_{\pi}$.

  2. (2)

The action of $\pi\in F$ is equal to the action of $\varphi_{\pi}^{f}$ on $D$.

###### Proof.

Let us define an $F$-action on the $\varphi$-module $D$. For any square matrix
$M\in M_{e}(F_{a})$, we denote by $\varphi_{F_{a}}(M)$ the matrix on which the
Frobenius $\varphi_{F_{a}}$ is applied to all entries. We then define the
diagonal bloc matrix

(A.2.1)
$\widetilde{M}\coloneqq\mathrm{Diag}(M,\varphi_{F_{a}}(M),\ldots,\varphi_{F_{a}}^{f-1}(M))\in\mathrm{M}_{ef}(F_{a}),$

where the diagonal blocs are the matrices $\varphi_{F_{a}}^{i}(M)$, for $i$
going from $0$ to $f-1$. The $F_{a}$-linear map $D\rightarrow D$ given by
$\widetilde{M}$ in the canonical basis $\mathscr{C}$ of $D$ commutes with the
(semi-linear) Frobenius $\varphi_{\pi}$ if and only if

$\widetilde{M}A=A\varphi_{F_{a}}(\widetilde{M}),$

by Definition A.1. It is elementary to check that this holds exactly when $M$
commutes with the matrix $C$ from (A.1.1). Since $C$ is cyclic, its commutator
in $M_{e}(F_{a})$ is precisely $F_{a}[C]\cong F$. This gives a $F_{a}$-linear
$F$-action on the $\varphi$-module $D$ and proves the statement (1) above.

We can now prove (2). First, note that by construction the action $\pi\in F$
is given by the matrix $\widetilde{C}$ from (A.2.1), since $C$ is the
companion matrix of the minimal polynomial of $\pi$.

Now, since $\varphi_{\pi}$ is semilinear, its $f$ power $\varphi_{\pi}^{f}$ is
$F_{a}$-linear and it is given in the canonical basis by the (twisted) product

(A.2.2) $A\cdot\varphi_{F_{a}}(A)\cdots\varphi_{F_{a}}^{f-1}(A).$

An elementary computation shows that the product (A.2.2) is equal to the
diagonal bloc matrix

(A.2.3)
$\mathrm{Diag}(C,\varphi_{F_{a}}(C),\ldots,\varphi_{F_{a}}^{f-1}(C))\in\mathrm{M}_{ef}(F_{a}),$

which is the matrix $\widetilde{C}$ from (A.2.1). As remarked above this shows
exactly that $\varphi_{\pi}^{f}$ acts as $\pi$. ∎

######  Lemma A.3.

  1. (1)

We have two field actions on $D$: the (left) $F$-action defined in Lemma
$\mathrm{\ref{lemma:F-action}}\eqref{lemma:F-action_1}$, which does commute
with the Frobenius; and another one of $F_{a}$ via its original structure of
$F_{a}$-vector space, which does not commute with the Frobenius. To
distinguish them we denote the latter as a right action as we did in
Proposition 6.5.

  2. (2)

Those actions coincide on $\mathbb{Q}_{p}\subset F$, and $D$ is a free
$F\otimes_{\mathbb{Q}_{p}}F_{a}$-module of rank one.

  3. (3)

Set

$D_{F}\coloneqq D\otimes_{F_{a}}F.$

It is a free $F\otimes_{\mathbb{Q}_{p}}F$-module of rank one. In particular
there is a unique $F$-line

$W\subset D_{F},$

where the right $F$-structure of $D_{F}$ coincide with the left one.

