# Arithmetic finiteness of very irregular varieties

Thomas Krämer  Thomas Krämer
Institut für Mathematik
Humboldt Universität zu Berlin
Rudower Chaussee 25, 12489 Berlin
Germany<EMAIL_ADDRESS>and  Marco Maculan  Marco Maculan
Institut de Mathématiques de Jussieu
Sorbonne Université
4, place Jussieu
75005 Paris
France<EMAIL_ADDRESS>
###### Abstract.

We prove the Shafarevich conjecture for very irregular varieties of dimension
less than half the dimension of their Albanese variety, subject to some mild
numerical conditions. Our proof relies on the Lawrence-Venkatesh method as in
the work of Lawrence-Sawin, together with the big monodromy criterion from our
work with Javanpeykar and Lehn.

###### Key words and phrases:

Shafarevich conjecture, Lawrence-Venkatesh method, irregular variety, Albanese
variety, good reduction, Galois representation, period map, monodromy

###### 2020 Mathematics Subject Classification:

11G35, 14G25 (primary), 11F80, 14D07, 14G05, 14K12 (secondary).

###### Contents

  1. 1 Introduction
  2. 2 Proof of theorems A, B and C
  3. 3 Buildings and filtrations on reductive groups
  4. 4 Realizations with extra structures
  5. 5 $p$-adic Hodge theory
  6. 6 Combinatorics on flag varieties
  7. 7 Reminder on period mappings
  8. 8 The Lawrence-Venkatesh method: Proof of D
  9. A Conjugacy classes of disconnected reductive pairs
  10. B The Hodge filtration on twisted cohomology

##  1\. Introduction

In this paper we obtain finiteness results à la Shafarevich for a large class
of varieties over number fields, combining the method of Lawrence-Sawin [LS20]
and Lawrence-Venkatesh [LV20] with the big monodromy criterion for
subvarieties on abelian varieties in our work with Javanpeykar and Lehn
[JKLM22].

###  1.1. Intrinsic results

Let $K$ be a number field and $\Sigma$ a finite set of places of $K$. Recall
that an abelian variety $A$ over $K$ has good reduction outside $\Sigma$ if it
is the generic fiber of an abelian scheme $\mathcal{A}$ over the ring of
$\Sigma$-integers $\mathcal{O}_{K,\Sigma}\subset K$. In this case we say that
a smooth subvariety $X\subset A$ has _good reduction outside $\Sigma$_ if the
Zariski closure of $X$ in $\mathcal{A}$ is smooth over
$\mathcal{O}_{K,\Sigma}$.

######  Definition.

A smooth projective variety $X$ over $K$ is _very irregular_ if
$X(K)\neq\emptyset$ and if for $x\in X(K)$ the Albanese morphism
$\mathrm{alb}_{x}\colon X\to\operatorname{Alb}(X)$ is a closed embedding with
ample normal bundle. In this case we say that $X$ has  _good reduction outside
$\Sigma$_ if the abelian variety $\operatorname{Alb}(X)$ and its subvariety
$X\hookrightarrow\operatorname{Alb}(X)$ do.

The above properties do not depend on the choice of the point $x$. Examples of
very irregular varieties include smooth projective curves of genus $\geqslant
2$ and complete intersections of ample divisors in an abelian variety,
provided that these varieties have $K$-rational points. Any very irregular
variety $X$ has ample canonical bundle, so the automorphism group scheme
$\operatorname{Aut}(X)$ is finite. We consider the natural linear action of
$\operatorname{Aut}(X)$ on $\textup{H}^{0}(X,\Omega^{1}_{X})$ and define

$\operatorname{Stab}(X)\;:=\;\operatorname{Ker}\bigl{(}\operatorname{Aut}(X)\to\operatorname{GL}(\textup{H}^{0}(X,\Omega^{1}_{X}))\bigr{)}.$

Looking at the induced action on the Albanese variety, one sees that the
quotient morphism $X\to X/\operatorname{Stab}(X)$ is finite and étale. In
particular, the order $|\operatorname{Stab}(X)|$ divides the topological Euler
characteristic $\chi_{\textup{top}}(X):=c_{d}(X)$ where $d=\dim X$. In the
case of surfaces our main result can then be stated as follows:

######  Theorem A.

Fix an integer $c\geqslant 1$. Then up to $K$-isomorphism there are only
finitely many very irregular surfaces $X$ with good reduction outside $\Sigma$
such that

  * •

$c_{2}(X)=c$,

  * •

$h^{0}(X,\Omega^{1}_{X})\geqslant 6$,

and such that $c_{2}(X)/|\operatorname{Stab}(X)|\neq 27$.

In particular there are only finitely many $K$-isomorphism classes of very
irregular surfaces $X$ with good reduction outside $\Sigma$, topological Euler
characteristic $c_{2}(X)=c$ and irregularity $h^{0}(X,\Omega_{X}^{1})\geqslant
11$, see for instance [JKLM22, lemma 2.13]. More generally we show the
following result for very irregular varieties of dimension less than half the
dimension of the Albanese variety:

######  Theorem B.

Let $P\in\mathbb{Q}[z]$ a polynomial of degree $d$. Then up to $K$-isomorphism
there are only finitely many smooth projective very irregular varieties $X$
with good reduction outside $\Sigma$ such that

  * •

$X$ has Hilbert polynomial $P$ with respect to the canonical bundle,

  * •

$g=h^{0}(X,\Omega^{1}_{X})\geqslant 2d+2$,

and such that (1.2) and (1.2) below hold with $A=\operatorname{Alb}(X)$,
$e=\chi_{\textup{top}}(X)/|\operatorname{Stab}(X)|$.

B applies in particular to all smooth projective curves of genus $\geqslant 4$
with a rational point. However, its proof relies on the Shafarevich conjecture
for abelian varieties proved by Faltings in [Fal83], thus B does not furnish a
new proof of the Shafarevich conjecture for curves. Instead, one can regard B
as a generalization of Faltings’ theorem to higher dimension.

###  1.2. Numerical conditions

In order to introduce the numerical conditions that enter in the statement of
B, let $A$ be an abelian variety of dimension $g$ over a field $k$ and
$X\subset A$ a smooth subvariety of dimension $d$. If $k$ is algebraically
closed, we say that $X$ is _symmetric up to translation_ if there is a point
$a\in A(k)$ such that $X=-a+X$. If $k$ is an arbitrary field, we say that $X$
is symmetric up to translation if its base change to an algebraic closure of
$k$ is so. By the _stabilizer_ of a subvariety $X\subset A$ we mean the
algebraic group $\operatorname{Stab}_{A}(X)$ whose points in a $k$-scheme $S$
are

$\operatorname{Stab}_{A}(X)(S)\;=\;\\{a\in A(S)\colon X_{S}+a\;=\;X_{S}\\}.$

When $A=\operatorname{Alb}(X)$ and $X$ is embedded via the Albanese morphism,
we have

$\operatorname{Stab}(X)=\operatorname{Stab}_{\operatorname{Alb}(X)}(X).$

To apply the big monodromy criterion in [JKLM22] we impose that the
topological Euler characteristic
$e=\chi_{\textup{top}}(X)/|\operatorname{Stab}_{A}(X)|$ of the quotient
$X/\operatorname{Stab}_{A}(X)$ satisfies the following conditions:

$\displaystyle|e|\;$ $\displaystyle\neq\;27$ $\displaystyle\textup{if
$d\geqslant 2$ and $X\subset A$ is not symmetric up to a translation},$ (1.1)
$\displaystyle|e|\;$ $\displaystyle\neq\;56$ $\displaystyle\textup{if
$d\geqslant 3$ is odd and $X\subset A$ is symmetric up to translation},$
$\displaystyle|e|\;$ $\displaystyle\neq\;2^{2m-1}$ if $d\geqslant(g-1)/4$,
$m\in\\{3,\dots,d\\}$ has the same parity as $d$ $\displaystyle\textup{and
$X\subset A$ is symmetric up to translation}.$

For $d<(g-1)/2$ we do not know any example of subvarieties $X$ with ample
normal bundle for which (1.2) fails. To deal with the semisimplification of
global Galois representations, we also need to assume that

(1.2) $2\chi(X\times X,\Omega^{d}_{X\times
X})\leqslant\chi_{\textup{top}}(X\times X).$

Here regardless of the dimension we do not know any example of a subvariety
$X$ with ample normal bundle for which (1.2) fails. We note that (1.2) holds
if $d$ is odd and also if $X$ is a surface with nef cotangent bundle, see
lemma B.4.

We take advantage of the general setup in this section to introduce two
notions that will be useful later:

######  Definition.

If $k$ is algebraically closed, we say that $X$ is

  * •

_divisible_ if $\operatorname{Stab}_{A}(X)\neq 0$;

  * •

a _product_ if there are smooth subvarieties $X_{1},X_{2}\subset A$ with $\dim
X_{i}>0$ such that the sum morphism $X_{1}\times X_{2}\to A$ is a closed
embedding with image $X$.

If $k$ is an arbitrary field, we say that $X$ is divisible resp. a product if
its base change to an algebraic closure of $k$ is so.

###  1.3. Extrinsic result

We will deduce B from a general result about subvarieties of arbitrary abelian
varieties. Let $A$ be an abelian variety of dimension $g$ over a number field
$K$, and $\Sigma$ a finite set of places of $K$ including all places where $A$
has bad reduction. We show:

######  Theorem C.

Fix a polynomial $P\in\mathbb{Q}[z]$ of degree $d<(g-1)/2$ and an ample line
bundle $L$ on $A$. Then up to translation by points in $A(K)$ there are only
finitely many smooth subvarieties $X\subset A$ over $K$ with ample normal
bundle and good reduction outside $\Sigma$ that are not a product, that have
Hilbert polynomial $P$ with respect to $L$, and that satisfy (1.2) and (1.2)
with $e=\chi_{\textup{top}}(X)/|\operatorname{Stab}_{A}(X)|$.

We call a subvariety $X\subset A$ a _geometric complete intersection of ample
divisors_ if its base change to an algebraic closure of $K$ is a complete
intersection of ample divisors. Smooth complete intersections of ample
divisors in an abelian variety are never a product [JKLM22, rem. 6.3], and
they satisfy (1.2) with $e=\chi_{\textup{top}}(X)$ by [JKLM22, prop. 2.16,
cor. 2.17]. Hence as a direct consequence of C we obtain:

######  Corollary.

Fix a polynomial $P\in\mathbb{Q}[z]$ of degree $d<(g-1)/2$ and an ample line
bundle $L$ on $A$. Then up to translation by points in $A(K)$ there exist only
finitely many smooth nondivisible geometric complete intersections of ample
divisors $X\subset A$ over $K$ with good reduction outside $\Sigma$ that have
Hilbert polynomial $P$ with respect to $L$ and satisfy (1.2).

The proof of C is based on the Lawrence-Venkatesh method [LV20] as elaborated
in Lawrence-Sawin [LS20]. To control the arising monodromy groups we will use
the big monodromy criterion from [JKLM22].

###  1.4. Big monodromy

In this section we temporarily work over $\mathbb{C}$. Let $S$ be a smooth
complex variety and $A$ a complex abelian variety of dimension $g$. Inside the
constant abelian scheme $A_{S}:=A\times S$ let $\mathcal{X}\subset A_{S}$ be a
subvariety which is smooth over $S$ with connected fibers of dimension $d$.
The situation is resumed in the following diagram:

${\mathcal{X}}$${A}$${A_{S}}$${S}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{\operatorname{pr}_{A}}$$\scriptstyle{\operatorname{pr}_{S}}$

Let $\pi_{1}(A,0)$ be the topological fundamental group and denote the group
of its characters by

$\Pi(A)=\operatorname{Hom}(\pi_{1}(A,0),\mathbb{C}^{\times}).$

In what follows, by a _linear subvariety_ we mean a subset
$\Pi(B,\mathbb{F})\subset\Pi(A)$ for an abelian quotient variety
$A\twoheadrightarrow B$ with $\dim B<\dim A$. We say that a statement holds
for most $\chi\in\Pi(A)$ if it holds for all $\chi$ outside a finite union of
torsion translates of linear subvarieties. For $\chi\in\Pi(A)$, let $L_{\chi}$
denote the associated rank one local system on $A$. It follows from generic
vanishing [BSS18, KW15, Sch15] that for most $\chi$ the higher direct images
$R^{i}f_{*}\pi^{*}L_{\chi}$ vanish in all degrees $i\neq d$; we consider the
local system

$V_{\chi}\;:=\;R^{d}f_{*}\pi^{*}L_{\chi}$

of rank $|e|$ where $e$ is the topological Euler characteristic of the fibers
of $\mathcal{X}\to S$. More generally, the study of finite étale covers of the
subvariety $\mathcal{X}\subset A_{S}$ induced by finite étale covers of $A$
leads to direct sums

$V_{\underline{\chi}}\;:=\;V_{\chi_{1}}\oplus\cdots\oplus V_{\chi_{n}}$

where $\underline{\chi}=(\chi_{1},\dots,\chi_{n})\in\Pi(A)^{n}$ is an
$n$-tuple of characters of the fundamental group. Using the natural
identification $\Pi(A,\mathbb{F})^{n}=\Pi(A^{n},\mathbb{F})$ we will also
apply the terminology most for such $n$-tuples of characters. Consider for
$s\in S(k)$ the monodromy representation

$\rho\colon\quad\pi_{1}(S,s)\;\longrightarrow\;\operatorname{GL}(V_{\underline{\chi},s})\quad\textnormal{on
the fiber}\quad
V_{\underline{\chi},s}\;=\;\bigoplus_{i=1}^{n}\textup{H}^{d}(\mathcal{X}_{s},L_{\chi_{i}}).$

The _algebraic monodromy group_ of the local system $V_{\underline{\chi}}$ is
the Zariski closure of the image of $\rho$. By construction it is an algebraic
subgroup of

$\operatorname{GL}(V_{\chi_{1},s})\times\cdots\times\operatorname{GL}(V_{\chi_{n},s})\;\subset\;\operatorname{GL}(V_{\underline{\chi},s}).$

We say that $\mathcal{X}\subset A_{S}$ is _symmetric up to translation_ if
there exists $a\colon S\to A$ such that
$\mathcal{X}_{t}=-\mathcal{X}_{t}+a(t)$ for all $t\in S(\mathbb{C})$. By
[JKLM22] then Poincaré duality gives for each $\chi\in\Pi(A)$ a nondegenerate
bilinear pairing

$\theta_{\chi,s}\colon\quad V_{\chi,s}\otimes V_{\chi,s}\longrightarrow
L_{\chi,a(s)}$

which is symmetric if $d$ is even, and alternating otherwise. Since the
pairing is compatible with the monodromy operation on the fiber, it follows
that the algebraic monodromy group of $V_{\underline{\chi}}$ is contained in
an orthogonal resp. symplectic group in the two cases. This leads to the
following definition:

######  Definition.

We say that $V_{\underline{\chi}}$ has big monodromy if its algebraic
monodromy group contains $G_{1}\times\cdots\times G_{n}$ as a normal subgroup,
where $G_{i}\subset\operatorname{GL}(V_{\chi_{i},s})$ is defined by

$G_{i}\;:=\;\begin{cases}\operatorname{SL}(V_{\chi_{i},s})&\textup{if
$\mathcal{X}$ is not symmetric up to translation},\\\
\operatorname{SO}(V_{\chi_{i},s},\theta_{\chi_{i},s})&\textup{if $\mathcal{X}$
is symmetric up to translation and $d$ is even},\\\
\operatorname{Sp}(V_{\chi_{i},s},\theta_{\chi_{i},s})&\textup{if $\mathcal{X}$
is symmetric up to translation and $d$ is odd}.\end{cases}$

We say that $\mathcal{X}\to S$ _has big monodromy for most tuples of torsion
characters_ if for each $n\geqslant 1$ and most torsion $n$-tuples
$\underline{\chi}\in\Pi(A)^{n}$ the local system $V_{\underline{\chi}}$ has
big monodromy.

If the fiber $\mathcal{X}_{\bar{\eta}}$ over a geometric generic point
$\bar{\eta}$ of $S$ is divisible or decomposes as a product, then the family
$\mathcal{X}\to S$ does not have big monodromy. There are two other cases
where the family cannot have big monodromy: We say that
$\mathcal{X}_{\bar{\eta}}$ is

  * •

_constant up to a translation_ if it is the translate of a subvariety
$Y\subset A$ along a point in $A(\bar{\eta})$;

  * •

a _symmetric power of a curve_ if there is a curve $C\subset A_{S,\bar{\eta}}$
such that the sum morphism $\operatorname{Sym}^{d}C\to A_{S,\bar{\eta}}$ is an
embedding with image $\mathcal{X}_{\bar{\eta}}$ and $d\geqslant 2$.

By [JKLM22] these are the only obstructions to big monodromy:

######  Big Monodromy Criterion.

Suppose $X:=\mathcal{X}_{\bar{\eta}}\subset A_{S,\bar{\eta}}$ has ample normal
bundle, dimension $d<(g-1)/2$, and satisfies the numerical assumption (1.2).
Then the following conditions are equivalent:

  * •

$X$ is nondivisible, not constant up to translation, not a symmetric power of
a curve and not a product.

  * •

$\mathcal{X}\to S$ has big monodromy for most torsion tuples of characters.

We will combine this with the Lawrence-Venkatesh method to obtain the main
nondensity result for integral points that is behind theorems A, B, C.

###  1.5. Nondensity of integral points

Let $K$ be a number field and $\Sigma$ a finite set of places of $K$. The
abelian variety in C has good reduction outside $\Sigma$ and hence it extends
uniquely to an abelian scheme over $\mathcal{O}_{K,\Sigma}$. For simplicity we
reset notation and write $A$ for this abelian scheme. Let $S$ be a smooth
separated finite type scheme over $\mathcal{O}_{K,\Sigma}$ with geometrically
connected generic fiber, and $\mathcal{X}\subset A_{S}$ be a closed subscheme
which is smooth over $S$ with geometrically connected fibers of dimension $d$.
We say that $\mathcal{X}\to S$ has big monodromy for most tuples of torsion
characters if for some (equivalently, any) embedding $\sigma\colon
K\hookrightarrow\mathbb{C}$ the family $\mathcal{X}_{\sigma}\to S_{\sigma}$
does so. In section 2 we will see that theorems A, B, C are a consequence of
the following general criterion for the nondensity of integral points:

######  Theorem D.

Suppose that

  * •

$\mathcal{X}\to S$ has big monodromy for most tuples of torsion characters,
and

  * •

every geometric fiber $X$ of $\mathcal{X}\to S$ satisfies (1.2).

Then $S(\mathcal{O}_{K,\Sigma})$ is not Zariski-dense in $S$.

The proof of this criterion will occupy sections 3 through 8. In the rest of
this introduction we will outline the main ideas, assuming for simplicity that
$K=\mathbb{Q}$ so that $\mathcal{O}_{K,\Sigma}=\mathbb{Z}[1/N]$ for some
integer $N\geqslant 1$. Pick a prime $p\nmid N$, let $\bar{\mathbb{Q}}_{p}$ be
an algebraic closure of $\mathbb{Q}_{p}$ and let
$\bar{\mathbb{Q}}\subset\bar{\mathbb{Q}}_{p}$ be the algebraic closure of
$\mathbb{Q}$. We then have an embedding

$\Gamma_{p}:=\operatorname{Gal}(\bar{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\;\lhook\joinrel\longrightarrow\;\Gamma:=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$

Suppose we can find a character
$\chi_{0}\colon\pi_{1}(A(\mathbb{C}),0)\to\mathbb{C}^{\times}$ of order $r$
with $p\nmid r$ such that

  1. (1)

$\textup{H}^{i}(\mathcal{X}_{s}(\mathbb{C}),L_{\chi})=0$ for all $i\neq d$ and
all $\chi\in\Gamma\chi_{0}$, $s\in S(\mathbb{C})$,

  2. (2)

$V_{\underline{\chi}}$ has big monodromy for the tuple $\underline{\chi}$ of
all characters $\chi\in\Gamma\chi_{0}$.

Here the assumption (2) is unrealistic and only serves to simplify the
presentation; we will explain how to bypass it in section 8. The orbit
$\Gamma\chi_{0}$ corresponds to a direct summand

$E\subset[r]_{\ast}\mathbb{Q}_{p,A}$

where $[r]\colon A\to A$ denotes the multiplication by $r$ and
$\mathbb{Q}_{p,A}$ is the constant $p$-adic étale sheaf. For each $s\in
S(\mathbb{Z}[1/N])$ the Galois representation

$V_{s}:=\textup{H}^{d}_{\textup{\'{e}t}}(A_{\bar{\mathbb{Q}}},E)$

of $\Gamma$ is pure of weight $d$ by [Del74]. It is conjectured to be
semisimple, but with our current state of knowledge a main task of this paper
will be to work without assuming this. For this we consider the
semisimplification filtration $V^{\bullet}_{s}$ on $V_{s}$. By Faltings’
finiteness of pure global Galois representations [Del85, th. 3.1], the image
of the composite map

${S(\mathbb{Z}[1/N])}$${{\left\\{\begin{array}[]{c}\textup{filtered global}\\\
\textup{Galois
rep's}\end{array}\right\\}/\cong}}$${{\left\\{\begin{array}[]{c}\textup{graded
global}\\\ \textup{Galois
rep's}\end{array}\right\\}/\cong}}$${s}$${{(V_{s},V^{\bullet}_{s})}}$${{\operatorname{gr}V^{\bullet}_{s}}}$

is finite. Hence to show that $S(\mathbb{Z}[1/N])$ is not Zariski dense in
$S$, it suffices to show that the fibers of the map

$S(\mathbb{Z}[1/N])\ni s\longmapsto\textup{isomorphism class of
}\operatorname{gr}V^{\bullet}_{s}$

are not Zariski dense. A natural attempt would be to restrict representations
to the local Galois group at a prime $\ell\neq p$ with $\ell\nmid N$. To see
why this does not suffice, fix an embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$ and consider the
inclusion

$\Gamma_{\ell}:=\operatorname{Gal}(\bar{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})\;\lhook\joinrel\longrightarrow\;\Gamma:=\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}).$

For $s\in S(\mathbb{Z}[1/N])$ let $V_{s,\ell}$ be the $p$-adic representation
of $\Gamma_{\ell}$ obtained from $V_{s}$ by restriction. By proper base change
in étale cohomology, the isomorphism class of $V_{s,\ell}$ only depends on the
image of $s$ in $S(\mathbb{F}_{\ell})$. In particular the set

$\left\\{V_{s,\ell}\mid s\in S(\mathbb{Z}[1/N])\right\\}/\cong$

is finite, so there is no hope of proving nondensity by restricting to the
subgroup $\Gamma_{\ell}$ for a single $\ell\neq p$. Instead the heart of the
Lawrence-Venkatesh method lies in the insight that the restriction of $V_{s}$
to $\Gamma_{p}$ moves enough when $s\in S(\mathbb{Z}[1/N])$ varies, provided
that the family $\mathcal{X}\to S$ has big monodromy.

To make this precise we will apply $p$-adic Hodge theory. Recall that $p$-adic
Hodge theory associates to every finite-dimensional _cristalline_
representation $V$ of $\Gamma_{p}$ a triple

$D_{\textup{cris}}(V):=(V_{\textup{dR}},\varphi_{V},h^{\bullet}V)$

of a finite-dimensional $\mathbb{Q}_{p}$-vector space $V_{\textup{dR}}$, an
endomorphism
$\varphi_{V}\in\operatorname{End}_{\mathbb{Q}_{p}}(V_{\textup{dR}})$ and a
filtration $h^{\bullet}V_{\textup{dR}}$ of $V_{\textup{dR}}$ by vector
subspaces (not necessarily stable under $\varphi_{V}$). Such triples are
called filtered isocristals, but in this introduction we will informally call
them $p$-adic Hodge structures. Rather than defining the functor
$D_{\textup{cris}}$ let us only mention what it does to the representations in
question. To do so consider the vector bundle $[r]_{\ast}\mathcal{O}_{A}$ with
the connection $\nabla$ induced by the canonical derivation on
$\mathcal{O}_{A}$. There is a well-defined direct summand

$\mathcal{E}\subset[r]_{\ast}\mathcal{O}_{A_{\mathbb{Z}_{p}}}$

associated to the local system $E$ and stable under $\nabla$. For $s\in
S(\mathbb{Z}[1/N])$ the representation $V_{s}$ is cristalline since
$\mathcal{X}_{s}$ has good reduction at $p$. Then Faltings’ étale-de Rham
comparison theorem [Fal89, th. 5.6] gives

$D_{\textup{cris}}(V_{s})\;=\;(\textup{H}^{d}_{\textup{dR}}(\mathcal{X}_{s}\times\mathbb{Q}_{p},\mathcal{E}),\;\varphi_{s},\;\textup{Hodge
filtration})$

where $\varphi_{s}$ is the endomorphism induced by the Frobenius operator on
cristalline cohomology via the de Rham-cristalline comparison theorem. This
should give the idea of how to prove that the representations $V_{s}$ move. In
a sense to be made precise below, we can identify de Rham cohomology groups
for $s$ and $s^{\prime}$ close enough by integrating the Gauss-Manin
connection. It then remains to show that the Hodge filtration moves enough,
i.e. the associated period mapping has large enough image, which will be
implied by the hypothesis of big monodromy.

Note that the semisimplication filtration $V^{\bullet}_{s}$ on $V_{s}$ induces
a filtration $D_{\textup{cris}}(V^{\bullet})$ on $D_{\textup{cris}}(V_{s})$
which is stable under the cristalline Frobenius. This operation is compatible
with taking the associated graded, leading to the following commutative
diagram:

${{\left\\{\begin{array}[]{c}\textup{filtered global}\\\ \textup{Galois
rep's}\\\ \textup{cristalline at
$p$}\end{array}\right\\}}}$${{\left\\{\begin{array}[]{c}\textup{graded
global}\\\ \textup{Galois rep's}\\\ \textup{cristalline at
$p$}\end{array}\right\\}}}$${{\left\\{\begin{array}[]{c}\textup{filtered
cristalline}\\\ \textup{local Galois
rep's}\end{array}\right\\}}}$${{\left\\{\begin{array}[]{c}\textup{graded
cristalline}\\\ \textup{local Galois
rep's}\end{array}\right\\}}}$${{\left\\{\begin{array}[]{c}\textup{filtered
$p$-adic}\\\ \textup{Hodge
structures}\end{array}\right\\}}}$${{\left\\{\begin{array}[]{c}\textup{graded
$p$-adic}\\\ \textup{Hodge structures}\end{array}\right\\}}}$restrict to
$\scriptstyle\Gamma_{p}$gradedrestrict to
$\scriptstyle\Gamma_{p}$graded$\scriptstyle{D_{\textup{cris}}}$$\scriptstyle{D_{\textup{cris}}}$graded

To prove that $S(\mathbb{Z}[1/N])$ is not Zariski dense, it will be enough to
show that the fibers of the map

$S(\mathbb{Z}[1/N])\ni s\longmapsto\textup{isomorphism class of
$\operatorname{gr}D_{\textup{cris}}(V^{\bullet}_{s})$}$

are not dense. Consider the $d$-th relative de Rham cohomology group

$\mathcal{V}\;:=\;\mathcal{H}_{\textup{dR}}^{d}(\mathcal{X}_{\mathbb{Z}_{p}}/S_{\mathbb{Z}_{p}};\pi^{\ast}(\mathcal{E},\nabla))$

where $\pi\colon\mathcal{X}\to A$ is the projection. By a careful choice of
the prime $p$ in the first place we may assume that $\mathcal{V}$ is a vector
bundle over $S_{\mathbb{Z}_{p}}$. This vector bundle $\mathcal{V}$ comes
equipped with the Gauss-Manin connection that we still denote $\nabla$. Fix a
point $o\in S(\mathbb{Z}[1/N])$ and consider the residue disk

$\Omega=\\{s\in S(\mathbb{Z}_{p})\mid s\equiv o\mod p\\}.$

Since $S(\mathbb{F}_{p})$ is finite, it suffices to show that the intersection
$\Omega\cap S(\mathbb{Z}[1/N])$ is not Zariski dense for any such residue
disk. Working on $\Omega$ has the advantage that the Gauss-Manin connection
can be integrated on $\Omega$: We have a natural isomorphism of $K$-vector
spaces

$f_{s}\colon\mathcal{V}_{s}=\textup{H}^{d}_{\textup{dR}}(\mathcal{X}_{s}\times\mathbb{Q}_{p},\mathcal{E})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{V}_{o}=\textup{H}^{d}_{\textup{dR}}(\mathcal{X}_{o}\times\mathbb{Q}_{p},\mathcal{E}),$

see section 8.5. The filtration on $\mathcal{V}_{o}$ induced by the Hodge
filtration on $\mathcal{V}_{s}$ via $f_{s}$ is by definition the image of $s$
via the $p$-adic period mapping

$\Phi_{p}\colon\quad\Omega\;\longrightarrow\;\operatorname{Flag}_{t}(\mathcal{V}_{o})$

where $\operatorname{Flag}_{t}(\mathcal{V}_{o})$ denotes the variety of flags
on $\mathcal{V}_{o}$ whose type $t$ is the type of the flag underlying the
Hodge filtration. Define an equivalence relation on $\Omega$ by putting

$s\sim
s^{\prime}\quad\textup{if}\quad\operatorname{gr}D_{\textup{cris}}(V_{s}^{\bullet})\cong\operatorname{gr}D_{\textup{cris}}(V_{s^{\prime}}^{\bullet}).$

The equivalence classes then give rise to certain constructible subsets
$Z\subset\operatorname{Flag}_{t}(\mathcal{V}_{o})$, and we win as soon as we
show that the preimage $\Phi_{p}^{-1}(Z)$ of each of these subsets is not
Zariski dense in $\Omega$. The Ax-Lindemann property of period mappings proved
by Bakker-Tsimerman [BT19], or rather its $p$-adic version given in
proposition 7.10 following [LV20, lemma 9.3], says that for this it will be
enough to show that

  * •

$\Phi_{p}$ has a Zariski dense image in
$\operatorname{Flag}_{t}(\mathcal{V}_{o})$, and

  * •

$\dim Z+\dim
S_{\mathbb{Q}}\leqslant\dim\operatorname{Flag}_{t}(\mathcal{V}_{o})$.

We now have to pay for the careless discussion given above: The period mapping
$\Phi_{p}$ will _never_ have a Zariski dense image in
$\operatorname{Flag}_{t}(\mathcal{V}_{o})$, for the simple reason that we
forgot all the extra structures that $\mathcal{E}$ comes with. Indeed, over a
finite extension $K$ of $\mathbb{Q}_{p}$ we have a splitting into a direct sum
of line bundles

$\mathcal{E}_{\rvert
A_{K}}=\bigoplus_{\chi\in\Gamma\chi_{0}}\mathcal{L}_{\chi}$

and the connection decomposes accordingly. So over $K$ the image of $\Phi_{p}$
is contained in

$\mathfrak{H}:=\prod_{\chi\in\Gamma\chi_{0}}\operatorname{Flag}_{t_{\chi}}(\textup{H}^{d}_{\textup{dR}}(\mathcal{X}_{s}\times
K,\mathcal{L}_{\chi})).$

In sections 3, LABEL: and 4 we set up an abstract framework to capture all the
relevant extra structures on étale and de Rham cohomology, and we explain what
a filtration and the associated graded means in this context. The abstract
nature of this framework should allow to apply it in situations where the
monodromy groups are not classical groups. In proposition 4.6 we prove a
version of Faltings’ finiteness theorem for pure global Galois representations
that keeps track of the extra structures, and in proposition 5.14 we show that
our general framework is compatible with $p$-adic Hodge theory in the sense
that the extra structures on étale cohomology are carried to those on de Rham
cohomology via Faltings’ comparison theorem.

Suppose now that with all the extra structures in place, we started the above
argument all over again. By assumption the family $\mathcal{X}\to S$ has big
monodromy with respect to the tuple of consisting of the characters in the
orbit $\Gamma\chi_{0}$. This implies that $\Phi_{p}$ has a Zariski dense image
in $\mathfrak{H}$. It remains to bound the dimension of the equivalence
classes $Z$ mentioned above. We do this in theorem 6.4 by giving such an upper
bound in terms of a combinatorial function and of the dimension of the
centralizer of the cristalline Frobenius. This generalizes [LV20, prop. 10.6]
and clarifies [LS20, lemma 8.20]. A crucial restriction on the dimension comes
from the purity of the global Galois representation we started with: The
filtration on each graded piece of
$\operatorname{gr}D_{\textup{cris}}(V_{s}^{\bullet})$ induced by the Hodge
filtration on $D_{\textup{cris}}$ has the same weight $d/2$ as the original
Hodge filtration. In the framework where extra structure is taken into account
such a property carries over in a weaker form; see section 3.5 and proposition
5.16. At this stage, to complete the proof we only need to bound the dimension
of the centralizer of the cristalline Frobenius $\varphi_{o}$. In order to
apply theorem 6.4 we will need that this centralizer is very small compared to
$\mathfrak{H}$. We will obtain this by letting $r$ go to infinity. This will
conclude the proof of D hence of theorems A, B and C.

##  2\. Proof of theorems A, B and C

In this section we explain how to deduce the theorems A, B, C from the main D.

###  2.1. The Hilbert scheme

Let $A$ be an abelian variety over a field $k$. Let $A$ act on its Hilbert
scheme $\operatorname{Hilb}_{A}$ by translation. This action preserves the
connected components. In particular for an ample line bundle $L$ on $A$ and
any $P\in\mathbb{Q}[z]$, the action preserves the projective variety

$\operatorname{Hilb}_{A,L,P}\;\subset\;\operatorname{Hilb}_{A}$

parametrizing the subschemes with Hilbert polynomial $P$ with respect to $L$.
Consider the subset

$\operatorname{Hilb}_{A,L,P}^{\star}\subset\operatorname{Hilb}_{A,L,P}$

consisting of the image of all geometric points
$\operatorname{Spec}\Omega\to\operatorname{Hilb}_{A}$ such that the
corresponding subvariety $X\subset A_{\Omega}$ is smooth, geometrically
connected, nondivisible, not a product, with ample normal bundle, with Hilbert
polynomial $P$ with respect to $L$ and such that (1.2) and (1.2) are satisfied
for $d=\dim X$ and $e=\chi_{\textup{top}}(X)$.

######  Lemma 2.1.

Let $P\in\mathbb{Q}[z]$ and let $L$ be an ample line bundle on $A$. Then the
subset $\operatorname{Hilb}_{A,L,P}^{\star}\subset\operatorname{Hilb}_{A,L,P}$
is constructible.

###### Proof.

Smoothness, geometric connectedness and ampleness are open conditions.
Therefore the locus $U$ where the universal family
$\mathcal{X}\to\operatorname{Hilb}_{A}$ has these properties is open. Let
$G\subset A_{U}$ the stabilizer of the $U$-scheme $\mathcal{X}_{U}\times A$,
namely, the equalizer of the morphisms $\sigma_{\rvert A\times U}$ and
$q_{\rvert A\times U}$, where

$\sigma,q\colon
A\times\operatorname{Hilb}_{A}\longrightarrow\operatorname{Hilb}_{A}$

are respectively the morphism defining the action of $A$ and the second
projection. Since each fiber of $\mathcal{X}_{U}\to U$ is of general type, the
subgroup $U$-scheme $G$ is finite. It follows that the locus where $G$ is
trivial is an open subset $V\subset U$. Euler characteristics are locally
constant whence (1.2) and (1.2) are satisfied exactly on a union of connected
components $W\subset V\cap\operatorname{Hilb}_{A,L,P}$. Summing up, it follows
that

$\operatorname{Hilb}^{\star}_{A,L,P}\subset V$

is the subset containing the image of the geometric points
$\operatorname{Spec}\Omega\to V$ such that the corresponding subvariety
$X\subset A_{\Omega}$ is not a product. To see that
$\operatorname{Hilb}^{\star}_{A,L,P}$ is constructible it suffices to show
that that its complement in $V$ is constructible. By Noetherian induction we
reduce to prove the following. Let $S\subset V$ be an integral subvariety with
generic point $\eta$ and $\Omega$ an algebraic closure of the residue field of
$\eta$. Suppose that the geometric point $\operatorname{Spec}\Omega\to S$ is
such that the corresponding subvariety $X\subset A_{\Omega}$ is a product.
Then there is a non-empty open subset $S^{\prime}\subset S$ such that for each
geometric point $\bar{s}$ of $S^{\prime}$ the subvariety
$\mathcal{X}_{\bar{s}}$ is a product. By spreading out we may suppose that
there are a quasi-finite étale morphism $\nu\colon\tilde{S}\to S$ and
subvarieties $\mathcal{X}_{1},\mathcal{X}_{2}\subset A_{\tilde{S}}$ such that
the sum induces an isomorphism

$\mathcal{X}_{1}\times_{\tilde{S}}\mathcal{X}_{2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{X}\times_{\operatorname{Hilb}_{A}}\tilde{S}$

of $\tilde{S}$-schemes. It follows the open subset $S^{\prime}=\nu(S)$ does
the job. ∎

###  2.2. Proof of C in the nondivisible case

Let $K$ be a number field and $\Sigma$ a finite set of places of $K$. As in
section 1.5 we let $A$ be an abelian scheme over $\mathcal{O}_{K,\Sigma}$ of
relative dimension $g$.

######  Lemma 2.2.

Suppose $g\geqslant 4$. Let $S$ be an integral $\mathcal{O}_{K,\Sigma}$-scheme
of finite type with smooth generic fiber. Let $\mathcal{C}\subset A_{S}$ be a
smooth relative curve over $S$ whose fiber $\mathcal{C}_{\bar{\eta}}$ over a
geometric generic point $\bar{\eta}$ of $S$ has ample normal bundle, is
nondivisible and not constant up to translation. Then
$S(\mathcal{O}_{K,\Sigma})$ is not Zariski dense in $S$.

###### Proof.

We may assume that $S(\mathcal{O}_{K,\Sigma})$ is nonempty, so that the
generic fiber of $S$ is geometrically connected. Clearly
$\mathcal{C}_{\bar{\eta}}$ is not a symmetric power of degree $\geqslant 2$ of
a curve, so the big monodromy criterion from section 1.4 implies that
$\mathcal{C}\to S$ has big monodromy for most tuples of torsion characters.
Now apply D. ∎

######  Theorem 2.3.

Fix a polynomial $P\in\mathbb{Q}[z]$ of degree $d<(g-1)/2$ and an ample line
bundle $L$ on $A_{K}$. Then up to translation by points in
$A(\mathcal{O}_{K,\Sigma})$ there are only finitely many smooth nondivisible
subvarieties $X\subset A_{K}$ over $K$ with ample normal bundle and good
reduction outside $\Sigma$ that are not a product, that have Hilbert
polynomial $P$ with respect to $L$, and that satisfy (1.2) and (1.2) with
$e=\chi_{\textup{top}}(X)$.