###### Proof.

The first statement is true by construction and it is also clear that the two
actions coincide on $\mathbb{Q}_{p}\subset F_{a}$, thus $D$ is
$F\otimes_{\mathbb{Q}_{p}}F_{a}$-module. Let us show that it is free of rank
one. Indeed the ring $R\coloneqq F\otimes_{\mathbb{Q}_{p}}F_{a}$ is a product
of fields, so the module $D$ splits as a product of vector spaces on those
fields. We claim that those vector spaces have all the same dimension, thus
$D$ is free as $R$-module by [MR1727221, AC.II, §5.3, Proposition 5, p.113];
furthermore, since $[F:\mathbb{Q}_{p}]=\dim_{F_{a}}(D)$, the rank of $D$ must
be one. Let us prove the claim. It follows from the fact that $D$ is endowed
with a Frobenius: if we twist the action of $F_{a}$ on $D$ by the absolute
Frobenius of $F_{a}$, then the Frobenius $\varphi_{\pi}$ of $D$ induces an
$R$-linear isomorphism $\Phi_{\pi}\colon\varphi^{*}D\coloneqq
D\otimes_{\varphi}F_{a}\rightarrow D$, mapping $m\otimes a$ to
$\varphi_{\pi}(m)\cdot a$. Since $F\supset F_{a}$ and $F_{a}/\mathbb{Q}_{p}$
is a Galois extension with Galois group generated by $\varphi$, we see that
the ring $R$ splits exactly as a product of $f$ fields (all abstractly
isomorphic to $F$); thus the Frobenius $\mathrm{Id}_{F}\otimes\varphi$ acts
transitively on $\operatorname{Spec}(R)$ as an $f$-cycle. Therefore the claim
follows from the existence of the isomorphism $\Phi_{\pi}$.

The statement (3) follows directly from (2). (In particular, the uniqueness of
the line $W$ comes from the fact that $D_{F}$ has rank one). ∎

######  Definition A.4 (The filtered $\varphi$-module).

Let us define a filtration on $D_{F}=D\otimes_{F_{a}}F.$ We set

$\mathrm{Fil}^{i}(D_{F})\coloneqq\begin{cases}D_{F}&\text{ if }i\leq 0;\\\
W&\text{ if }i=1;\\\ 0&\text{ if }i\geq 2;\end{cases}$

where the $F$-line $W$ is constructed in Lemma A.3(3). The datum

$D_{\pi}\coloneqq(D,\varphi_{\pi},\mathrm{Fil}^{\bullet}(D_{F}))$

altogether with the $F$-action, defined in Lemmas A.2 and A.3, form a filtered
$\varphi$-module $D_{\pi}$ over $F$ with $F$-coefficients, cf.
[MR1944572]444In their definition, set $N\coloneqq 0$ as monodromy operator.,
that we call _the Lubin–Tate filtered $\varphi$-module associated with_
$\pi\in F$. We may denote it $D$ for brevity.

######  Proposition A.5.

The Lubin–Tate filtered $\varphi$-module $D$ is admissible. Its Newton and
Hodge polygon are given as in the picture below.

###### Proof..

By the definition of the filtration, the Hodge–Tate slopes are: the slope zero
with multiplicity $ef-1$ and the slope one with multiplicity one. Let us
compute the Newton slopes. The slope of the Frobenius $\varphi_{\pi}$ are
equal to the slopes of $\varphi_{\pi}^{f}$ divided by $f$, cf. [MR0498577,
2.1.3]. Since $\varphi_{\pi}^{f}$ is a $F_{a}$-linear endomorphism of $D$, its
slopes are given by the $p$-adic valuation of the eigenvalues of
$\varphi_{\pi}^{f}$, where the $p$-adic valuation $v$ is normalized by
$v(F_{a}^{\times})=\mathbb{Z}$, cf. loc.cit.. The eigenvalues of the matrix
(A.2.3) are of the eigenvalues of the diagonal bloc matrices
$\varphi_{F_{a}}^{i}(C)$. The matrix $\varphi_{F_{a}}^{i}(C)$ is the companion
matrix of the polynomial $\varphi_{F_{a}}^{i}(E_{\pi}(x))\in F_{a}[x]$ and its
eigenvalues are the roots of it. For any $i$, this is an Eisenstein polynomial
of degree $e$; whence, by using the theory of the Newton polygon (for
polynomials with coefficients on local fields, cf. [MR1274045, Ch. I, §6,
Theorem 6.1]), we get that all the roots of $\varphi_{F_{a}}^{i}(E_{\pi}(x))$
have the same valuation $1/e$. Therefore, the Frobenius $\varphi_{\pi}$ has
only one slope, equal to $1/ef$, with multiplicity $ef$, and the
$\varphi$-module $D$ must be absolutely irreducible: indeed its Newton polygon
does not meet points with integral coordinates, except for the vertexes
$(0,0)$ and $(ef,1)$, see the picture above. Finally, the Newton polygon of
$D$ has the same height as the Hodge polygon and lies above it; by
irreducibility there are no nontrivial submodules; thus $D$ verifies the
condition of weakly admissibility. Therefore it is admissible by [MR1779803].
∎