###### Proof.

Let $\operatorname{Hilb}_{A}$ the Hilbert scheme of $A$ and
$H\subset\operatorname{Hilb}_{A}$ the open subset where the universal family
$\pi\colon\mathcal{X}\to\operatorname{Hilb}_{A}$ is a smooth morphism. With
notation as in lemma 2.1 let

$F\subset H(\mathcal{O}_{K,\Sigma})\cap\operatorname{Hilb}_{A_{K},P}^{\star}$

and $S$ an irreducible component of the Zariski closure of $F$ in $H$. It
suffices to show that there are a nonempty open subset $S^{\prime}_{K}\subset
S_{K}$, a morphism $a\colon S^{\prime}_{K}\to A_{K}$ and a subvariety
$X\subset A_{K}$ such that

$\mathcal{X}_{S^{\prime}_{K}}=X+a.$

By [JKLM22, cor. 4.8] the last property is equivalent to saying that
$\mathcal{X}_{\bar{\eta}}$ is constant up to translation (in loc. cit. this is
stated for families over an algebraically closed field, but the argument works
in general). Let $S^{\prime}_{K}\subset S_{K}^{\textup{reg}}$ be a nonempty
open subset and $S^{\prime}\subset S$ the maximal open subset with generic
fiber $S^{\prime}_{K}$, that is,

$S^{\prime}\;:=\;S\;\smallsetminus\;\textup{Zariski closure of
$(S_{K}\smallsetminus S^{\prime}_{K})$ in $S$}.$

By construction $S^{\prime}(\mathcal{O}_{K,\Sigma})$ is Zariski-dense in
$S^{\prime}$. Arguing by contradiction, suppose $\mathcal{X}_{\bar{\eta}}$ is
not constant up to a translation (hence $\dim S_{K}>0$). We are going to show
that this implies that $S^{\prime}(\mathcal{O}_{K,\Sigma})$ is not Zariski
dense yielding the desired contradiction.

Suppose that $\mathcal{X}_{\bar{\eta}}$ is not a symmetric power of a curve.
Then the big monodromy criterion from section 1.4 implies that the family
$\mathcal{X}_{S^{\prime}}\to S^{\prime}$ has big monodromy for most tuples of
torsion characters. We conclude the proof by D.

Suppose that $\mathcal{X}_{\bar{\eta}}$ is a symmetric power of a curve. Up to
enlarging $\Sigma$ and shrinking $S^{\prime}_{K}$ we may assume that there are
a finite étale morphism $\nu\colon\tilde{S}\to S^{\prime}$ and a smooth
relative curve $\mathcal{C}\subset A_{\tilde{S}}$ over $\tilde{S}$ such that
the sum morphism induces an isomorphism

$\operatorname{Sym}^{d}_{\tilde{S}}\mathcal{C}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{X}_{\tilde{S}}=\mathcal{X}_{S^{\prime}}\times_{S^{\prime}}\tilde{S}.$

By the Chevalley-Weil theorem [CTZ22, cor. 1.3], there is a finite extension
$K^{\prime}$ of $K$ and a finite set of places $\Sigma^{\prime}$ of
$K^{\prime}$ containing those above $\Sigma$ such that induced map
$\tilde{S}(\mathcal{O}_{K^{\prime},\Sigma^{\prime}})\to
S^{\prime}(\mathcal{O}_{K,\Sigma})$ is surjective. The relative curve
$\mathcal{C}$ is not constant up to translation because otherwise
$\mathcal{X}_{\tilde{S}}$ would be so. Therefore lemma 2.2 implies that the
set $\tilde{S}(\mathcal{O}_{K^{\prime},\Sigma^{\prime}})$ is not Zariski
dense, thus $S^{\prime}(\mathcal{O}_{K,\Sigma})$ is not Zariski dense either.
∎

###  2.3. Proof of C in the general case

We go back to the notation of C, thus $A$ is an abelian variety of dimension
$g$ over $K$ equipped with an ample line bundle $L$. Pick $P\in\mathbb{Q}[z]$
of degree $d<(g-1)/2$ and consider the set $S$ of smooth subvarieties
$X\subset A$ over $K$ that are geometrically not a product, that have

  * •

ample normal bundle,

  * •

Hilbert polynomial $P$ with respect to $L$,

  * •

good reduction outside $\Sigma$,

and that satisfy (1.2) and (1.2) with
$e=\chi_{\textup{top}}(X)/|\operatorname{Stab}_{A}(X)|$.

Let $H\subset\operatorname{Hilb}_{A}$ be the locus of the Hilbert scheme of
$A$ over which the universal family $\mathcal{X}\to\operatorname{Hilb}_{A}$ is
smooth, with geometrically connected fibers and Hilbert polynomial $P$ with
respect to $L$. The function $H\to\mathbb{N}$,
$h\mapsto\omega_{\mathcal{X}_{h}}^{d}$ is locally constant, where
$\omega_{\mathcal{X}_{h}}$ is the canonical bundle of $\mathcal{X}_{h}$ and
$\omega_{\mathcal{X}_{h}}^{d}$ is its top self intersection. The scheme $H$ is
of finite type over $K$ thus the function
$h\mapsto\omega_{\mathcal{X}_{h}}^{d}$ takes finitely many values. For each
$h\in H$ the positive integer $\omega_{\mathcal{X}_{h}}^{d}$ is divisible by
$|\operatorname{Stab}(\mathcal{X}_{h})|$ hence the set

$\mathcal{G}:=\\{\operatorname{Stab}_{A}(\mathcal{X}_{h})\mid h\in H(K)\\}$

is finite. Up to enlarging $\Sigma$ we may suppose that it contains all the
places above the prime divisors of $\omega_{\mathcal{X}_{h}}^{d}$ for all
$h\in H$. Then for $G\in\mathcal{G}$ the abelian variety $A/G$ has dimension
$g$ and good reduction outside $\Sigma$. Similarly for all $h\in H(K)$ the
subvariety $X:=\mathcal{X}_{h}\subset A$ has good reduction outside $\Sigma$
if and only if the subvariety

$X/\operatorname{Stab}_{A}(X)\lhook\joinrel\longrightarrow
A/\operatorname{Stab}_{A}(X)$

does. Since $\mathcal{G}$ is finite it suffices to prove the statement for
varieties $X\in S$ such that $\operatorname{Stab}_{A}(X)$ equals one fixed
$G\in\mathcal{G}$. In this case pick an ample line bundle $M$ on the abelian
variety $B:=A/G$. The line bundle $\pi^{\ast}M$ on $A$ is ample where
$\pi\colon A\to B$ is the projection. Moreover the function
$H\to\mathbb{Q}[z]$ associating to $h\in H$ the Hilbert polynomial of
$\mathcal{X}_{h}$ with respect to $\pi^{\ast}M$ takes finitely many values.
Again by treating each polynomial separately we may suppose that all varieties
$X\in S$ have the same Hilbert polynomial $Q$ with respect to $\pi^{\ast}M$.
Then the subvarieties $X/G\hookrightarrow B$ are nondivisible and have Hilbert
polynomial $Q/|G|$ with respect to $M$. It follows from theorem 2.3 applied to
(the unique abelian scheme over $\mathcal{O}_{K,\Sigma}$ extending) $B$ that
the set

$\\{X/G\subset B\mid X\in S\\}/B(K)$

is finite, which concludes the proof. ∎

###  2.4. Proof of theorems A and B

Recall that by Matsusaka’s Big Theorem there is an integer $N\geqslant 1$
depending only on $P$ with the following property: For every smooth projective
variety $X$ over a field $k$, every ample line bundle $L$ on $X$ with Hilbert
polynomial $P$ and every $n\geqslant N$, the line bundle $L^{\otimes n}$ is
very ample and $\textup{H}^{i}(X,L^{\otimes n})=0$ for all $i\geqslant 1$. We
now proceed in the following steps:

_Step 1._ In the first step we show that the varieties in question are
parametrized by a $K$-scheme of finite type. Let $V$ be a $k$-vector space of
dimension $P(N)$. Consider the functor

$\mathcal{H}_{0}\colon(\textup{Sch}/K)\longrightarrow(\textup{Sets}),\quad
S\longmapsto(f\colon X\to S,\;\varphi\colon V\otimes_{K}\mathcal{O}_{S}\to
f_{\ast}\omega^{\otimes N}_{X/S},\;x\colon S\to X)$

where

  * •

$f$ is a proper smooth morphism of relative dimension $d=\deg P$ with
geometrically connected fibers and relative canonical bundle
$\omega_{X/S}=\det\Omega_{X/S}$,

  * •

$\varphi$ an isomorphism of $\mathcal{O}_{S}$-modules,

  * •

$x$ is a section of $S$,

such that the Hilbert polynomial of the fibers of $f$ with respect to
$\omega_{X/S}$ is $P$ and for each $s\in S$ the line bundle
$\omega_{X_{s}}^{\otimes n}$ on $X_{s}$ is very ample. Then $\mathcal{H}_{0}$
is represented by a scheme $H_{0}$ of finite type over $K$. Indeed the functor
$\mathcal{H}^{\prime}_{0}$ obtained from $\mathcal{H}_{0}$ by dropping the
datum of the section $x\colon S\to X$ is represented by a quasi-projective
variety $H^{\prime}_{0}$ over $K$; see [Vie95, th. 1.46]. Then the functor
$\mathcal{H}_{0}$ is represented by the universal family $H_{0}$ over
$H_{0}^{\prime}$. Let

$\mathcal{X}_{0}=H_{0}\times_{H^{\prime}_{0}}H_{0}\to H_{0}$

be the universal family. With this notation the universal section $x_{0}\colon
H_{0}\to\mathcal{X}_{0}$ is just the diagonal. The relative Picard scheme
$\operatorname{Pic}^{0}(\mathcal{X}_{0}/H_{0})$ is representable and we
consider its dual abelian scheme $\operatorname{Alb}(\mathcal{X}_{0}/H_{0})$.
The universal section $x_{0}$ permits to define the Albanese morphism

$a_{0}\colon\mathcal{X}_{0}\longrightarrow\operatorname{Alb}(X_{0}/H_{0}).$

The locus $H\subset H_{0}$ where $a_{0}$ is a closed embedding with ample
normal bundle is open. Let $\mathcal{X}=\mathcal{X}_{0}\times_{H_{0}}H$ be the
universal family over $H$, $x\colon H\to\mathcal{X}$ the universal section and
$a\colon\mathcal{X}\hookrightarrow\operatorname{Alb}(\mathcal{X}/H)$ the
Albanese morphism.

_Step 2._ In order to apply C we need a polarization on the Albanese and not
only on the variety itself. Note that line bundles on an Albanese embedded
variety do not necessarily extend to the whole Albanese. Nonetheless we can
define a natural polarization on $\operatorname{Alb}(\mathcal{X}/H)$. Let
$\mathcal{P}$ be the Poincaré bundle and $p$, $q$ the first and second
projection on $\mathcal{X}\times_{H}\operatorname{Pic}^{0}(\mathcal{X}/H)$.
Since the line bundle $p^{\ast}\omega_{X/S}$ is relatively ample with respect
to $q$, there exists an integer $n\geqslant N$ such that

$\textup{R}^{i}q_{\ast}(p^{\ast}\omega^{\otimes
n}_{\mathcal{X}/H}\otimes\mathcal{P})=0\quad\textup{for all}\quad i\geqslant
1.$

It follows that the coherent sheaf

$\mathcal{F}=q_{\ast}(p^{\ast}\omega^{\otimes
n}_{\mathcal{X}/H}\otimes\mathcal{P})$

over $\operatorname{Pic}^{0}(\mathcal{X}/H)$ is locally free and [Laz04, th.
6.3.48] implies that $\det\mathcal{F}^{\vee}$ is a line bundle which is
relatively ample with respect to $\operatorname{Pic}^{0}(\mathcal{X}/H)\to H$.
We then consider its dual polarization $\mathcal{L}$ on
$\operatorname{Alb}(\mathcal{X}/H)$. The function $H\to\mathbb{Q}[z]$
associating to $h\in H$ the Hilbert polynomial of $\mathcal{X}_{h}$ with
respect to $\mathcal{L}$ is locally constant. Since $H$ has finitely many
connected components, by treating each of them separately we may assume that
$H$ is _connected_. Therefore each fiber of $\mathcal{X}\to H$ has Hilbert
polynomial with respect to $\mathcal{L}$ equal to some fixed
$Q\in\mathbb{Q}[z]$.

_Step 3._ In the third step we are going to reduce to some fixed ambient
abelian variety. Let $S$ be the set of $h\in H(K)$ for which
$\operatorname{Alb}(\mathcal{X}_{h}/K)$ and
$\mathcal{X}_{h}\hookrightarrow\operatorname{Alb}(\mathcal{X}_{h}/K)$ have
good reduction outside $\Sigma$. As $H$ is connected, the degree of the
polarization $\mathcal{L}$ on $\operatorname{Alb}(\mathcal{X}/H)$ is constant
and equal to $\operatorname{rk}\pi_{\ast}\omega_{X/S}^{\otimes n}$ where
$\pi\colon\mathcal{X}\to S$ denotes the structure morphism. Then Faltings’
theorem for polarized abelian varieties [Fal83] implies

$|\\{(\operatorname{Alb}(\mathcal{X}_{h}/K),\mathcal{L}_{h})\mid h\in
S\\}/\cong|<+\infty$

where isomorphisms are understood to be over $K$. By treating each isomorphism
class separately, we may assume

$(\operatorname{Alb}(\mathcal{X}_{h}/K),\mathcal{L}_{h})\cong(A,L)\textup{ for
all $h\in S$}$

for a fixed polarized abelian variety $(A,L)$ over $K$. If we embed
$\mathcal{X}_{h}$ in $A$ via such an isomorphism, then the Hilbert polynomial
of $\mathcal{X}_{h}$ with respect to $L$ is $Q$ and we conclude by applying C.
∎

##  3\. Buildings and filtrations on reductive groups

In this section we gather some generalities about reductive groups that will
be used throughout the rest of the paper. Let $\mathbb{F}$ be a perfect field.

###  3.1. Reductive groups

In this paper a reductive group over $\mathbb{F}$ is an affine algebraic group
whose category of representations is semisimple. For a linear algebraic group
$G$ let $\operatorname{rad}G$ be its unipotent radical. Then $G$ is reductive
if and only if $\operatorname{rad}G$ is trivial. Note that $G$ is not
necessarily connected. Let $G$ be a reductive group over $\mathbb{F}$. Given a
cocharacter $\lambda\colon\mathbb{G}_{m}\to G$ consider the algebraic subgroup
$P_{\lambda}\subset G$ whose points with values in a $\mathbb{F}$-scheme $S$
are those $g\in G(S)$ such that the limit

$\lim_{t\to 0}\lambda(t)g\lambda(t)^{-1}$

exists, that is, the morphism $\lambda g\lambda^{-1}\colon\mathbb{G}_{m}\times
S\to G$ extends to $\mathbb{A}^{1}\times S$. The connected component of
$P_{\lambda}$ is a parabolic subgroup of $G^{\circ}$. Consider the subgroup
$U_{\lambda}$ of $P_{\lambda}$ made of those $g$ for which $\lim_{t\to
0}\lambda(t)g\lambda(t)^{-1}$ is the neutral element of $G$. The subgroup
$U_{\lambda}$ is connected by definition and coincides with the unipotent
radical $\operatorname{rad}P_{\lambda}$ of $P_{\lambda}$. If $L_{\lambda}$ is
the centralizer of the image of $\lambda$ in $G$, then the projection onto
$P_{\lambda}/\operatorname{rad}P_{\lambda}$ induces an isomorphism

$L_{\lambda}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}P_{\lambda}/\operatorname{rad}P_{\lambda}.$

See [Mar03, prop. 5.2] for these facts. When $G$ is connected, parabolic
subgroups of $G$ are parametrized by a proper smooth $\mathbb{F}$-scheme
$\operatorname{Par}(G)$. Its Stein factorization
$\operatorname{Type}(G)=\operatorname{Spec}\Gamma(\operatorname{Par}(G),\mathcal{O}_{\operatorname{Par}(G)})$
is finite étale over $\mathbb{F}$ and the preimage $\operatorname{Par}_{t}(G)$
of $t\in\operatorname{Type}(G)(\mathbb{F})$ via the natural map
$\operatorname{Par}(G)\to\operatorname{Type}(G)$ is a geometrically connected
component. A parabolic subgroup $P\subset G$ is said of type $t(P)=t$ if the
point $[P]\in\operatorname{Par}(G)(\mathbb{F})$ defined by $P$ lies in
$\operatorname{Par}_{t}(G)$.

###  3.2. The rational building

We recall the notion of the rational (or vectorial) building of a reductive
group $G$ over $\mathbb{F}$; when $G$ is connected this can be found in
[Rou81, §2]. Write $X_{\ast}(G)$ for the set of cocharacters of $G$.

######  Definition 3.1.

The _rational building_ $\mathcal{B}(G,\mathbb{F})$ of $G$ is the quotient

$\mathcal{B}(G,\mathbb{F})\;:=\;(X_{\ast}(G)\times(\mathbb{N}\smallsetminus\\{0\\}))/\sim$

where $(\lambda,n)\sim(\lambda^{\prime},n^{\prime})$ if there is $g\in
P_{\lambda}(\mathbb{F})$ such that $\lambda^{\prime
n}=g\lambda^{n^{\prime}}g^{-1}$.

The equivalence class of $(\lambda,n)$ is written $[\tfrac{\lambda}{n}]$. For
$a,b\in\mathbb{Z}$ with $b\geqslant 1$ and $x=[\tfrac{\lambda}{n}]$ write
$\tfrac{a}{b}x=[\tfrac{\lambda^{a}}{bn}]$. The group $G(\mathbb{F})$ acts on
the rational building by conjugating cocharacters. The stabilizer of a point
$x\in\mathcal{B}(G,\mathbb{F})$ is $P_{x}(\mathbb{F})$ where
$P_{x}:=P_{\lambda}$ and $x=[\tfrac{\lambda}{n}]$. We write $t(x)$ for the
type of the parabolic subgroup $P_{x}^{\circ}\subset G^{\circ}$. The
construction of the rational building is functorial: for a morphism of
algebraic groups $f\colon G\to G^{\prime}$ with $G^{\prime}$ reductive,
composing with $f$ gives rise to map

$f_{\ast}\colon\mathcal{B}(G,\mathbb{F})\longrightarrow\mathcal{B}(G^{\prime},\mathbb{F}).$

######  Lemma 3.2.

The natural map
$i\colon\mathcal{B}(G^{\circ},\mathbb{F})\to\mathcal{B}(G,\mathbb{F})$ is
bijective.

###### Proof.

Since any cocharacter takes values in $G^{\circ}$ the map $i$ is clearly
surjective. To see that $i$ is injective let $(\lambda,n)$ and
$(\lambda^{\prime},n^{\prime})\in X_{\ast}(G)$ be couples made of a cocharater
and a positive integer such that there is $g\in P_{\lambda}(\mathbb{F})$ with
$\lambda^{\prime n}=g\lambda^{n^{\prime}}g^{-1}$. We have to prove that we can
choose such an element $g$ in the connected component of $P_{\lambda}$. The
image $\bar{g}$ of $g$ in $L_{\lambda}\cong
P_{\lambda}/\operatorname{rad}P_{\lambda}$ commutes with $\lambda$ thus

$\lambda^{\prime
n}=g\lambda^{n^{\prime}}g^{-1}=g\bar{g}^{-1}\lambda^{n^{\prime}}\bar{g}g^{-1}.$

This concludes the proof because
$g\bar{g}^{-1}\in\operatorname{rad}P_{\lambda}\subset P_{\lambda}^{\circ}$. ∎

The map $f_{\ast}$ is injective resp. surjective if and only if $f$ has finite
kernel resp. is surjective. The _apartment_ associated with a maximal split
torus $T\subset G$ is the subset
$\mathcal{B}(T,\mathbb{F})=X_{\ast}(T)\otimes_{\mathbb{Z}}\mathbb{Q}\subset\mathcal{B}(G,\mathbb{F})$.