######  Remark A.6.

The $\varphi$-module $D$ is actually a $F_{a}$-form of the irreducible
$\varphi$-module of slope $1/ef$ over $\widehat{F}^{\mathrm{nr}}$, in the
Dieudonne–Manin classification, cf. for example [MR0498577, Theorem 2]. Here
$\widehat{F}^{\mathrm{nr}}$ denotes the completion of maximal unramified
extension of $F$. Its endomorphisms as $\varphi$-module form the central
division algebra over $\mathbb{Q}_{p}$ with invariant $1/{ef}$, cf.
[MR0338002, Chapitre VI, §3, Lemma 3.3.2.2]; whereas its endomorphisms as
_filtered_ $\varphi$-module gives the maximal abelian subfield $F$ of this
algebra.

### A.7.

Before attaching a period to $D$, let us recall the contravariant Fontaine
formalism, see [MR1293972, 5.3.7]. Set

$V_{\mathrm{cris}}^{\vee}(D)\coloneqq\mathrm{Hom}_{\text{fil-}\varphi\text{-mod}}\bigl{(}D,\mathrm{B}_{\mathrm{cris}}\bigr{)},$

where the morphisms are taken in the category of filtered $\varphi$-modules
over $F$, cf. loc. cit. 4.3.3. The $\mathbb{Q}_{p}$-vector space
$V_{\mathrm{cris}}^{\vee}(D)$ is a crystalline representation of $G_{F}$ which
is of dimension $ef$ over $\mathbb{Q}_{p}$, since $D$ is admissible. Moreover,
it is endowed with an action of $F$, thus it is an $F$-vector space
necessarily of dimension one.

###  A.8. Definition of the period

Keep previous notation; moreover, we use the notation of 8.1 and 8.2, excepted
that we are not assuming anymore that $F$ is endowed with a non-trivial
involution.

Let us attach a period $\alpha_{\pi}$ to $D=D_{\pi}$. Any morphism $x\in
V_{\mathrm{cris}}^{\vee}(D)$ can be seen as the element

$x_{F}\coloneqq x\otimes\iota\in
D^{\vee}_{\mathrm{dR}}\coloneqq\mathrm{Hom}_{\mathrm{B}_{\mathrm{dR}}}\bigl{(}D_{F},\mathrm{B}_{\mathrm{dR}}\bigr{)},$

where $\iota\colon F\hookrightarrow\mathrm{B}_{\mathrm{dR}}$ is the embedding
fixed in 8.1. It is clear by construction that for every $x\in
V_{\mathrm{cris}}^{\vee}(D)$, we have
$x_{F}(D_{F})\subset\mathrm{B}_{\mathrm{cris},F}\cap\mathrm{B}_{\mathrm{dR}}^{+}$.
Let us chose a basis $w$ of the line $W=\mathrm{Fil}^{1}(D_{F})$ over $F$ and
a basis $x\in V_{\mathrm{cris}}^{\vee}(D)$ over $F$. We define a period

(A.8.1) $\alpha_{\pi}\coloneqq
x_{F}(w)\in\mathrm{B}_{\mathrm{cris},F}\cap\mathrm{Fil}^{1}\mathrm{B}_{\mathrm{dR}}.$

It does not depend on the choices of $w$ and $x$ up to a constant in
$F^{\times}$ and we call it the period associated with the filtered
$\varphi$-module $D_{\pi}$. We will see in Proposition A.11 that it is a
Lubin–Tate period.