######  Example 3.3.

Suppose $G=\operatorname{GL}(V)$ for a finite-dimensional $\mathbb{F}$-vector
space $V$. For a point
$x=[\tfrac{\lambda}{n}]\in\mathcal{B}(\operatorname{GL}(V),\mathbb{F})$ let
$V=\bigoplus_{i\in\mathbb{Z}}V_{i}$ be the decomposition of $V$ in isotypical
factors under the action of $\lambda$; that is $V_{i}$ is the subspace where
$\lambda(t)$ acts as $v\mapsto t^{i}v$. For $p\in\mathbb{Q}$, set

$\textup{F}^{p}V=\bigoplus_{i\geqslant pn}V_{i}.$

This defines a separated exhaustive nonincreasing filtration $F^{\bullet}V$ on
$V$ indexed by rational numbers. This procedure sets up a
$\operatorname{GL}(V,\mathbb{F})$-equivariant bijection

$\mathcal{B}(\operatorname{GL}(V),\mathbb{F})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\left\\{\begin{array}[]{c}\textup{separated
exhaustive nonincreasing filtrations }\\\ \textup{of $V$ indexed by rational
numbers}\end{array}\right\\}.$

Let $G=\operatorname{GO}(V,\omega)$ resp. $G=\operatorname{GSp}(V,\omega)$ for
a nondegenerate pairing $\omega$ on $V$ which is symmetric resp. alternating.
Then
$\mathcal{B}(G,\mathbb{F})\subset\mathcal{B}(\operatorname{GL}(V),\mathbb{F})$
is the subset of _self-dual_ filtrations, i.e. such that there is
$c\in\mathbb{Q}$ such that $(F^{\alpha}V)^{\bot}=F^{c-\alpha}V$ for each
$\alpha\in\mathbb{Q}$.

######  Example 3.4.

(1) For reductive groups $G_{1}$, $G_{2}$ over $\mathbb{F}$ we have

$\mathcal{B}(G_{1}\times
G_{2},\mathbb{F})=\mathcal{B}(G_{1},\mathbb{F})\times\mathcal{B}(G_{2},\mathbb{F}).$

(2) Given a finite extension $\mathbb{F}^{\prime}$ of $\mathbb{F}$ and
$G^{\prime}$ a reductive group over $\mathbb{F}^{\prime}$ we have

$\mathcal{B}(G,\mathbb{F})=\mathcal{B}(G^{\prime},\mathbb{F}^{\prime})$

where $G:=\operatorname{Res}_{\mathbb{F}^{\prime}/\mathbb{F}}G^{\prime}$ is
the Weil restriction.

###  3.3. Filtrations

We will need a similar description for points in a general reductive group in
terms of filtrations. In order to do this let $G$ be an affine algebraic group
over $\mathbb{F}$. Recall that a representation of $G$ is a
$\mathbb{F}$-vector space $V$ together with the structure of a comodule over
the Hopf algebra $\mathbb{F}[G]:=\Gamma(G,\mathcal{O}_{G})$. Such a structure
is given by a map $\sigma\colon V\to V\otimes\mathbb{F}[G]$ called
_comultiplication_ henceforth. When $V$ is finite-dimensional this is
equivalent to the datum of a morphism of algebraic groups
$G\to\operatorname{GL}(V)$. Let $\textup{Rep}(G)$ be the category of finite-
dimensional representations. A _filtration functor_ on $\textup{Rep}(G)$ is a
collection $F^{\bullet}=(F^{\alpha})_{\alpha\in\mathbb{Q}}$ of exact functors
$F^{\alpha}\colon\textup{Rep}(G)\to\operatorname{Vect}_{\mathbb{F}}$ such that

  * (F1)

for each $V\in\textup{Rep}(G)$ the collection
$F^{\bullet}V=(F^{\alpha}V)_{\alpha\in\mathbb{Q}}$ is a non-increasing,
exhausting and separated filtration on $V$;

  * (F2)

the associated graded functor $\operatorname{gr}^{\bullet}$ is exact;

  * (F3)

for $V,W\in\textup{Rep}(G)$ and $\alpha\in\mathbb{Q}$,

$F^{\alpha}(V\otimes W)=\sum_{\alpha=\beta+\gamma}F^{\beta}V\otimes
F^{\gamma}W.$

Note that in fact there is an integer $n\geqslant 1$ for which
$\operatorname{gr}^{\alpha}=0$ if $\alpha n\not\in\mathbb{Z}$: this follows
from the existence of a tensor generator for $\textup{Rep}_{\mathbb{F}}(G)$,
see [DMOS82, Tannakian categories, prop. 2.20 (b)]. This is what is called an
_exact $\otimes$-filtration_ in [SR72, IV.2.1] except for the fact that
filtrations are only indexed by integers in there. For instance a cocharacter
$\lambda\colon\mathbb{G}_{m}\to G$ determines a filtration functor by example
3.3. Conversely, every filtration functor with $\operatorname{gr}^{\alpha}=0$
if $\alpha\not\in\mathbb{Z}$ arise in this way when
$\operatorname{char}(\mathbb{F})=0$ [SR72, prop. IV.2.2.2]. The datum of a
filtration functor is encoded by a suitable filtration induced on
$\mathbb{F}[G]$. More precisely, let
$\mu\colon\mathbb{F}[G]\to\mathbb{F}[G]\otimes\mathbb{F}[G]$ be the
comultiplication of the Hopf algebra $\mathbb{F}[G]$ and

${\nu\colon\mathbb{F}[G]\otimes\mathbb{F}[G]}$${(\mathbb{F}[G]\otimes\mathbb{F}[G])\otimes(\mathbb{F}[G]\otimes\mathbb{F}[G])}$${\mathbb{F}[G]\otimes\mathbb{F}[G]\otimes\mathbb{F}[G]}$${a\otimes
f\otimes b\otimes g}$${a\otimes b\otimes fg.}$$\scriptstyle{\mu\otimes\mu}$

######  Definition 3.5.

A filtration
$\mathbb{F}[G]^{\bullet}=(\mathbb{F}[G]^{\alpha})_{\alpha\in\mathbb{Q}}$ on
$\mathbb{F}[G]$ is _Hopf_ if it is non-increasing, exhausting, separated and
for all $\alpha\in\mathbb{Q}$ we have:

(HF1) $\displaystyle\mathbb{F}[G]^{\alpha}$
$\displaystyle\;=\;\mu^{-1}(\mathbb{F}[G]\otimes\mathbb{F}[G]^{\alpha})$ (HF2)
$\displaystyle\sum_{\alpha=\beta+\gamma}\mathbb{F}[G]^{\beta}\otimes\mathbb{F}[G]^{\gamma}$
$\displaystyle\;=\;\nu^{-1}(\mathbb{F}[G]\otimes\mathbb{F}[G]\otimes\mathbb{F}[G]^{\alpha}).$

Let $X=\operatorname{Spec}A$ be a variety over $\mathbb{F}$ with a right
action of $G$. The action is defined by an $\mathbb{F}$-algebra morphism
$\sigma\colon A\to A\otimes\mathbb{F}[G]$. With $\sigma$ as comultiplication
$A$ is a representation of $G$ and as such is an increasing union of finite-
dimensional representations. Given a filtration functor $F^{\bullet}$ on
$\textup{Rep}(G)$, the $\mathbb{F}$-algebra $A$ inherits a filtration
$F^{\bullet}A$. Let $G$ act on itself by right multiplication; the arising
representation on $\mathbb{F}[G]$ is called the _regular representation_. The
preceding discussion with $A=\mathbb{F}[G]$ yields a filtration
$F^{\bullet}\mathbb{F}[G]$ on $\mathbb{F}[G]$ which is Hopf. Indeed (HF1)
holds because the group law $G\times G\to G$ is equivariant with respect the
right action $(x,y)g=(x,yg)$ of $G$ on $G\times G$. On other hand (HF2) holds
because the filtration functor $F^{\bullet}$ is compatible with tensor
product.

Conversely, let $\mathbb{F}[G]^{\bullet}$ be a Hopf filtration on
$\mathbb{F}[G]$ and $V$ a representation with comultiplication $\sigma\colon
V\to V\otimes\mathbb{F}[G]$. For $\alpha\in\mathbb{Q}$ set

$F^{\alpha}V:=\sigma^{-1}(V\otimes\mathbb{F}[G]^{\alpha}).$

Note that the construction of the filtration
$F^{\bullet}V=(F^{\alpha}V)_{\alpha\in\mathbb{Q}}$ is functorial with respect
to $G$-equivariant $\mathbb{F}$-linear maps.

######  Proposition 3.6.

The collection of functors $F^{\alpha}$ is a filtration functor on
$\textup{Rep}(G)$. Moreover, the above construction are mutually inverse and
set up a bijection

$\\{\textup{filtration functors on
$\textup{Rep}(G)$}\\}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\\{\textup{Hopf
filtrations on $\mathbb{F}[G]$}\\}.$

###### Proof.

For a morphism $\varphi\colon V\to W$ of representations we have, for all
$\alpha\in\mathbb{Q}$,

$F^{\alpha}V=\varphi^{-1}(F^{\alpha}W)\quad\textup{if $\varphi$ is
injective},\qquad F^{\alpha}W=\varphi(F^{\alpha}V)\quad\textup{if $\varphi$ is
surjective}.$

It follows that the functors
$F^{\alpha}\colon\textup{Rep}(G)\to\operatorname{Vect}_{\mathbb{F}}$ are exact
and that the collection of functors
$F^{\bullet}=(F^{\alpha})_{\alpha\in\mathbb{Q}}$ satisfies (F1) and (F2). To
prove (F3) consider representations $V$ and $W$ with comultiplications
$\sigma_{V}\colon V\to V\otimes\mathbb{F}[G]$ and $\sigma_{W}\colon W\to
W\otimes\mathbb{F}[G]$ respectively. The comultiplication on $V\otimes W$ is
then defined by

${\sigma_{V\otimes W}\colon V\otimes W}$${V\otimes\mathbb{F}[G]\otimes
W\otimes\mathbb{F}[G]}$${V\otimes
W\otimes\mathbb{F}[G]}$$\scriptstyle{\sigma_{V}\otimes\sigma_{W}}$

where the latter arrow is $v\otimes f\otimes w\otimes g\mapsto v\otimes
w\otimes fg$. Consider the following commutative diagram:

${V\otimes W}$${V\otimes
W\otimes\mathbb{F}[G]}$${(V\otimes\mathbb{F}[G])\otimes(W\otimes\mathbb{F}[G])}$${(V\otimes\mathbb{F}[G])\otimes(W\otimes\mathbb{F}[G])\otimes\mathbb{F}[G]}$${V\otimes
W\otimes\mathbb{F}[G]\otimes\mathbb{F}[G]}$${V\otimes
W\otimes\mathbb{F}[G]\otimes\mathbb{F}[G]\otimes\mathbb{F}[G]}$$\scriptstyle{\sigma_{V\otimes
W}}$$\scriptstyle{\sigma_{V}\otimes\sigma_{W}}$$\scriptstyle{\sigma_{V}\otimes\sigma_{W}\otimes\operatorname{id}}$$\scriptstyle{s}$$\scriptstyle{s\otimes\operatorname{id}}$$\scriptstyle{\operatorname{id}_{V\otimes
W}\otimes\sigma_{\mathbb{F}[G]\otimes\mathbb{F}[G]}}$

where $s(v\otimes f\otimes w\otimes g)=v\otimes w\otimes f\otimes g$. Then by
(HF2) we have:

$(V\otimes W)^{\alpha}=\sum_{\alpha=\beta+\gamma}V^{\beta}\otimes W^{\gamma}.$

Therefore $F^{\bullet}$ is a filtration functor. Property (HF1) implies
$F^{\alpha}\mathbb{F}[G]=\mathbb{F}[G]^{\alpha}$ for each
$\alpha\in\mathbb{Q}$, thus the map $H$ is surjective. In order to prove that
$H$ is injective, let now $F^{\bullet}$ be a filtration functor on
$\textup{Rep}(G)$. Given a representation $V$ the comultiplication
$V\to\mathbb{F}[G]\times V$ is $G$-equivariant with $G$ acting on
$V\otimes\mathbb{F}[G]$ trivially on $V$ and by left-multiplication on
$\mathbb{F}[G]$. Since the comultiplication is injective, the exactness of
$F^{\bullet}$ implies $F^{\alpha}V=\sigma^{-1}(V\otimes
F^{\alpha}\mathbb{F}[G])$ for all $\alpha\in\mathbb{Q}$ and all finite-
dimensional representations $V$. This shows that $H$ is injective and
concludes the proof. ∎

######  Proposition 3.7.

Let $f\colon G\to H$ be a morphism of affine algebraic groups over
$\mathbb{F}$. Then the natural map

$\\{\textup{filtration functors on
$\textup{Rep}(G)$}\\}\;\longrightarrow\;\\{\textup{filtration functors on
$\textup{Rep}(H)$}\\}$

is injective if $f$ is injective, and is bijective if the morphism
$G^{\circ}\to H^{\circ}$ induced by $f$ is an isomorphism.

###### Proof.

Suppose $f$ injective. Then $f$ is a closed immersion and the induced morphism
of $\mathbb{F}$-algebras $\varphi\colon\mathbb{F}[H]\to\mathbb{F}[G]$ is
surjective. By proposition 3.6 it suffices to show that the corresponding map

$\\{\textup{Hopf functors on
$\textup{Rep}(G)$}\\}\;\longrightarrow\;\\{\textup{Hopf functors on
$\textup{Rep}(H)$}\\}$

is injective. The above map associates to a Hopf filtration
$\mathbb{F}[G]^{\bullet}$ on $\mathbb{F}[G]$, the filtration
$\mathbb{F}[H]^{\bullet}$ on $\mathbb{F}[H]$ induced by the action of right
multiplication of $G$ on $H$. Now since $f$ is $G$-equivariant with respect to
right multiplication of $G$, we deduce that $\mathbb{F}[G]^{\bullet}$ is the
image of $\mathbb{F}[H]^{\bullet}$ via $\varphi$. Thus
$\mathbb{F}[G]^{\bullet}$ can be recovered $\mathbb{F}[H]^{\bullet}$, showing
that desired injectivity.

For the second statement, it suffices to show that the natural map

$\\{\textup{filtration functors on
$\textup{Rep}(G^{\circ})$}\\}\;\longrightarrow\;\\{\textup{filtration functors
on $\textup{Rep}(G)$}\\}$

is surjective, as we already know that it is injective. Given a filtration
functor $F^{\bullet}$ on $\textup{Rep}(G)$ up to rescaling we may assume that
$\operatorname{gr}^{\alpha}=0$ if $\alpha\not\in\mathbb{Z}$. Then
$F^{\bullet}$ comes from a cocharacter. But a cocharacter has necessarily
values in $G^{\circ}$ as $\mathbb{G}_{m}$ is connected. Thus $F^{\bullet}$
comes from a filtration functor on $G^{\circ}$. ∎

Given a filtration functor $F^{\bullet}$ consider the algebraic subgroup
$P\subset G$ whose points with values in a $\mathbb{F}$-scheme $S$ are those
$g\in G(S)$ such that

$g(F^{\alpha}V\otimes_{\mathbb{F}}\mathcal{O}_{S})=F^{\alpha}V\otimes_{\mathbb{F}}\mathcal{O}_{S}\quad\textup{for
all $\alpha\in\mathbb{Q}$}.$

When $G$ is a reductive group, the subgroup $P$ is parabolic [SR72, Prop.
IV.2.2.5] and, if $F^{\bullet}$ is given by a cocharacter $\lambda$, then $P$
is the subgroup $P_{\lambda}$ defined in the previous section. The filtration
functors associated to cocharacters $\lambda$, $\lambda^{\prime}$ then
coincide if and only if $\lambda^{\prime}$ is conjugate to $\lambda$ under
$P_{\lambda}$. Summing up:

######  Corollary 3.8.

Suppose $\operatorname{char}(\mathbb{F})=0$ and $G$ reductive. Then the above
constructions set up bijections

$\mathcal{B}(G,\mathbb{F})\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\\{\textup{filtration
functors on
$\textup{Rep}(G)$}\\}\stackrel{{\scriptstyle\sim}}{{\longleftrightarrow}}\\{\textup{Hopf
filtrations on $\mathbb{F}[G]$}\\}.$

###  3.4. Projection onto Levi quotients

Let $G$ be a reductive group and $Q\subset G^{\circ}$ a parabolic subgroup of
its connected component. For any $x\in\mathcal{B}(G,\mathbb{F})$ there is a
cocharacter $\lambda$ with values in $Q$ such that $x=[\tfrac{\lambda}{n}]$
for some $n\geqslant 1$: Indeed $P^{\circ}_{x}\cap Q$ contains a maximal split
torus, and by [BT65, prop. 11.6] all maximal split tori are conjugate in
$P_{x}^{\circ}$. Let

$\pi_{Q}\colon\quad
Q\;\longrightarrow\;Q^{\textup{ss}}\;:=\;Q/\operatorname{rad}Q$

be the projection onto the Levi quotient. Then the point

$\operatorname{pr}_{Q}(x):=[\tfrac{\pi_{Q}\circ\lambda}{n}]\in\mathcal{B}(Q^{\textup{ss}},\mathbb{F})$

does not depend on the chosen $\lambda$. The so-defined map
$\operatorname{pr}_{Q}\colon\mathcal{B}(G,\mathbb{F})\to\mathcal{B}(Q^{\textup{ss}},\mathbb{F})$
is $Q(\mathbb{F})$-equivariant and surjective.

######  Example 3.9.

Let $V$ be a finite-dimensional $\mathbb{F}$-vector space and
$G=\operatorname{GL}(V)$. The parabolic subgroup $Q$ then is the stabilizer of
a flag

$W_{0}=0\subset W_{1}\subset\cdots\subset W_{n}\subset W_{n+1}=V.$

The reductive quotient $Q^{\textup{ss}}$ is identified with
$\prod_{i=0}^{n}\operatorname{GL}(\bar{W}_{i})$ where
$\bar{W}_{i}=W_{i+1}/W_{i}$. If $F^{\bullet}V$ is the filtration of $V$
induced by $x$, then the point $\operatorname{pr}_{Q}(x)$ can be seen as the
tuple of filtrations $(F^{\bullet}\bar{W}_{i})_{i}$ where

$F^{\alpha}\bar{W}_{i}=F^{\alpha}V\cap W_{i+1}\;/\;F^{\alpha}V\cap
W_{i}\quad\text{for $\alpha\in\mathbb{Q}$}.$

######  Remark 3.10.

In general seeing points of the building as Hopf filtrations, the Hopf
filtration on $\mathbb{F}[Q^{\textup{ss}}]$ corresponding to
$\operatorname{pr}_{Q}(x)$ is obtained as follows: First take the image under
the surjection $\mathbb{F}[G]\to\mathbb{F}[Q]$ of the filtration on
$\mathbb{F}[G]$ given by $x$; then intersect the resulting filtration with
$\mathbb{F}[Q^{\textup{ss}}]\subset\mathbb{F}[Q]$.

Fibers of $\operatorname{pr}_{Q}$ are described in the following:

######  Lemma 3.11.

For $x\in\mathcal{B}(G,\mathbb{F})$ the map
$\operatorname{rad}Q(\mathbb{F})\to\mathcal{B}(G,\mathbb{F})$, $g\mapsto gx$
has values in $\operatorname{pr}_{Q}^{-1}(\operatorname{pr}_{Q}(x))$ and
induces a bijection

$f\colon(\operatorname{rad}Q/\operatorname{rad}Q\cap
P_{x})(\mathbb{F})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{pr}_{Q}^{-1}(\operatorname{pr}_{Q}(x)).$

In particular we have
$\operatorname{Stab}_{Q^{\textup{ss}}}(\operatorname{pr}_{Q}(x))=(Q\cap
P_{x})/(\operatorname{rad}Q\cap P_{x})$.

###### Proof.

First of all notice that the group $\operatorname{rad}Q\cap P_{x}$ is
unipotent thus the vanishing of the first Galois cohomology group yields

$(\operatorname{rad}Q/\operatorname{rad}Q\cap
P_{x})(\mathbb{F})=\operatorname{rad}Q(\mathbb{F})/(\operatorname{rad}Q(\mathbb{F})\cap
P_{x}(\mathbb{F})).$

The $Q(\mathbb{F})$-equivariance of $\operatorname{pr}_{Q}$ implies that $f$
has values in $\operatorname{pr}_{Q}^{-1}(\operatorname{pr}_{Q}(x))$ and is
injective. To prove that $f$ is surjective let
$x^{\prime}\in\operatorname{pr}_{Q}^{-1}(\operatorname{pr}_{Q}(x))$ and
$L,L^{\prime}\subset Q$ Levi factors such that
$\mathcal{B}(L,\mathbb{F}),\mathcal{B}(L^{\prime},\mathbb{F})\subset\mathcal{B}(G,\mathbb{F})$
contain respectively $x$ and $x^{\prime}$. Since all Levi factors of $Q$ are
conjugated under $\operatorname{rad}Q$ there is
$u\in\operatorname{rad}Q(\mathbb{F})$ such that $L=uL^{\prime}u^{-1}$.
Therefore $ux^{\prime}$ belongs to $\mathcal{B}(L,\mathbb{F})$ and

$\operatorname{pr}_{Q}(ux^{\prime})=\operatorname{pr}_{Q}(x^{\prime})=\operatorname{pr}_{Q}(x).$

The projection $\pi_{Q}\colon Q\to Q^{\textup{ss}}$ induces an isomorphism
$L\cong Q^{\textup{ss}}$ thus the composite map

$\mathcal{B}(L,\mathbb{F})\lhook\joinrel\longrightarrow\mathcal{B}(G,\mathbb{F})\stackrel{{\scriptstyle\operatorname{pr}_{Q}}}{{\longrightarrow}}\mathcal{B}(Q^{\textup{ss}},\mathbb{F})$

is a bijection. Therefore $x=ux^{\prime}$ as desired. ∎

###  3.5. Positivity

Let $V$ be a finite-dimensional $\mathbb{F}$-vector space,
$y\in\mathcal{B}(\operatorname{GL}(V),\mathbb{F})$ and $F^{\bullet}V$ the
filtration on $V$ associated to $y$ as in example 3.3. The _weight_ of $y$ (or
equivalently of the filtration $F^{\bullet}V$) is the rational number

$\operatorname{wt}_{V}(y):=\frac{1}{\dim V}\langle
x,\det\rangle=\sum_{\alpha\in\mathbb{Q}}\alpha\frac{\dim\operatorname{gr}^{\alpha}V}{\dim
V}.$

Let $Q\subset G^{\circ}$ be a parabolic subgroup. The adjoint representation
of $Q$ restricts to a representation $\rho\colon
Q\to\operatorname{GL}(\operatorname{Lie}\operatorname{rad}Q)$ because the
unipotent radical is a normal subgroup. The determinant of $\rho$ is trivial
on $\operatorname{rad}Q$ thus it defines a character

$\delta_{Q}\colon Q^{\textup{ss}}\longrightarrow\mathbb{G}_{m}$

called the _modular character_ henceforth.111This differs from [LS20] where
the modular character is the inverse of the above. Let
$x\in\mathcal{B}(G,\mathbb{F})$ and let $L\subset G$ be a Levi factor such
that $x\in\mathcal{B}(L,\mathbb{F})$. Unwinding the definition we find that
the weight of the point
$\rho_{\ast}x\in\mathcal{B}(\operatorname{GL}(\operatorname{Lie}\operatorname{rad}Q),\mathbb{F})$
is

$\operatorname{wt}_{V}(\rho_{\ast}x)=\langle\operatorname{pr}_{Q}(x),\delta_{Q}\rangle,$

where $V=\operatorname{Lie}\operatorname{rad}Q$. Note that the right-hand side
of the previous identity does not depend on the chosen $L$.

######  Definition 3.12.

The point $x$ is _positive with respect to $Q$_ if
$\langle\operatorname{pr}_{Q}(x),\delta_{Q}\rangle\geqslant 0$. In this case
we write $(x,Q)\succeq 0$. We will as well say that $x$ is $Q$-positive or
that the couple $(x,Q)$ is positive.

######  Example 3.13.

Let $G=\operatorname{GL}(V)$ for some finite-dimensional $\mathbb{F}$-vector
space $V$ and let $Q\subset G$ be the parabolic subgroup stabilizing a flag

$0=W_{0}\subset W_{1}\subset\cdots\subset W_{r}\subset W_{r+1}=V$

Then $Q^{\textup{ss}}=\prod_{i=0}^{r}\operatorname{GL}(\bar{W}_{i})$ where
$\bar{W}_{i}=W_{i+1}/W_{i}$. Let $x\in\mathcal{B}(G,\mathbb{F})$, and let
$F^{\bullet}V$ be the associated filtration. Since the building is compatible
with products by example 3.4, we have

$\operatorname{pr}_{Q}(x)\;=\;(\bar{x}_{0},\dots,\bar{x}_{r})\quad\text{with}\quad\bar{x}_{i}\;\in\;\mathcal{B}(\operatorname{GL}(\bar{W}_{i}),\mathbb{F}).$

The filtration on $\bar{W}_{i}$ defined by $\bar{x}_{i}$ is given by

$F^{\alpha}\bar{W}_{i}\;=\;(W_{i+1}\cap F^{\alpha}V)/(W_{i}\cap
F^{\alpha}V)\quad\text{for}\quad\alpha\in\mathbb{Q}.$

With this notation the point $x$ is $Q$-positive if and only if

$\sum_{i<j}\dim\bar{W}_{i}\dim\bar{W}_{j}(\operatorname{wt}(F^{\bullet}\bar{W}_{i})-\operatorname{wt}(F^{\bullet}\bar{W}_{j}))\geqslant
0.$

Indeed, as a vector space

$\operatorname{Lie}\operatorname{rad}Q\;=\;\bigoplus_{i<j}\operatorname{Hom}(\bar{W}_{j},\bar{W}_{i}),$

and hence

$\det(\operatorname{Lie}\operatorname{rad}Q)\;=\;\bigotimes_{i<j}\left(\det(\bar{W}_{j}^{\vee})\right)^{\otimes\dim\bar{W}_{i}}\otimes\left(\det(\bar{W}_{i})\right)^{\otimes\dim\bar{W}_{j}}.$

If we denote by $\delta_{i}\colon Q^{\textup{ss}}\to\mathbb{G}_{m}$ the
character defined by the action on $\det(\bar{W}_{i})$, it follows that

$\delta_{Q}\;=\;\sum_{i<j}\left(\dim\bar{W}_{j}\cdot\delta_{i}-\dim\bar{W}_{i}\cdot\delta_{j}\right)$

where we use additive notation for characters in the group
$\operatorname{Hom}(Q^{\textup{ss}},\mathbb{G}_{m})$. Here we have
$\langle\operatorname{pr}_{Q}(x),\delta_{i}\rangle=\dim\bar{W}_{i}\cdot\operatorname{wt}(F^{\bullet}\bar{W}_{i})$
by the definition of the weight of a filtration, hence

$\langle\operatorname{pr}_{Q}(x),\delta_{Q}\rangle\;=\;\sum_{i<j}\dim\bar{W}_{i}\dim\bar{W}_{j}\left(\operatorname{wt}(F^{\bullet}\bar{W}_{i})-\operatorname{wt}(F^{\bullet}\bar{W}_{j})\right)$

and the claimed characterisation of $Q$-positivity follows.

######  Example 3.14.

If $G$ is the product of reductive groups $G_{i}$ for $i\in I$, then
$x=(x_{i})_{i\in I}$ with $x_{i}\in\mathcal{B}(G_{i},\mathbb{F})$ and
$Q=\prod_{i\in I}Q_{i}$ with parabolic subgroups $Q_{i}\subset G_{i}^{\circ}$.
In this case

$\langle\operatorname{pr}_{Q}(x),\delta_{Q}\rangle\;=\;\sum_{i\in
I}\langle\operatorname{pr}_{Q_{i}}(x),\delta_{Q_{i}}\rangle.$

In particular, if $x$ is $Q$-positive, then $x_{i}$ is $Q_{i}$-positive for at
least one $i\in I$.

######  Example 3.15.

If $G=\operatorname{Res}_{\mathbb{F}^{\prime}/\mathbb{F}}G^{\prime}$ is the
Weil restriction of a reductive group $G^{\prime}$ over a finite extension
$\mathbb{F}^{\prime}$ of $\mathbb{F}$, then we have
$Q=\operatorname{Res}_{\mathbb{F}^{\prime}/\mathbb{F}}Q^{\prime}$ for some
parabolic subgroup $Q^{\prime}\subset G^{\prime\circ}$. The identification
$\mathcal{B}(G,\mathbb{F})=\mathcal{B}(G^{\prime},\mathbb{F}^{\prime})$ of
buildings then leads to

$\langle\operatorname{pr}_{Q}(x),\delta_{Q}\rangle\;=\;[\mathbb{F}^{\prime}:\mathbb{F}]\,\langle\operatorname{pr}_{Q^{\prime}}(x),\delta_{Q^{\prime}}\rangle.$

In particular we see that $x$ is $Q$-positive if and only if it is
$Q^{\prime}$-positive.

###  3.6. Full subgroups

Let $V$ be a finite-dimensional $\mathbb{F}$-vector space and
$G\subset\operatorname{GL}(V)$ a reductive subgroup. Any point
$x\in\mathcal{B}(G,\mathbb{F})$ defines a filtration on $V$ indexed by
rational numbers; we denote the filtration steps by $V_{x}^{\alpha}$ for
$\alpha\in\mathbb{Q}$ and the associated graded by

$\operatorname{gr}V_{x}\;=\;\bigoplus_{\alpha\in\mathbb{Q}}V_{x}^{(\alpha)}\quad\text{where}\quad
V_{x}^{(\alpha)}\;=\;V_{x}^{\alpha}/V_{x}^{<\alpha}.$

By construction the filtration is preserved by the stabilizer $Q_{x}\subset G$
of $x$. Recall that this stabilizer may be disconnected; its connected
component is the parabolic subgroup $Q_{x}^{\circ}=Q_{x}\cap G^{\circ}$ of
$G^{\circ}$ [Mar03, prop. 5.2]. Let $\operatorname{rad}Q_{x}$ be the unipotent
radical of this connected component; the associated Levi quotient
$Q_{x}^{\textup{ss}}=Q_{x}/\operatorname{rad}Q_{x}$ faithfully acts on
$\operatorname{gr}V_{x}$ in a way preserving each graded piece, i.e. we have a
natural embedding

$\varepsilon_{x}\colon\quad
Q_{x}^{\textup{ss}}\;\lhook\joinrel\longrightarrow\;\prod_{\alpha\in\mathbb{Q}}\operatorname{GL}(V_{x}^{(\alpha)}).$

For a finite subset $I\subset\mathbb{Q}$ we consider the graded vector space
$\operatorname{gr}_{I}V_{x}:=\bigoplus_{\alpha\in I}V_{x}^{(\alpha)}$ and
denote by

$\varepsilon_{x,I}\colon\quad
Q_{x}^{\textup{ss}}\;\longrightarrow\;\operatorname{gr}_{I}\operatorname{GL}(V_{x})\;:=\;\prod_{\alpha\in
I}\operatorname{GL}(V_{x}^{(\alpha)}),$

the composite of the morphism $\varepsilon_{x}$ with the projection to the
factors that are indexed by the subset $I$. For
$x,y\in\mathcal{B}(G,\mathbb{F})$ and $I\subset\mathbb{Q}$, let

$\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y)\subset\operatorname{Iso}(\operatorname{gr}_{I}V_{x},\operatorname{gr}_{I}V_{y})$

be the closed subvariety of all graded linear isomorphisms
$f\colon\operatorname{gr}_{I}V_{x}\to\operatorname{gr}_{I}V_{y}$ such that

$f(\operatorname{Im}\varepsilon_{x,I})f^{-1}=\operatorname{Im}\varepsilon_{y,I}.$

The stabilizer $Q_{x}$ acts freely on the transporter
$\operatorname{Tran}_{G}(x,y)$ by right multiplication.

######  Lemma 3.16.

If $\varepsilon_{x,I}$ and $\varepsilon_{y,I}$ are injective, then the
morphism

$\operatorname{Tran}_{G}(x,y)\longrightarrow\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y),\quad
g\longmapsto\operatorname{gr}_{I}g$

is invariant under the action of $\operatorname{rad}Q_{x}$ on the source and
factors through a closed immersion

$\operatorname{Tran}_{G}(x,y)/\operatorname{rad}Q_{x}\;\hookrightarrow\;\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y).$

###### Proof.

Immediate from the definitions. ∎

In particular, if the morphism
$\operatorname{Tran}_{G}(x,y)\to\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y)$
is surjective, then it is a principal bundle under $\operatorname{rad}Q_{x}$
and therefore surjective on $\mathbb{F}$-points by vanishing of the first
Galois cohomology group of smooth connected unipotent groups over perfect
fields.

######  Definition 3.17.

If $\mathbb{F}$ is algebraically closed, we say that the subgroup
$G\subset\operatorname{GL}(V)$ is _full_ if for every
$x,y\in\mathcal{B}(G,\mathbb{F})$ that are conjugate under
$\operatorname{GL}(V)(\mathbb{F})$, there exists a subset $I\subset\mathbb{Q}$
with the property that $\varepsilon_{x,I}$ and $\varepsilon_{y,I}$ are
embeddings and the natural morphism

$\operatorname{Tran}_{G}(x,y)\longrightarrow\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y)$

is surjective. If $\mathbb{F}$ is arbitrary, we say that the subgroup
$G\subset\operatorname{GL}(V)$ is _full_ if it becomes a full subgroup after
extending scalars to an algebraic closure of $\mathbb{F}$.

As we will see below in example 3.19 it is necessary to have $I\neq\mathbb{Q}$
in order to allow orthogonal and symplectic groups to be full.

######  Example 3.18.

Taking $x=y\in\mathcal{B}(G,\mathbb{F})$ to be the trivial cocharacter and
looking at the action scalar matrices, one sees that the determinant
$\det\colon\operatorname{GL}(V)\to\mathbb{G}_{m}$ must restrict to a
surjective character on every full subgroup. In particular the subgroup
$G=\operatorname{SL}(V)\subset\operatorname{GL}(V)$ is not full, although
$G=\operatorname{GL}(V)$ is full.

######  Example 3.19.

Let $\omega$ be a nondegenerate symmetric or alternating bilinear form on $V$,
and let

$G\;=\;\begin{cases}\operatorname{GSp}(V,\omega)&\text{if $\omega$ is
alternating},\\\ \operatorname{GO}(V,\omega)&\text{if $\omega$ is
symmetric}.\end{cases}$

Then $G\subset\operatorname{GL}(V)$ is a full subgroup. Indeed, by definition
we may assume that $\mathbb{F}$ is algebraically closed. Since any parabolic
subgroup of $G^{\circ}$ is the stabilizer of an isotropic flag, up to
relabelling indices the filtration induced by any
$x\in\mathcal{B}(G,\mathbb{F})$ has the form

$0=V_{x}^{0}\subset V_{x}^{1}\subset V_{x}^{2}\subset\cdots\subset
V_{x}^{r}\subset(V_{x}^{r})^{\perp}\subset\cdots\subset(V_{x}^{1})^{\perp}\subset(V_{x}^{0})^{\perp}=V$

with isotropic subspaces $V^{i}_{x}\subset V$ and $r=r(x)$ depending on $x$.
Here we allow that $(V_{x}^{r})^{\perp}=V_{x}^{r}$ but all the other
inclusions in the above filtration are assumed to be strict. On the quotient
$U_{x}:=(V_{x}^{r})^{\perp}/V_{x}^{r}$ the bilinear form $\omega$ induces a
nondegenerate bilinear form $\omega_{x}$ which is symmetric or alternating if
$\omega$ is so, and we put

$H_{x}\;:=\;\begin{cases}\operatorname{GSp}(U_{x},\omega_{x})&\text{if
$\omega$ is alternating},\\\ \operatorname{GO}(U_{x},\omega_{x})&\text{if
$\omega$ is symmetric}.\end{cases}$

Put $W_{x}^{i}:=V_{x}^{i}/V_{x}^{i-1}$ for $i=1,\dots,r$. Then

$\operatorname{gr}V_{x}\;=\;W_{x}^{1}\oplus\cdots\oplus W_{x}^{r}\oplus
U_{x}\oplus(W_{x}^{r})^{\vee}\oplus\cdots\oplus(W_{x}^{1})^{\vee}$

where by construction each direct summand sits in a different graded piece,
and the Levi quotient

$Q_{x}^{\textup{ss}}\;=\;\operatorname{GL}(W_{x}^{1})\times\cdots\times\operatorname{GL}(W_{x}^{r})\times
H_{x}$

acts on $\operatorname{gr}V_{x}$ in the natural way preserving the grading. If
we choose $I\subset\mathbb{Q}$ such that

$\operatorname{gr}_{I}V_{x}\;=\;W_{x}^{1}\oplus\cdots\oplus W_{x}^{r}\oplus
U_{x}$

is the direct sum of the first half of the graded pieces including the middle
piece, then

$\varepsilon_{x,I}\colon\quad
Q_{x}^{\textup{ss}}\;\hookrightarrow\;\operatorname{gr}_{I}\operatorname{GL}(V_{x})\;=\;\operatorname{GL}(W_{x}^{1})\times\cdots\times\operatorname{GL}(W_{x}^{r})\times\operatorname{GL}(U_{x})$

is an embedding because the action on the remaining graded pieces is
determined by duality. Now $x,y\in\mathcal{B}(G,\mathbb{F})$ be two points
conjugate under $\operatorname{GL}(V)(\mathbb{F})$. Then in particular the
associated filtrations have the same length, so they are of the above shape
with $r=r(x)=r(y)$. Then by definition any point of
$\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y)$ is a
direct sum $\bar{g}=\bar{g}_{1}\oplus\cdots\oplus\bar{g}_{r}\oplus\bar{h}$ of
isomorphisms

$\bar{g}_{i}\colon\quad
W_{x}^{i}\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;W_{y}^{i}\quad\text{and}\quad\bar{h}\colon\quad
U_{x}\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;U_{y}\quad\text{with}\quad\bar{h}H_{x}\bar{h}^{-1}=H_{y}.$

Now by Witt’s theorem [Art57, th. III.3.9], there exists an element $g\in
G(\mathbb{F})$ such that

  * •

$g(V_{x}^{i})\;=\;V_{y}^{i}$ and hence also
$g((V_{x}^{i})^{\perp})=(V_{y}^{i})^{\perp}$ for $i=1,\dots,r$, and

  * •

$g$ induces the given isomorphisms $\bar{g}_{1},\dots,\bar{g}_{r},\bar{h}$
between graded pieces.

Then $g\in\operatorname{Tran}_{G}(x,y)$ lifts the element
$\bar{g}\in\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x,y)$
as required.

Note that it is essential that we used only half of the graded pieces. To make
it simple suppose $x=y$ so we can drop the subscript $x$ in the notation
above. Assume moreover that the filtration associated with $x$ is made of one
single Lagrangian subspace $V^{1}=W$. In this case

$\varepsilon\colon
Q^{\textup{ss}}=\operatorname{GL}(W)\lhook\joinrel\longrightarrow\operatorname{GL}(W)\times\operatorname{GL}(W^{\vee}).$

The normalizer of $Q^{\textup{ss}}$ in
$\operatorname{GL}(W)\times\operatorname{GL}(W^{\vee})$ is strictly larger
than $Q^{\textup{ss}}$ as it contains the center
$\mathbb{G}_{m}\times\mathbb{G}_{m}$. Composing $\varepsilon$ with the
projection onto one of the two factors of
$\operatorname{GL}(W)\times\operatorname{GL}(W^{\vee})$ permits to get rid of
this problem.

##  4\. Realizations with extra structures

In this section we introduce the étale and de Rham realizations to which the
Lawrence-Venkatesh method will be applied, that we call respectively _étale
data_ and _de Rham data_. A particular accent is put on the extra symmetries
that these data satisfy, as they will be crucial for the method to work. We
formulate Faltings’ finiteness theorem in the context of semisimple pure étale
data and discuss the effect of semisimplification on de Rham data. Let
$\mathbb{F}$ be a field of characteristic $0$ and $p$ a prime number.