### A.9.

Recall from 8.2 that we have denoted by
$\tilde{F}\subset\mathrm{B}_{\mathrm{dR}}$ the Galois closure of $F$ and by
$\Gamma$ the set of $\mathbb{Q}_{p}$-embeddings of $F$ in $\tilde{F}$. For any
$\sigma\in\Gamma$, consider the map

$\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma\colon D\otimes_{F_{a}}F\rightarrow
D\otimes_{F_{a}}\tilde{F},$

and set $w_{\sigma}\coloneqq(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(w)\in
D_{\tilde{F}}\coloneqq D_{F}\otimes_{F}\tilde{F}$, where $w$ is the basis of
$\mathrm{Fil}^{1}(D_{F})$ chosen in (A.8.1). We have $w_{\sigma}\not=0$ since
$w\not=0$ and $\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma$ is injective. By
definition of $\iota$ we get $w_{\iota}=w\otimes 1$.

######  Lemma A.10.

For any $z\in F$ and $d\in D_{\tilde{F}}=D\otimes_{F_{a}}\tilde{F}$ denote by
$z(d)$ the action induced by the action of $F$ on $D$ and by $d\cdot z$ the
action by multiplication by the scalar $z\in\tilde{F}$ on the right. Let
$w_{\sigma}\in D_{\tilde{F}}$ be the element defined A.9. The following are
true.

  1. (1)

For every $\sigma\in\Gamma$, we have
$\varphi_{\pi}^{f}(w_{\sigma})=\sigma(\pi)w_{\sigma}$.

  2. (2)

For every $\sigma\in\Gamma$ and for every $z\in F$, we have
$z(w_{\sigma})=w_{\sigma}\cdot\sigma(z)$.

  3. (3)

The family $(w_{\sigma})_{\sigma\in\Gamma}$ forms a basis of $D_{\tilde{F}}$
as vector space over $\tilde{F}$.

###### Proof..

The proof of $\eqref{lemma:normal_basis_1}$ is a formal argument:

$\varphi_{\pi}^{f}(w_{\sigma})=(\varphi_{\pi}^{f+\nu(\sigma)}\otimes\sigma)(w)=(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(\varphi_{\pi}^{f}(w))=(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(\pi
w)=\sigma(\pi)w_{\sigma}.$

Let us prove (2):

$z(w_{\sigma})=z((\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(w))=(z\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(w)=(\varphi_{\pi}^{\nu(\sigma)}z\otimes\sigma)(w)=(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(z(w)),$

since the $F$-action on $D$ commutes with $\varphi_{\pi}$ by construction. Now
by definition of $w$, we have $z(w)=w\cdot z$, hence we finish by

$(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(z(w))=(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma)(w\cdot
z)=w_{\sigma}\cdot\sigma(z).$

Point (3) follows from (2) : the $w_{\sigma}$ are eigenvectors of multiplicity
one diagonalizing the left action of $F$ on $D_{\tilde{F}}$. ∎

######  Proposition A.11.

Let $\pi\in F$ be a uniformizer and $D$ the associated Lubin–Tate filtered
$\varphi$-module from Definition A.4. Then the element
$\alpha\coloneqq\alpha_{\pi}\in\mathrm{B}_{\mathrm{cris},F}\cap\mathrm{Fil}^{1}\mathrm{B}_{\mathrm{dR}}$
associated with it in (A.8.1) is a Lubin–Tate period relative to $\pi\in F$ in
the sense of Definition 9.1.