###  4.1. Semilinear automorphisms and Weil restriction

Let $Z$ be a finite étale scheme over $\mathbb{F}$. The $\mathbb{F}$-algebra
$R=\Gamma(Z,\mathcal{O}_{Z})$ is the product for $z\in Z$ of finite extensions
$\mathbb{F}_{z}$ of $\mathbb{F}$. An $\mathcal{O}_{Z}$-module $V$ on $Z$ is
determined by its fibers, thus it is equivalent to the datum for each $z\in Z$
of a $\mathbb{F}_{z}$-vector space $V_{z}$; in particular it is always locally
free. We say that a $\mathbb{F}$-linear endomorphism $f$ of $V$ is
_semilinear_ if there is an automorphism $\sigma$ of the $\mathbb{F}$-algebra
$R$ such that $f(\lambda x)=\sigma(\lambda)f(x)$ for any $x\in V$ and
$\lambda\in R$. In general $\sigma$ is not unique. Nonetheless when $f$ is
bijective there is a natural choice for $\sigma$: on the support of $V$ it is
uniquely determined by $f$ and outside it can be taken to be the identity.
When $V$ is a vector bundle on $Z$, semilinear automorphisms can be seen as
$\mathbb{F}$-points of the algebraic group
$\operatorname{GL}^{\textup{s}}_{Z/\mathbb{F}}(V)$ over $\mathbb{F}$
constructed as follows. First we have an inclusion

$\operatorname{Res}_{Z/\mathbb{F}}\operatorname{GL}(V)\subset\operatorname{GL}(V_{0})$

where $V_{0}$ is $V$ seen merely as $\mathbb{F}$-vector space. Then
$\operatorname{GL}^{\textup{s}}_{Z/\mathbb{F}}(V)$ is defined as the
normalizer of $\operatorname{Res}_{Z/\mathbb{F}}\operatorname{GL}(V)$ in
$\operatorname{GL}(V_{0})$. We have an exact sequence

(4.1)
$1\longrightarrow\operatorname{Res}_{Z/\mathbb{F}}\operatorname{GL}(V)\longrightarrow\operatorname{GL}^{\textup{s}}_{Z/\mathbb{F}}(V)\longrightarrow\operatorname{Aut}_{\mathbb{F}}(Z)$

of algebraic groups over $\mathbb{F}$. Conjugation by a semilinear
automorphism $f$ of $V$ induces an automorphism of $\operatorname{GL}(V)$ as a
$\mathbb{Q}_{p}$-scheme such that the following diagram commutes:

${\operatorname{GL}(V)}$${\operatorname{GL}(V)}$${Z}$${Z}$$\scriptstyle{g\mapsto
fgf^{-1}}$$\scriptstyle{\operatorname{Spec}\sigma}$

where $\sigma$ is the associated automorphism of $R$. We say that $f$
normalizes a closed subscheme $X\subset\operatorname{GL}(V)$ if $fXf^{-1}=X$.

The datum of an $Z$-scheme $X$ is equivalent to the datum for each $z\in Z$ of
an $\mathbb{F}_{z}$-scheme $X_{z}$; in particular all $Z$-schemes are flat. In
what follows we will adopt freely both point of view. For this reason we can
perform on group $Z$-schemes of finite type the usual operations available on
algebraic groups (quotients, normalizers, etc.). Moreover we have the
following relation between Weil restrictions

$\operatorname{Res}_{Z/\mathbb{F}}X=\prod_{z\in
Z}\operatorname{Res}_{\mathbb{F}_{z}/\mathbb{F}}X_{z}$

as soon as they exists (for example if $X$ is affine). We profit of the setup
to prove the following result that will be repeatedly used in this section:

######  Lemma 4.1.

Let $V$ be a vector bundle over $Z$, $G\subset\operatorname{GL}(V)$ a subgroup
$Z$-scheme and $N$ its normalizer. Then
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}N\subset\operatorname{GL}^{\textup{s}}_{Z/\mathbb{Q}_{p}}(V)$
is of finite index in its normalizer.

###### Proof.

For a vector bundle $E$ on $Z$ write $E_{0}$ when seen as a
$\mathbb{Q}_{p}$-vector space. With this notation for a smooth group
$Z$-scheme $M$ we have
$\operatorname{Lie}\operatorname{Res}_{Z/\mathbb{Q}_{p}}M=(\operatorname{Lie}M)_{0}$.
In order to prove the statement it suffices to show that
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}N$ and $N^{\prime}$ have same Lie
algebra. For, we have the following chain of identities:

$\displaystyle\operatorname{Lie}\operatorname{Res}_{Z/\mathbb{Q}_{p}}N$
$\displaystyle=(\operatorname{Lie}N)_{0}$
$\displaystyle=\\{X\in\mathfrak{gl}(V)\mid[X,\operatorname{Lie}G]\subset\operatorname{Lie}G\\}_{0}$
$\displaystyle=\\{X\in\mathfrak{gl}(V)_{0}\mid[X,(\operatorname{Lie}G)_{0}]\subset(\operatorname{Lie}G)_{0}\\}$
$\displaystyle=\\{X\in\mathfrak{gl}(V)_{0}\mid[X,\operatorname{Lie}\operatorname{Res}_{Z/\mathbb{Q}_{p}}G]\subset\operatorname{Lie}\operatorname{Res}_{Z/\mathbb{Q}_{p}}G\\}$
$\displaystyle=\operatorname{Lie}N^{\prime},$

where $\mathfrak{gl}(V)=\operatorname{Lie}\operatorname{GL}(V)$ and
$N^{\prime}$ is the normalizer of
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}G\subset\operatorname{GL}^{\textup{s}}_{Z/\mathbb{Q}_{p}}(V)$.
This concludes the proof. ∎

We say that a group $Z$-scheme $G$ is _reductive_ if $G_{z}$ is reductive
linear algebraic group over $\mathbb{F}_{z}$ for each $z\in Z$. Note that this
differs from the notion in [GP11] where reductive group schemes are assumed to
have connected fibers. Let $G$ be a reductive group $Z$-scheme. We write
$G^{\circ}$ for the group subscheme of $G$ whose at $z$ is $G_{z}^{\circ}$. We
also define

$\mathcal{B}(G,Z):=\mathcal{B}(\operatorname{Res}_{Z/\mathbb{F}}G,\mathbb{F})=\prod_{z\in
Z}\mathcal{B}(\operatorname{Res}_{\mathbb{F}_{z}/\mathbb{F}}X_{z},\mathbb{F})=\prod_{z\in
Z}\mathcal{B}(X_{z},\mathbb{F}_{z})$

where the equalities come from example 3.4. For a point $x=(x_{z})_{z\in
Z}\in\mathcal{B}(G,Z)$ we write $\operatorname{Stab}_{G}(x)$ for the
$Z$-scheme whose fiber at $z$ is $\operatorname{Stab}_{G_{z}}(x_{z})$. This
formalism will be applied in two contexts below.

###  4.2. Étale data

Take $\mathbb{F}=\mathbb{Q}_{p}$. Let $Z$ be a finite étale scheme over
$\mathbb{Q}_{p}$ and $\Gamma$ a topological group acting continuously on $Z$
via $\mathbb{Q}_{p}$-scheme isomorphisms. By a  _semilinear representation_ of
$\Gamma$ on $Z$ we mean a couple $(V,\rho)$ where $V$ is a vector bundle on
$Z$ and $\rho\colon\Gamma\to\operatorname{GL}^{\textup{s}}_{Z/\mathbb{F}}(V)$
is a continuous group morphism such that the following diagram is commutative:

${\Gamma}$${\operatorname{GL}_{Z/\mathbb{Q}_{p}}^{\textup{s}}(V)}$${\Gamma}$${\operatorname{Aut}(Z/\mathbb{Q}_{p})}$$\scriptstyle{\rho}$

Geometrically the commutativity of the above diagram means that the action of
$\Gamma$ on $V$ is equivariant with respect the action on $Z$. A closed
subscheme $X\subset\operatorname{GL}(V)$ is normalized by $\Gamma$ if
$\rho(\gamma)$ normalizes $X$ for each $\gamma\in\Gamma$.

######  Definition 4.2.

A _$\Gamma$ -étale datum on $Z$_ is a triple $D=(V,G,\rho)$ where

  * •

$(V,\rho)$ is a semilinear representation of $\Gamma$ on $Z$,

  * •

$G\subset\operatorname{GL}(V)$ is a reductive group $Z$-scheme normalized by
$\Gamma$.

If the set $Z$ is clear from the context we will call $D$ a $\Gamma$-étale
datum. The $\Gamma$-étale datum $D$ is _semisimple_ if the Zariski closure of
$\operatorname{Im}(\rho\colon\Gamma\to\operatorname{GL}(V_{0}))$ is a
reductive subgroup, where $V_{0}$ stands for $V$ as a $\mathbb{Q}_{p}$-vector
space.

By a _filtered_ resp.  _graded $\Gamma$-étale datum_ we mean a $\Gamma$-étale
datum $D=(V,G,\rho)$ together with a filtration resp. a grading where

  * •

a _filtration_ is a $\Gamma$-fixed point $x\in\mathcal{B}(G,Z)$;

  * •

a _grading_ is given by a structure of graded vector space
$V_{z}=\bigoplus_{\alpha\in\mathbb{Q}}V_{z}^{(\alpha)}$ on $V_{z}$ (for each
$z\in Z$) such that
$G_{z}\subset\prod_{\alpha\in\mathbb{Q}}\operatorname{GL}(V_{z}^{(\alpha)})$
and $\Gamma$ preserves the subspace

$\sum_{\gamma\in\Gamma}V_{\gamma z}^{(\alpha)}\quad\text{for
each}\quad\alpha\;\in\;\mathbb{Q}.$

We denote such a graded $\Gamma$-étale datum by $(V^{(\bullet)},G,\rho)$.

An _isomorphism_ between $\Gamma$-étale data $(V,G,\rho)$ and
$(V^{\prime},G^{\prime},\rho^{\prime})$ is a $\Gamma$-equivariant isomorphism
of vector bundles

$f\colon\quad
V\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;V^{\prime}\quad\text{with}\quad
G^{\prime}=fGf^{-1}.$

By an isomorphism of filtered $\Gamma$-étale data we mean an isomorphism of
$\Gamma$-étale data which preserves the corresponding points in the building.
An isomorphism between graded $\Gamma$-étale data $(V^{(\bullet)},G,\rho)$ and
$(V^{\prime(\bullet)},G^{\prime},\rho^{\prime})$ is an isomorphism between the
underlying $\Gamma$-étale data that preserves the grading in the sense that
$f(V_{z}^{(\alpha)})=V^{\prime(\alpha)}_{z}$ for all $\alpha\in\mathbb{Q}$ and
all $s\in Z$. The _pull-back_ of $\Gamma$-étale data along a
$\Gamma$-equivariant morphism $Z^{\prime}\to Z$ is defined in the obvious way.

######  Example 4.3.

Let $A$ be an abelian variety over a field $K$ of characteristic $0$. Given an
integer $r\geqslant 1$ consider the $p$-adic sheaf

$E:=[r]_{\ast}\mathbb{Q}_{p,A}$

on $A$ where $[r]$ is multiplication by $r$ on $A$ and $\mathbb{Q}_{p,A}$ the
trivial $p$-adic sheaf on $A$. To decompose $E$ let $\bar{K}$ be an algebraic
closure of $K$ and $A[r](\bar{K})$ the $r$-torsion subgroup of $A(\bar{K})$.
We identify $A[r](\bar{K})$ with a constant group scheme over $\mathbb{Q}_{p}$
and consider its Cartier dual

$Z:=A[r](\bar{K})^{\ast}=\operatorname{Hom}(A[r](\bar{K}),\mathbb{G}_{m}).$

The Galois group $\Gamma:=\operatorname{Gal}(\bar{K}/K)$ acts naturally on $Z$
via its action on $A[r](\bar{K})$. The points of $Z$ with values in a
$\mathbb{Q}_{p}$-algebra $B$ are morphisms $A[r](\bar{K})\to B^{\times}$ of
(abstract) groups. For $z\in Z$ let $\mathbb{F}_{z}$ be the residue field in
$z$ and

$\chi_{z}\colon A[r](\bar{K})\longrightarrow\mathbb{F}_{z}^{\times}$

the corresponding character. By precomposing $\chi_{z}$ with the canonical
projection

$\pi_{1}^{\textup{\'{e}t}}(A_{\bar{K}},0)=\projlim_{n\in\mathbb{N}}A[n](\bar{K})\longrightarrow
A[r](\bar{K})$

we obtain a character of geometric étale fundamental group of $A_{\bar{K}}$
hence a $p$-adic rank $1$ local system on $A_{\bar{K}}$. Then

$E_{\rvert A_{\bar{K}}}=\bigoplus_{z\in Z}L_{z}$

Let $X\subset A$ be a smooth subvariety of dimension $d$. Then the
$\mathbb{Q}_{p}$-vector space

$V:=\textup{H}_{\textup{\'{e}t}}^{d}(X_{\bar{K}},E)=\bigoplus_{z\in
Z}\textup{H}_{\textup{\'{e}t}}^{d}(X_{\bar{K}},L_{z})$

is finite-dimensional, inherits a natural structure of vector bundle over $Z$
and the natural $\mathbb{Q}_{p}$-linear action of $\Gamma$ on $V$ is
semilinear. For each $z\in Z$ set

$G_{z}=\begin{cases}\operatorname{GL}(V_{z})&\textup{if $X$ is not symmetric
up to translation,}\\\ \operatorname{GO}(V_{z},\omega_{z})&\textup{if $X$ is
symmetric up to translation and $d$ is even,}\\\
\operatorname{GSp}(V_{z},\omega_{z})&\textup{if $X$ is symmetric up to
translation and $d$ is odd}.\end{cases}$

Here _symmetric up to a translation_ means that $X=-X+a$ for some $a\in A(K)$,
and we then write $\omega_{z}\colon V_{z}\otimes V_{z}\to L_{z,a}$ for the
perfect pairing induced by Poincaré duality. Since the action of $\Gamma$
preserves the Poincaré pairing we conclude that

$D=(V,G,\rho)$

is a $\Gamma$-étale datum over $Z$.

######  Definition 4.4.

For any filtered $\Gamma$-étale datum $D=(V,G,\rho,x)$, we define the
associated graded

$\operatorname{gr}D\;=\;(\bar{V}^{(\bullet)},\bar{G},\bar{\rho})$

as follows: For each $s\in Z$, let $V_{z}^{\bullet}$ be the filtration induced
by $x_{z}$. Let $Q_{z}\subset G_{z}$ be the stabilizer of $x_{z}$, and
consider

$\bar{G}_{z}:=Q_{z}/\operatorname{rad}Q_{z}\lhook\joinrel\longrightarrow\prod_{\alpha}\operatorname{GL}(\bar{V}_{z}^{(\alpha)})\quad\text{where}\quad\bar{V}_{z}^{(\alpha)}=\operatorname{gr}^{\alpha}V_{z}.$

Since $x\in\mathcal{B}(G,Z)$ is fixed under $\Gamma$, the subspace
$\sum_{\gamma\in\Gamma}V_{\gamma z}^{\alpha}$ is stable under $\Gamma$ for
each $\alpha\in\mathbb{Q}$. Thus $\sum_{\gamma\in\Gamma}V_{\gamma
z}^{(\alpha)}$ inherits a natural linear action of $\Gamma$. The induced
representation
$\bar{\rho}\colon\Gamma\to\operatorname{GL}_{Z/\mathbb{Q}_{p}}(\bar{V})$ is
semilinear, where $\bar{V}=\bigoplus_{z\in Z}\operatorname{gr}V_{z}$.

###  4.3. Finiteness

Let $K$ be a number field and $\bar{K}$ an algebraic closure of $K$. Here we
consider étale data in the above sense with
$\Gamma=\operatorname{Gal}(\bar{K}/K)$. For a place $v$ of $K$ let $k_{v}$ the
residue field at $v$. Fix a finite subset $\Sigma$ of places of $K$.

######  Definition 4.5.

Let $V_{0}$ be a finite-dimensional $\mathbb{Q}_{p}$-vector space and
$w\in\mathbb{Z}$. A continuous representation
$\rho\colon\Gamma\to\operatorname{GL}(V_{0})$ is said to be _pure of weight
$w$_ (with respect to $\Sigma$) if it is unramified outside
$\Sigma\cup\\{p\\}$ and for each place $v\not\in\Sigma\cup\\{p\\}$ the
characteristic polynomial of any Frobenius element at $v$ has integral
coefficients and roots of absolute value $|k_{v}|^{w/2}$. A $\Gamma$-étale
datum $(V,G,\rho)$ is _pure of weight $w$_ if so is the representation
$\Gamma\to\operatorname{GL}_{Z/\mathbb{Q}_{p}}(V)$.

Let $W$ be a vector bundle on $Z$ and $H\subset\operatorname{GL}(W)$ a
reductive group $Z$-scheme. The first result of this section is the following
form of Faltings’ finiteness of pure semisimple Galois representations:

######  Proposition 4.6.

For $w\in\mathbb{Z}$, there are up to isomorphism only finitely many
$\Gamma$-étale data that are semisimple, pure of weight $w$ and such that
$(G,V)\cong(H,W)$.

###### Proof.

We argue as in [LV20, lemma 2.6]. We may assume $(G,V)=(H,W)$. By [Del85, th.
3.1] there are up to isomorphism only finitely many continuous representations
$\Gamma\to\operatorname{GL}(W_{0})$ that are semisimple and pure of weight $w$
with respect to $\Sigma$, where $W_{0}$ stays for $W$ seen as
$\mathbb{Q}_{p}$-vector space. Let $\rho$ be such a representation. Let $N$ be
the normalizer of $H\subset\operatorname{GL}(W)$. It suffices to check that
there are only finitely many $N(Z)$-conjugacy classes in the isomorphism class
of $\rho$. The image of $\rho$ is contained in the normalizer $N^{\prime}$ of
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}H\subset\operatorname{GL}(W_{0})$. Let
$L\subset N^{\prime}$ be the Zariski closure of the image of $\rho$. It
suffices to show that

$\mathcal{C}\;:=\;\\{L^{\prime}\subset N^{\prime}\mid
L^{\prime}=gLg^{-1},g\in\operatorname{GL}(W_{0})(\mathbb{Q}_{p})\\}$

is a finite union of $N(Z)$-orbits. Now the subgroup
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}N\subset N^{\prime}$ is of finite index
by lemma 4.1 below. Therefore, resuming the proof, it suffices to prove that
$\mathcal{C}$ is a finite union of $N^{\prime}(\mathbb{Q}_{p})$-orbits. The
representation $\rho$ is semisimple by hypothesis, thus the connected
component of $L$ is reductive. According to proposition A.1, the set

$\\{L^{\prime}\subset(N^{\prime}\times_{\mathbb{Q}_{p}}{\bar{\mathbb{Q}}_{p}})\mid
L^{\prime}=gLg^{-1},g\in\operatorname{GL}(W_{0})(\bar{\mathbb{Q}}_{p})\\}$

is a finite union of $N^{\prime}(\bar{\mathbb{Q}}_{p})$-orbits, where
$\bar{\mathbb{Q}}_{p}$ is an algebraic closure of $\mathbb{Q}_{p}$. We then
conclude by [Ser94, th. 4.4.5]. ∎

######  Corollary 4.7.

Let $w\in\mathbb{Z}$ and $\mathcal{I}\subset\mathcal{B}(H,Z)/H^{\circ}(Z)$ a
finite subset. Then up to isomorphism there are only finitely many graded
$\Gamma$-étale data on $Z$ arising as the graded filtered étale data
$D=(V,G,\rho,x)$ such that

  * •

$D$ is pure of weight $w$,

  * •

$\operatorname{gr}D$ is semisimple, and

  * •

there is an isomorphism $(V,G)\cong(W,H)$ sending $x$ in $\mathcal{I}$.