###### Proof..

We compute

$\varphi_{\mathrm{cris}}^{f}(\alpha)=((\varphi_{\mathrm{cris}}^{f}\otimes\mathrm{Id}_{F})\circ(x\otimes\iota))(w)=((x\otimes\iota)\circ(\varphi_{\pi}^{f}\otimes\mathrm{Id}_{F}))(w)=(x\otimes\iota)(w\cdot\pi)=\pi\alpha,$

so the condition (3) of Definition 9.1 holds.

Let us check the other properties of Definition 9.1. By construction
$v_{\mathrm{dR}}(\alpha)\geq 1$ and, for all
$\sigma\in\Gamma\backslash\\{\iota\\}$, we compute

$\sigma_{\mathrm{cris}}(\alpha)=((\varphi_{\mathrm{cris}}^{\nu(\sigma)}\otimes\sigma)\circ(x\otimes\iota))(w)=((x\otimes\mathrm{Id}_{\tilde{F}})\circ(\varphi_{\pi}^{\nu(\sigma)}\otimes\sigma))(w)=(x\otimes\mathrm{Id}_{\tilde{F}})(w_{\sigma}),$

and in particular $v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(\alpha))\geq 0$,
since
$(x\otimes\mathrm{Id}_{\tilde{F}})(D_{\tilde{F}})\subset\mathrm{B}_{\mathrm{cris},\tilde{F}}\cap\mathrm{B}_{\mathrm{dR}}^{+}$.
The top wedge product of $D$ as $F_{a}$-vector space

$\operatorname{det}(D)\coloneqq\bigwedge^{ef}D$

is a rank one admissible filtered $\varphi$-module $D$ of slope one (both
Newton and Hodge). Hence $V_{\mathrm{cris}}^{\vee}(\operatorname{det}(D))$ is
a rank one crystalline representation. Recall that we denoted by
$\mathscr{C}=(\mathfrak{e}_{1},\ldots,\mathfrak{e}_{ef})$ the canonical base
of $D$ in Definition A.1 and that $(w_{\sigma})_{\sigma\in\Gamma}$ is a basis
of $D_{\tilde{F}}$ by Lemma A.10(3). Thus we have

$P\coloneqq\prod_{\sigma\in\Gamma}\sigma_{\mathrm{cris}}(\alpha)=(x\otimes\mathrm{Id}_{\tilde{F}})(\wedge_{\sigma\in\Gamma}w_{\sigma})\in\tilde{F}^{\times}\cdot
x(\wedge_{j=0}^{ef}\mathfrak{e}_{j})\subset\tilde{F}^{\times}\cdot\mathrm{B}_{\mathrm{cris}}\subset\mathrm{B}_{\mathrm{cris},\tilde{F}}.$

Consider the Galois group $G\coloneqq\mathrm{Gal}({\tilde{F}}/{F^{\prime}})$,
where $F^{\prime}\coloneqq\tilde{F}_{a}F$. Let $G$ act on
$\mathrm{B}_{\mathrm{cris},\tilde{F}}=\mathrm{B}_{\mathrm{cris}}\otimes_{\tilde{F}_{a}}\tilde{F}$
as the identity on $\mathrm{B}_{\mathrm{cris}}$ and via its natural action on
the right. The action of $G$ on $\Gamma$ by composition is bijective ($G$ is a
group), hence $P$ is stable under $G$ and it belongs to

$(\mathrm{B}_{\mathrm{cris},\tilde{F}})^{G}=B_{\mathrm{cris},F^{\prime}}=\mathrm{B}_{\mathrm{cris}}\otimes_{\tilde{F}_{a}}\tilde{F}_{a}F=\mathrm{B}_{\mathrm{cris}}\otimes_{F_{a}}F=\mathrm{B}_{\mathrm{cris},F}.$