###### Proof.

It suffices to show the statement when $\mathcal{I}=\\{[x]\\}$ for some
$x\in\mathcal{B}(H,Z)$. In this case, up to isomorphism we may write any
filtered $\Gamma$-étale datum in question as a quadruple $(W,H,\rho,x)$ where
now only $\rho$ is variable. Let $Q$ be the stabilizer of $x$ and
$W^{\bullet}$ the filtration induced on $W$ by $x$. We conclude by applying
proposition 4.6 with $Q^{\textup{ss}}$ and $\operatorname{gr}W^{\bullet}$. ∎

In order to apply corollary 4.7 it will be important to have a finite subset
$\mathcal{I}$ permitting to construct the semisimplification of all
representations in question:

######  Lemma 4.8.

Suppose that the subgroup $H\subset\operatorname{GL}(V)$ is of finite index in
its normalizer. Then there exists a finite subset

$\mathcal{I}\subset\mathcal{B}(H,Z)/H(Z)$

with the following property: for each semilinear representation
$\rho\colon\Gamma\to\operatorname{GL}_{Z/\mathbb{Q}_{p}}^{\textup{s}}(W)$
normalizing $H$ there is $x\in\mathcal{B}(H,Z)$ fixed under $\Gamma$ with
image in $\mathcal{I}$ such that the graded of the filtered $\Gamma$-étale
datum $(W,H,\rho,x)$ is semisimple.

###### Proof.

Let $N\subset\operatorname{GL}(W)$ be the normalizer of $H$ and
$N^{\prime}\subset\operatorname{GL}_{Z/\mathbb{Q}_{p}}^{\textup{s}}(W)$ the
normalizer of $\operatorname{Res}_{Z/\mathbb{Q}_{p}}H$. Since $H\subset N$ is
of finite index the subgroup $\operatorname{Res}_{Z/\mathbb{Q}_{p}}N\subset
N^{\prime}$ is of finite index by lemma 4.1. Therefore, we have the following
chains of identities:

$\operatorname{Res}_{Z/\mathbb{Q}_{p}}H^{\circ}=\operatorname{Res}_{Z/\mathbb{Q}_{p}}N^{\circ}=N^{\prime\circ},\qquad\mathcal{B}(H,Z)=\mathcal{B}(N,Z)=\mathcal{B}(N^{\prime},\mathbb{Q}_{p}).$

On the other hand the set

$\operatorname{Par}(H^{\circ})(Z)/H(Z)=\prod_{z\in
Z}\operatorname{Par}(H^{\circ}_{z})(K_{z})/H_{z}(K_{z}),$

is finite, where $K_{z}$ is the residue field at $z$. Let
$\mathcal{P}\subset\operatorname{Par}(H^{\circ})(Z)$ be a finite set of
representatives of the preceding quotient. According to [BMR20, lemma 2.1] for
each parabolic subgroup $P\in\mathcal{P}$ there is $x_{P}\in\mathcal{B}(H,Z)$
such that $\operatorname{Stab}_{N^{\prime}}(x_{P})$ is the normalizer of
$\operatorname{Res}_{Z/\mathbb{Q}_{p}}P$ in $N^{\prime}$. We claim that the
set

$\mathcal{I}=\\{[x_{P}]\mid P\in\mathcal{P}\\}\subset\mathcal{B}(H,Z)/H(Z)$

does the job. Indeed let
$\rho\colon\Gamma\to\operatorname{GL}_{Z/\mathbb{F}}^{\textup{s}}(W)$ be a
semilinear representation normalizing $H$. By [BMR20, corollary 3.5] there is
a point $y$ in $\mathcal{B}(H,Z)=\mathcal{B}(N^{\prime},\mathbb{Q}_{p})$ fixed
under $\Gamma$ such that the graded of the filtered $\Gamma$-étale datum
$(W,H,\rho,y)$ is semisimple. Then
$\operatorname{Stab}_{H}(y)^{\circ}=hPh^{-1}$ for some $P\in\mathcal{P}$ and
$h\in H(Z)$. By construction the stabilizer of $x:=hx_{P}$ is the normalizer
of $\operatorname{Res}_{Z/\mathbb{Q}_{p}}\operatorname{Stab}_{H}(y)^{\circ}$.
Since $\Gamma$ fixes $y$ the image of $\rho$ is contained in
$\operatorname{Stab}_{N^{\prime}}(x)$ hence $\Gamma$ fixes $x$ too. Moreover

$\operatorname{rad}\operatorname{Res}_{Z/\mathbb{Q}_{p}}\operatorname{Stab}_{H}(y)=\operatorname{rad}\operatorname{Stab}_{N^{\prime}}(x)$

thus the graded of filtered $\Gamma$-étale datum $(W,H,\rho,x)$ is semisimple
by [BMR20, example 4.9]. This concludes the proof. ∎

###  4.4. de Rham data

Let $R=\mathcal{R}[\tfrac{1}{p}]$ where $\mathcal{R}$ is a finite étale
$\mathbb{Z}_{p}$-algebra. Equivalently $R$ is a finite product of finite
unramified extensions of $\mathbb{Q}_{p}$. The ring $\mathcal{R}$ coincides
with the ring of Witt vectors of its reduction modulo $p$ and therefore
inherits a Frobenius endomorphism $\sigma$. We denote again by $\sigma$ the
unique ring endomorphism of $R$ extending $\sigma$. Let $V$ be a vector bundle
on $Z=\operatorname{Spec}R$. In this section we always suppose semilinear
endomorphisms $\varphi$ of $V$ to be $\mathbb{Q}_{p}$-linear endomorphism such
that $\varphi(\lambda v)=\sigma(\lambda)\varphi(v)$ for all $\lambda\in R$ and
$v\in V$.

######  Definition 4.9.

A _de Rham datum on $Z$_ is a quadruple $D=(V,G,h,\varphi)$ where

  * •

$V$ is a vector bundle over $Z$,

  * •

$G\subset\operatorname{GL}(V)$ is a reductive group $Z$-scheme,

  * •

$h\in\mathcal{B}(G,Z)$,

  * •

$\varphi$ is a bijective semilinear endomorphism of $V$ normalizing $G$.

If the scheme $Z$ is clear from the context we will simply call $D$ a de Rham
datum. By a  _filtered_ resp. _graded de Rham datum_ we mean a de Rham datum
$D=(V,G,h,\varphi)$ together with a filtration resp. a grading, where

  * •

a _filtration_ is a point $x\in\mathcal{B}(G,Z)$ fixed under $\varphi$;

  * •

a _grading_ is the datum for each $z\in Z$ of a structure of graded
$K_{z}$-vector space $V_{z}=\bigoplus_{\alpha\in\mathbb{Q}}V_{z}^{(\alpha)}$
such that the graded pieces are stable under $\varphi_{z}$ and such that

$G_{z}\subset\prod_{\alpha\in\mathbb{Q}}\operatorname{GL}(V_{z}^{(\alpha)}).$

An _isomorphism_ between de Rham data $(V,G,h,\varphi)$ and
$(V^{\prime},G^{\prime},h^{\prime},\varphi^{\prime})$ is an isomorphism of
vector bundles

$f\colon\quad
V\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;V^{\prime}\quad\text{with}\quad
G^{\prime}=fGf^{-1},\quad\varphi^{\prime}=f\varphi f^{-1},\quad
h^{\prime}=F_{\ast}h,$

where $F\colon G\to G^{\prime}$ is the conjugation by $f$. By an isomorphism
of filtered de Rham data we mean an isomorphism of de Rham data which
preserves the given points in the building. An isomorphism between graded de
Rham data $(V^{(\bullet)},G,h,\varphi)$ and
$(V^{\prime(\bullet)},G^{\prime},h^{\prime},\varphi^{\prime})$ is an
isomorphism between the underlying de Rham data that preserves the grading in
the sense that $f(V_{z}^{(\alpha)})=V^{\prime(\alpha)}_{z}$ for all
$\alpha\in\mathbb{Q}$ and all $z\in Z$. The _pull-back_ of de Rham data along
a finite étale morphism $Z^{\prime}\to Z$ is defined in the obvious way.

######  Example 4.10.

Let $K$ be an unramified extension of $\mathbb{Q}_{p}$ with ring of integers
$\mathcal{O}_{K}$ and $\mathcal{A}$ an abelian scheme over $\mathcal{O}_{K}$.
Given an integer $r\geqslant 1$ nondivisible by $p$ consider the vector bundle

$\mathcal{E}:=[r]_{\ast}\mathcal{O}_{\mathcal{A}}$

on $\mathcal{A}$ where $[r]$ is the multiplication by $r$ on $\mathcal{A}$.
The morphism $[r]$ is finite étale thus the vector bundle $\mathcal{E}$ comes
equipped with an integrable connection $\nabla$ induced by the canonical
derivation on $\mathcal{O}_{\mathcal{A}}$. To decompose $\mathcal{E}$ let
$\mathcal{A}^{\natural}$ the universal vector extension of the dual abelian
scheme $\mathcal{A}^{\ast}$. From a modular point of view
$\mathcal{A}^{\natural}$ parametrizes rank $1$ connections on $\mathcal{A}$,
that is, couples $(\mathcal{L},\nabla)$ made of a line bundle $\mathcal{L}$ on
$\mathcal{A}$ and a (necessarily integrable) connection $\nabla$ on
$\mathcal{L}$. Tensor product of line bundles together with a connection endow
$\mathcal{A}^{\natural}$ with a structure of group scheme over
$\mathcal{O}_{K}$. Consider the universal rank $1$ connection
$(\mathcal{L},\nabla)$ on $A^{\natural}\times A$. The closed subscheme of
$r$-torsion points $\mathcal{Z}=\mathcal{A}^{\natural}[r]$ is finite étale
over $\mathcal{O}_{K}$. Then

$(\mathcal{E},\nabla)=\pi_{\ast}(\mathcal{L},\nabla)\quad\textup{where
$\pi\colon\mathcal{Z}\to\operatorname{Spec}\mathcal{O}_{K}$ is structural
morphism}.$

Let $\mathcal{X}\subset\mathcal{A}$ be a closed subscheme which is smooth of
relative dimension $d$ over $\mathcal{O}_{K}$. Set
$\mathcal{Y}:=\mathcal{X}\times_{\mathcal{O}_{K}}\mathcal{Z}$ and consider the
$d$-th relative de Rham cohomology group

$\mathcal{V}:=\mathcal{H}^{d}(\mathcal{X}/\mathcal{O}_{K};\mathcal{E},\nabla)=\mathcal{H}^{d}_{\textup{dR}}(\mathcal{Y}/\mathcal{Z};\mathcal{L},\nabla)=\textup{R}^{d}f_{\ast}(\Omega^{\bullet}_{\mathcal{Y}/\mathcal{Z}}\otimes\mathcal{L})$

where $\Omega^{\bullet}_{\mathcal{Y}/\mathcal{Z}}\otimes\mathcal{L}$ is the de
Rham complex of $(\mathcal{L},\nabla)$ and $f\colon\mathcal{Y}\to\mathcal{Z}$
the projection onto the second factor. The cristalline-de Rham comparison
theorem furnishes a canonical isomorphism of $\mathcal{O}_{K}$-modules

$\mathcal{V}\cong\textup{H}^{d}_{\textup{cris}}(\mathcal{Y}_{p}/\mathcal{Z};\mathcal{L}_{p},\nabla_{p})$

where the subscript $p$ stays for the reduction modulo $p$. By functoriality
of cristalline cohomology $\mathcal{V}$ inherits a Frobenius operator
$\varphi$. Let $A$, $X$, $Y$ and $Z$ be the generic fiber respectively of
$\mathcal{A}$, $\mathcal{X}$, $\mathcal{Y}$ and $\mathcal{Z}$. Then

$V:=\mathcal{V}\otimes_{\mathcal{O}_{K}}K=\mathcal{H}^{d}_{\textup{dR}}(Y/Z;\mathcal{L},\nabla)$

is a vector bundle over $Z$ and as $K$-vector spaces we have

$V_{0}=\textup{H}^{d}_{\textup{dR}}(X/K;\mathcal{E},\nabla).$

The Frobenius operator $\varphi$ extends to a semilinear automorphism of $V$
denoted $\varphi$ again. For each $z\in Z$ set

$G_{z}=\begin{cases}\operatorname{GL}(V_{z})&\textup{if $X$ is not symmetric
up to translation,}\\\ \operatorname{GO}(V_{z},\omega_{z})&\textup{if $X$ is
symmetric up to translation and $d$ is even,}\\\
\operatorname{GSp}(V_{z},\omega_{z})&\textup{if $X$ is symmetric up to
translation and $d$ is odd}.\end{cases}$

Here _symmetric up to a translation_ means that $X=-X+a$ for some $a\in A(K)$,
and we then write $\omega_{z}\colon V_{z}\otimes V_{z}\to\mathcal{L}_{z,a}$
for the perfect pairing induced by Poincaré duality. The Frobenius operator
$\varphi$ normalizes $G$ because the cristalline-de Rham comparison
isomorphism is compatible with the Poincaré pairing and the Frobenius operator
preserves it up to a scalar multiple. The vector bundle $V$ comes with the
Hodge filtration

$F^{\bullet}\;:\quad
V\;=\;F^{0}\;\supset\;F^{1}\;\supset\;\cdots\;\supset\;F^{d}\;\supset\;F^{d+1}\;=\;0,$

where $F^{i}=\operatorname{Im}(\textup{R}^{d}f_{\ast}(\Omega^{\bullet\geqslant
i}_{Y/Z}\otimes_{\mathcal{O}_{Y}}\mathcal{L})\to V)$. Moreover, if $X$ is
symmetric up to a translation, then the Hodge filtration is autodual with
respect to the Poincaré pairing in the sense that $(F^{i})^{\perp}=F^{d+1-i}$
for each $i$ (this can be checked on fibers at geometric points of $Z$, so it
follows from the statement over the complex numbers in corollary B.3). Thus
the Hodge filtration $F^{\bullet}$ defines a point $h\in\mathcal{B}(G,Z)$, in
other words the quadruple

$D=(V,G,h,\varphi)$

is a de Rham datum on $Z$.

######  Definition 4.11.

The associated graded
$\operatorname{gr}D=(\bar{V},\bar{G},\bar{h},\bar{\varphi},\bar{V}^{(\bullet)})$
of a filtered de Rham datum $D=(V,G,h,\varphi,x)$ is defined as follows. For
each point $z\in Z$ let $V_{z}^{\bullet}$ be the filtration which is induced
by $x_{z}$, and let $Q_{z}\subset G_{z}$ be the parabolic subgroup stabilizing
$x_{z}$. Then we put

$\bar{G}_{z}:=Q_{z}/\operatorname{rad}Q_{z}\lhook\joinrel\longrightarrow\prod_{\alpha\in\mathbb{Q}}\operatorname{GL}(V^{(\alpha)}_{z})$

where $V^{(\alpha)}_{z}:=\operatorname{gr}^{\alpha}V_{z}$, and we define
$\bar{h}_{z}:=\operatorname{pr}_{Q_{z}}(h_{z})$ and
$\bar{\varphi}_{z}:=\operatorname{gr}\varphi_{z}$.

###  4.5. From graded to filtered data

Let $D_{i}=(V_{i},G_{i},h_{i},\varphi_{i},x_{i})$ for $i=1,2$ be two filtered
de Rham data. We want to understand the relation between them if the
associated graded data are isomorphic. As a reference datum we fix
$(V,G,\varphi)$ where

  * •

$V$ is a vector bundle on $Z$,

  * •

$G\subset\operatorname{GL}(V)$ is a reductive subgroup scheme over $Z$,

  * •

$\varphi\colon V\to V$ is a semilinear endomorphism normalizing $G$.

Suppose that for $i=1,2$ we are given isomorphisms of vector bundles

$f_{i}\colon\quad
V_{i}\;\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\;V\quad\text{such
that}\quad
G\;=\;f_{i}G_{i}f_{i}^{-1}\quad\text{and}\quad\varphi\;=\;f_{i}\varphi_{i}f_{i}^{-1}.$

Via these isomorphisms we will identify the points
$x_{i},h_{i}\in\mathcal{B}(G_{i},\mathbb{F})$ with points in
$\mathcal{B}(G,Z)$ which we again denote $x_{i},h_{i}$. Let $Q_{i}\subset G$
denote the stabilizer of $x_{i}$, with Levi quotient
$Q_{i}^{\textup{ss}}=Q_{i}/\operatorname{rad}Q_{i}$. With notation as in
section 3.4 consider the projection

$\operatorname{pr}_{i}\colon\quad\mathcal{B}(G,Z)\;=\;\prod_{z\in
Z}\mathcal{B}(G_{z},K_{z})\;\longrightarrow\;\mathcal{B}(Q_{i}^{\textup{ss}},Z)\;=\;\prod_{z\in
Z}\mathcal{B}(Q_{i,z}^{\textup{ss}},K_{z}).$

In general an isomorphism between graded cannot be lifted to an isomorphism of
the original filtered one. Nonetheless the filtered data satisfy a weaker
relation (to be compared with definition 6.2):

######  Proposition 4.12.

If in the above setup the graded de Rham data $\operatorname{gr}D_{1}$ and
$\operatorname{gr}D_{2}$ are isomorphic and for every $z\in Z$ the subgroup
$G_{z}\subset\operatorname{GL}(V_{z})$ is full in the sense of definition
3.17, then there exists $g\in G(Z)$ such that

$\displaystyle x_{2}=gx_{1},$ $\displaystyle g\varphi
g^{-1}\varphi^{-1}\in\operatorname{rad}Q_{2},$
$\displaystyle\operatorname{pr}_{2}(h_{2})=\operatorname{pr}_{2}(gh_{1}).$

###### Proof.

By reasoning pointwise we may assume $Z$ is a singleton. Then
$Z=\operatorname{Spec}K$ for a finite unramified extension $K$ of
$\mathbb{Q}_{p}$. Via the isomorphisms $f_{i}$ we make the identifications

$V_{i}=V,\quad
G_{i}=G\quad\text{and}\quad\varphi_{i}=\varphi\quad\text{for}\quad i=1,2.$

Let $V_{i}^{\alpha}\subset V$ be the steps of the filtration induced by
$x_{i}$ and let $V_{i}^{(\alpha)}=V_{i}^{\alpha}/V_{i}^{<\alpha}$ be the
associated graded pieces, for each $\alpha\in\mathbb{Q}$. By assumption there
exists an isomorphism $\operatorname{gr}D_{1}\simeq\operatorname{gr}D_{2}$ of
graded de Rham data, which by definition consists of a collection of
$K$-linear isomorphisms $\bar{g}^{(\alpha)}\colon V_{1}^{(\alpha)}\to
V_{2}^{(\alpha)}$ whose direct sum $\bar{g}$ has the following properties:

  1. (1)

$Q_{2}^{\textup{ss}}=\bar{g}Q_{1}^{\textup{ss}}\bar{g}^{-1}$

  2. (2)

$\bar{\varphi}=\bar{g}\bar{\varphi}\bar{g}^{-1}$

  3. (3)

$\operatorname{pr}_{2}(h_{2})=\bar{g}_{\ast}\operatorname{pr}_{1}(h_{1})$

We now use the assumption that the subgroup $G\subset\operatorname{GL}(V)$ is
full. Note that the filtrations on $V$ induced by the two points $x_{1},x_{2}$
are conjugate under $\operatorname{GL}(V)(\mathbb{F})$ since their filtered
pieces have the same dimension $\dim V_{1}^{\alpha}=\dim V_{2}^{\alpha}$.
Since the corresponding weights $\alpha\in\mathbb{Q}$ also match, it follows
that the two points $x_{1},x_{2}$ are conjugate under
$\operatorname{GL}(V)(\mathbb{F})$. By the definition of full subgroups we
then have a finite subset $I\subset\mathbb{Q}$ such that

$\varepsilon_{x_{i},I}\colon\quad
Q_{i}^{\textup{ss}}\;\longrightarrow\;\prod_{\alpha\in
I}\operatorname{GL}(V_{i}^{(\alpha)})$

is an embedding and
$\operatorname{Tran}_{G}(x_{1},x_{2})\to\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x_{1},x_{2})$
is surjective, so the map on $\mathbb{F}$-points

$\operatorname{Tran}_{G}(x_{1},x_{2})(\mathbb{F})\longrightarrow\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x_{1},x_{2})(\mathbb{F})$

is also surjective by the remark after lemma 3.16. Since
$\bar{g}\in\operatorname{gr}_{I}\operatorname{Tran}_{\operatorname{GL}(V)}(x_{1},x_{2})(\mathbb{F})$
by (1), this gives an element
$g\in\operatorname{Tran}_{G}(x_{1},x_{2})(\mathbb{F})\subset G(\mathbb{F})$
lifting $\bar{g}$. By definition of the transporter
$\operatorname{Tran}_{G}(x_{1},x_{2})$ we have $gx_{1}=x_{2}$. To prove
$g\varphi g^{-1}\varphi^{-1}\in\operatorname{rad}Q_{2}$ note that $g\varphi
g^{-1}\varphi^{-1}$ belongs to $Q_{2}$ because $\varphi$ fixes $x_{1}$ and
$x_{2}$. The wanted assertion then follows from condition (2) and the
injectivity of $\varepsilon_{x_{2},I}$. Finally from (3) we get
$\operatorname{pr}_{2}(h_{2})=\operatorname{pr}_{2}(gh_{1})$ which concludes
the proof. ∎

##  5\. $p$-adic Hodge theory

We now discuss how to pass from étale to de Rham data via $p$-adic Hodge
theory, and how global purity on the étale side leads to positivity in the
sense of section 3.5 on the de Rham side.