Since $\operatorname{det}(D)$ is admissible of rank one, the image of the
basis $\wedge_{j=0}^{ef}\mathfrak{e}_{j}$ through $x\not=0$ is invertible in
$\mathrm{B}_{\mathrm{cris},F}$, therefore
$P\in\mathrm{B}_{\mathrm{cris},F}^{\times}$. Furthermore, since the slope of
$\operatorname{det}(D)$ is one, we get $v_{\mathrm{dR}}(P)=1$. The equality
$\sum_{\sigma\in\Gamma}v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(\alpha))=v_{\mathrm{dR}}(P)=1$
implies that the inequalities we have shown above on the
$v_{\mathrm{dR}}(\sigma_{\mathrm{cris}}(\alpha))$ must be equalities.
Moreover, $\alpha$ is invertible in $\mathrm{B}_{\mathrm{cris},F}$, since it
divides $P$ which is invertible. We have thus verified that $\alpha$ fulfils
the four properties of Definition 9.1. ∎

######  Remark A.12.

By taking the determinant of the matrix $A$ of $\varphi_{\pi}$, cf. (A.1.1),
or by recalling that rank one crystalline representations are all of the form
$\mathbb{Q}_{p}(1)\otimes\eta$, where $\eta$ is an unramified character, we
can actually prove

$\prod_{\sigma\in\Gamma}\sigma_{\mathrm{cris}}(\alpha)=uc_{\eta}t\in\mathrm{B}_{\mathrm{cris},F}^{\times},$

where $u\in{F}^{\times}$, $t$ is the period of $\mathbb{Q}_{p}(1)$, cf.
Convention 2.6(6), and $c_{\eta}\in W(\overline{\mathbb{F}}_{p})^{\times}$ is
the period of $\eta$, cf. Convention 2.1.

######  Proposition A.13.

Let $\pi\in F$ be a uniformizer and $D$ the associated Lubin–Tate filtered
$\varphi$-module, from Definition _A.4_. Then the crystalline representation
$V_{\mathrm{cris}}^{\vee}(D)$ is given by the Lubin–Tate character associated
with $\pi\in F$.

###### Proof..

Let $\mathfrak{F}$ be a Lubin–Tate group over $\mathcal{O}_{F}$ associated
with $\pi$. Let $T$ be its Tate module, given by the Lubin–Tate character
$\psi\colon G_{F}\rightarrow\mathcal{O}^{\times}_{F}$; set $V\coloneqq
T[1/p]$. We want to show that $V_{\mathrm{cris}}^{\vee}(D)=V$. Colmez
constructs an element $t_{\mathfrak{F}}\in\mathrm{B}_{\mathrm{cris},F}$,
satisfying Definition 9.1 and such that for every $g\in G$,
$g(t_{\mathfrak{F}})=\psi(g)t_{\mathfrak{F}}$, see for example
[fourquaux_these, §3.6]. Let
$\alpha=\alpha_{\pi}\in\mathrm{B}_{\mathrm{cris},F}^{\times}$ the Lubin–Tate
period in Proposition A.11. By Lemma 8.5 the period
$t_{\mathfrak{F}}\alpha^{-1}$ belongs to $F^{\times}$, whence we have also,
for every $g\in G_{F}$, $g(\alpha)=\psi(g)\alpha$. The map
$V_{\mathrm{cris}}^{\vee}(D)\rightarrow\mathrm{B}_{\mathrm{cris},F}$, defined
by $x\mapsto(x\otimes\iota)(w)$, is $G_{F}$-equivariant by definition of the
action of $G_{F}$ on $V_{\mathrm{cris}}^{\vee}(D)$ and it is $F$-linear by
definition of $w$, cf. (A.8.1). Therefore this map identifies
$V_{\mathrm{cris}}^{\vee}(D)$ to
$F\cdot\alpha\subset\mathrm{B}_{\mathrm{cris},F}$ and that completes the
proof. ∎

######  Remark A.14.

The representation $V_{\mathrm{cris}}^{\vee}(D)$ is an example of
$F$-crystalline representation, cf. [MR2565906]*Introduction. For a more
general construction relating $F$-crystalline representations to their
filtered $\varphi$-modules see [MR2565906]*§(3.3) [MR3505135]*§4.1.

## References