###  5.1. Cristalline representations

Let $K$ be a finite unramified extension of $\mathbb{Q}_{p}$ with ring of
integers $\mathcal{O}_{K}$, $\bar{K}$ an algebraic closure of $K$ and
$\Gamma=\operatorname{Gal}(\bar{K}/K)$. The ring of cristalline periods

$B:=B_{\textup{cris}}$

is a field extension of $\bar{K}$ and comes equipped with an action of
$\Gamma$, a Frobenius operator $\varphi$ and a (non-increasing, exhausting,
separated) filtration $h^{\bullet}B=(h^{\alpha}B)_{\alpha\in\mathbb{Z}}$
stable under $\Gamma$ (but in general not under $\varphi$).

######  Definition 5.1.

Let $V$ be a $\mathbb{Q}_{p}$-vector space endowed with a linear action of
$\Gamma$. When $V$ is finite-dimensional, we say that the representation $V$
is _cristalline_ if it is continuous and
$\dim_{K}(V\otimes_{\mathbb{Q}_{p}}B)^{\Gamma}=\dim_{\mathbb{Q}_{p}}V$. When
$V$ is infinite-dimensional, we say that $V$ is _cristalline_ if it is
increasing union of finite-dimensional cristalline subrepresentations.

We then set

$V_{\textup{dR}}:=(V\otimes_{\mathbb{Q}_{p}}B)^{\Gamma},\qquad
h^{\alpha}V:=(V\otimes_{\mathbb{Q}_{p}}h^{\alpha}B)^{\Gamma},$

so we have a natural isomorphism $V\otimes_{\mathbb{Q}_{p}}B\cong
V_{\textup{dR}}\otimes_{K}B$. The filtration $h_{V}:=h^{\bullet}V$ is non-
increasing, exhausting and separated; it is called the _Hodge filtration_ of
$V_{\textup{dR}}$. The operator $\varphi$ induces a bijective semilinear
endomorphism $\varphi_{V}$ of $V_{\textup{dR}}$ which we call again the
Frobenius operator. The class of cristalline representations is stable under
subquotients and tensor products. For a morphism of cristalline
representations $f\colon V\to W$ the resulting $K$-linear map
$f_{\textup{dR}}\colon V_{\textup{dR}}\to W_{\textup{dR}}$ is compatible with
the induced Frobenius operators on $V_{\textup{dR}}$ and $W_{\textup{dR}}$.
The functor $V\rightsquigarrow V_{\textup{dR}}$ is exact and if $f$ is
injective resp. surjective then $h_{V}$ is the restriction to $V$ of $h_{W}$
resp. $h_{W}$ is the quotient of $h_{V}$. We will refer to this property as
_exactness of the Hodge filtration_. When $V$ is finite-dimensional the
filtration $h_{V}$ can be seen as a point of
$\mathcal{B}(\operatorname{GL}(V_{\textup{dR}}),K)$ by example 3.3.

######  Example 5.2.

A representation $\rho\colon\Gamma\to\operatorname{GL}(V)$ with finite image
is cristalline. Indeed
$\operatorname{Ker}\rho=\Gamma^{\prime}:=\operatorname{Gal}(\bar{K}/K^{\prime})$
for some finite Galois extension $K^{\prime}$ of $K$ and

$\displaystyle(V\otimes_{\mathbb{Q}_{p}}B)^{\Gamma}$
$\displaystyle=[(V\otimes_{\mathbb{Q}_{p}}B)^{\Gamma^{\prime}}]^{\Gamma^{\prime}/\Gamma}=(V\otimes_{\mathbb{Q}_{p}}B^{\Gamma^{\prime}})^{\operatorname{Gal}(K^{\prime}/K)}=(V\otimes_{\mathbb{Q}_{p}}K^{\prime})^{\operatorname{Gal}(K^{\prime}/K)}.$

Galois descent then implies
$\dim_{K}(V\otimes_{\mathbb{Q}_{p}}K^{\prime})^{\operatorname{Gal}(K^{\prime}/K)}=\dim_{K^{\prime}}V\otimes_{\mathbb{Q}_{p}}K^{\prime}=\dim_{\mathbb{Q}_{p}}V$.

Let $A$ be a $\mathbb{Q}_{p}$-algebra endowed with a cristalline action of
$\Gamma$ compatible with the ring structure of $A$, that is
$\gamma(ab)=(\gamma a)(\gamma b)$ for all $a,b\in A$ and $\gamma\in\Gamma$. A
$\mathbb{Q}_{p}$-linear action of $\Gamma$ on an $A$-module $M$ is
_semilinear_ if for all $\gamma\in\Gamma$, $a\in A$ and $m\in M$ we have
$\gamma(am)=(\gamma a)(\gamma m)$. Given an $A$-module $M$ with a cristalline
semilinear action of $\Gamma$ the $K$-vector space $M_{\textup{dR}}$ inherits
a natural structure of $A_{\textup{dR}}$-module. For a $\Gamma$-equivariant
morphism of $A$-modules $f\colon M\to N$ the induced map
$f_{\textup{dR}}\colon M_{\textup{dR}}\to N_{\textup{dR}}$ is a morphism of
$A_{\textup{dR}}$-modules.

######  Lemma 5.3.

The so-defined functor

$\left\\{\begin{array}[]{c}\textup{$A$-modules with a cristalline}\\\
\textup{semilinear action of
$\Gamma$}\end{array}\right\\}\;\longrightarrow\;\left\\{\begin{array}[]{c}\textup{$A_{\textup{dR}}$-modules}\end{array}\right\\}$

is exact, symmetric monoidal and compatible with pull-back along
$\Gamma$-equivariant morphisms of affine $\mathbb{Q}_{p}$-schemes with a
cristalline action of $\Gamma$. An $A$-module $M$ with a cristalline
semilinear action of $\Gamma$ is finitely generated if and only if
$M_{\textup{dR}}$ is.

###### Proof.

The exactness comes from that of the functor $V\mapsto V_{\textup{dR}}$. Given
$A$-modules $M$ and $N$ with a cristalline semilinear action of $\Gamma$, we
have

$\displaystyle(M\otimes_{A}N)_{\textup{dR}}\otimes_{K}B$
$\displaystyle\cong(M\otimes_{A}N)\otimes_{\mathbb{Q}_{p}}B$
$\displaystyle=(M\otimes_{\mathbb{Q}_{p}}B)\otimes_{A\otimes_{\mathbb{Q}_{p}}B}(N\otimes_{\mathbb{Q}_{p}}B)$
$\displaystyle\cong(M_{\textup{dR}}\otimes_{K}B)\otimes_{A_{\textup{dR}}\otimes_{K}B}(N_{\textup{dR}}\otimes_{K}B)$
$\displaystyle=(M_{\textup{dR}}\otimes_{A_{\textup{dR}}}N_{\textup{dR}})\otimes_{K}B$

Passing to $\Gamma$-invariants we obtain a natural isomorphism

$\alpha_{M,N}\colon(M\otimes_{A}N)_{\textup{dR}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M_{\textup{dR}}\otimes_{A_{\textup{dR}}}N_{\textup{dR}}.$

This also show that the functor $M\mapsto M_{\textup{dR}}$ is compatible with
pull-back. It remains to prove that the functor is symmetric. For, notice that
the commutativity of the diagram

${M_{\textup{dR}}\otimes_{A_{\textup{dR}}}N_{\textup{dR}}}$${(M\otimes_{A}N)_{\textup{dR}}}$${N_{\textup{dR}}\otimes_{A_{\textup{dR}}}M_{\textup{dR}}}$${(N\otimes_{A}M)_{\textup{dR}}}$$\scriptstyle{\alpha_{M,N}}$$\scriptstyle{x\otimes
y\mapsto y\otimes x}$$\scriptstyle{x\otimes y\mapsto y\otimes
x}$$\scriptstyle{\alpha_{N,M}}$

can be tested after extending scalars to $B$ where it becomes clear. The last
statement follows from the fact that finite generation can be tested after
extending scalars to $B$; see [Sta22, Theorem 08XD]. ∎

Let $X$ be an affine $\mathbb{Q}_{p}$-scheme endowed with an action of
$\Gamma$ defined by a group morphism
$\sigma\colon\Gamma\to\operatorname{Aut}(X)$. Then

$\Gamma\times\Gamma(X,\mathcal{O}_{X})\longrightarrow\Gamma(X,\mathcal{O}_{X}),\quad(\gamma,f)\longmapsto\sigma(\gamma^{-1})^{\ast}f$

defines a linear action of $\Gamma$ on $\Gamma(X,\mathcal{O}_{X})$ compatible
with the ring structure.

######  Definition 5.4.

We say that the action of $\Gamma$ on $X$ is _cristalline_ if
$\Gamma(X,\mathcal{O}_{X})$ is a cristalline representation. If the action of
$\Gamma$ on $X$ is cristalline, then the $K$-scheme

$X_{\textup{dR}}:=\operatorname{Spec}\Gamma(X,\mathcal{O}_{X})_{\textup{dR}}$

comes with a natural isomorphism $X\times_{\mathbb{Q}_{p}}B\cong
X_{\textup{dR}}\times_{K}B$ of $B$-schemes. The Frobenius operator on
$\Gamma(X,\mathcal{O}_{X})_{\textup{dR}}$ induces an automorphism
$\varphi_{X}$ of $X_{\textup{dR}}$ as a $\mathbb{Q}_{p}$-scheme. We write
$h_{X}$ for the Hodge filtration on $\Gamma(X,\mathcal{O}_{X})_{\textup{dR}}$.

The construction is functorial: For a $\Gamma$-equivariant morphism $f\colon
X\to Y$ between affine $\mathbb{Q}_{p}$-schemes endowed with a cristalline
action of $\mathbb{Q}_{p}$ let $f_{\textup{dR}}\colon X_{\textup{dR}}\to
Y_{\textup{dR}}$ be the induced morphism.

######  Lemma 5.5.

The so-defined functor

$\begin{array}[]{rcl}{\left\\{\begin{array}[]{c}\textup{affine
$\mathbb{Q}_{p}$-schemes}\\\ \textup{with a cristalline action of
$\Gamma$}\end{array}\right\\}}&\longrightarrow&{\left\\{\begin{array}[]{c}\textup{affine
$K$-schemes}\end{array}\right\\}}\\\ X&\longmapsto&X_{\textup{dR}}\end{array}$

is compatible with fiber products. Moreover, let $f\colon X\to Y$ be a
$\Gamma$-equivariant morphism between affine $\mathbb{Q}_{p}$-schemes endowed
with a cristalline action of $\Gamma$. Then,

  1. (1)

given a property $\mathsf{P}$ of schemes morphisms that can be checked after
faithfully flat base change, the morphism $f$ has the property $\mathsf{P}$ if
and only if $f_{\textup{dR}}$ does;

  2. (2)

the following square is commutative:

${X_{\textup{dR}}}$${Y_{\textup{dR}}}$${X_{\textup{dR}}}$${Y_{\textup{dR}}}$$\scriptstyle{f_{\textup{dR}}}$$\scriptstyle{\varphi_{X}}$$\scriptstyle{\varphi_{Y}}$$\scriptstyle{f_{\textup{dR}}}$

###### Proof.

Compatibility with tensor product comes from lemma 5.3 while (2) is just the
fonctoriality of the Frobenius endomorphism. Statement (1) is clear because
the injections $\mathbb{Q}_{p}\hookrightarrow B$ and $K\hookrightarrow B$ are
faithfully flat ring morphisms. ∎

######  Example 5.6.

Let $X$ be a finite $\mathbb{Q}_{p}$-scheme with an action of $\Gamma$. Since
$\operatorname{Aut}(X)$ is finite, the representation
$\rho\colon\Gamma\to\operatorname{GL}(\Gamma(X,\mathcal{O}_{X}))$ has finite
image. As in example 5.2 write
$\operatorname{Ker}\rho=\operatorname{Gal}(\bar{K}/K^{\prime})$ for some
finite Galois extension $K^{\prime}$ of $K$. Then

$X_{\textup{dR}}=X_{K^{\prime}}/\operatorname{Gal}(K^{\prime}/K)$

where $\operatorname{Gal}(K^{\prime}/K)$ acts on
$X_{K^{\prime}}=\operatorname{Spec}\Gamma(X,\mathcal{O}_{X})\otimes_{\mathbb{Q}_{p}}K^{\prime}$
via its action on both $K^{\prime}$ and $\Gamma(X,\mathcal{O}_{X})$.

It will be useful to have a handy condition to prove that the action on an
affine scheme is cristalline.

######  Lemma 5.7.

Let $X=\operatorname{Spec}A$ be an affine $\mathbb{Q}_{p}$-scheme with a
cristalline action of $\Gamma$ and let $Y$ be an affine $X$-scheme of finite
type with an action of $\Gamma$ such that the structural morphism $Y\to X$ is
$\Gamma$-equivariant. Then, the action on $Y$ is cristalline if and only if
there are an $A$-module $M$ of finite type with a semilinear cristalline
action of $\Gamma$ and a $\Gamma$-equivariant closed embedding
$Y\hookrightarrow\operatorname{Spec}(\operatorname{Sym}_{A}M)$ of $X$-schemes.

###### Proof.

Suppose that the action of $\Gamma$ on $R=\Gamma(Y,\mathcal{O}_{Y})$ is
cristalline. Let $V\subset R$ be a finite-dimensional subrepresentation
generating $R$ as an $A$-algebra. The $A$-submodule $M\subset R$ generated by
$V$ is stable under $\Gamma$ and finitely generated as an $A$-module. Then $M$
does the job. Conversely let $M$ be an $A$-module with a cristalline
semilinear action of $\Gamma$ and $f\colon S:=\operatorname{Sym}_{A}M\to R$ a
$\Gamma$-equivariant surjective morphism of $A$-algebras. For each integer
$n\geqslant 0$ the representation

$S_{n}:=\bigoplus_{i=0}^{n}\operatorname{Sym}^{i}_{A}M$

is cristalline, thus so $R_{n}:=f(S_{n})\subset R$. Since
$S=\bigcup_{n=0}^{\infty}S_{n}$ and $\varphi$ is surjective we have
$R=\bigcup_{n=0}^{\infty}R_{n}$ which concludes the proof. ∎

######  Lemma 5.8.

Let $X\to T$ and $\pi\colon T\to S$ be $\Gamma$-equivariant morphisms between
affine varieties over $\mathbb{Q}_{p}$ endowed with a cristalline action of
$\Gamma$. Suppose that $\pi$ is finite flat. Then the natural action of
$\Gamma$ on $\operatorname{Res}_{T/S}X$ is cristalline and

$(\operatorname{Res}_{T/S}X)_{\textup{dR}}=\operatorname{Res}_{T_{\textup{dR}}/S_{\textup{dR}}}X_{\textup{dR}}.$

###### Proof.

By lemma 5.7 there is a coherent $\mathcal{O}_{T}$-module $V$ with a
cristalline semilinear action on $\Gamma(T,V)$ and a $\Gamma$-equivariant
closed embedding $i\colon
X\hookrightarrow\operatorname{Spec}_{T}{\operatorname{Sym}V}$. Then the
morphism induced by Weil restriction

$j\colon\operatorname{Res}_{T/S}X\lhook\joinrel\longrightarrow\operatorname{Res}_{T/S}(\operatorname{Spec}_{T}{\operatorname{Sym}V})=\operatorname{Spec}_{S}{\operatorname{Sym}\pi_{\ast}T}$

is a closed immersion. The group $\Gamma$ acts naturally on
$\operatorname{Res}_{T/S}X$ and the closed immersion $j$ is
$\Gamma$-equivariant. Lemma 5.7 then implies that the action of $\Gamma$ on
$\operatorname{Res}_{T/S}X$ is cristalline. Now note that there is a natural
morphism of $S_{\textup{dR}}$-schemes

$f\colon(\operatorname{Res}_{T/S}X)_{\textup{dR}}\longrightarrow\operatorname{Res}_{T_{\textup{dR}}/S_{\textup{dR}}}X_{\textup{dR}}.$

Indeed by universal property of the Weil restriction such a morphism $f$ is
given by a morphism of $T_{\textup{dR}}$-schemes
$(\operatorname{Res}_{T/S}X)_{\textup{dR}}\times_{S_{\textup{dR}}}T_{\textup{dR}}\to
X_{\textup{dR}}$ still denoted $f$. Similarly the identity of
$\operatorname{Res}_{T/S}X$ is given by a morphism of $T$-schemes

$g\colon\operatorname{Res}_{T/S}(X)\times_{S}T\longrightarrow X.$

The morphism $g$ is $\Gamma$-equivariant and we set

$f:=g_{\textup{dR}}\colon(\operatorname{Res}_{T/S}X)_{\textup{dR}}\times_{S}T_{\textup{dR}}=(\operatorname{Res}_{T/S}(X)\times_{S}T)_{\textup{dR}}\longrightarrow
X_{\textup{dR}}$

where the identity comes from lemma 5.5. In order to conclude the proof it
suffices to show that $f$ is an isomorphism after extending scalars to $B$.
The construction of the Weil restriction is compatible with scalars extension
thus

$\displaystyle(\operatorname{Res}_{T/S}X)_{\textup{dR}}\times_{K}B$
$\displaystyle\cong\operatorname{Res}_{T/S}(X)\times_{\mathbb{Q}_{p}}B=\operatorname{Res}_{T_{B}/S_{B}}(X_{B}),$
$\displaystyle(\operatorname{Res}_{T_{\textup{dR}}/S_{\textup{dR}}}X_{\textup{dR}})\times_{K}B$
$\displaystyle=\operatorname{Res}_{T_{\textup{dR},B}/S_{\textup{dR},B}}(X_{\textup{dR},B})\cong\operatorname{Res}_{T_{B}/S_{B}}(X_{B}).$

The endomorphism of $\operatorname{Res}_{T_{B}/S_{B}}(X_{B})$ induced by $f$
via the preceding isomorphisms is just the identity, which concludes the
proof. ∎

######  Remark 5.9 (A variation on the construction).

Let $V$ be a $K$-vector space with a cristalline $K$-linear action of
$\Gamma$. Write $V_{0}$ for $V$ seen simply as a $\mathbb{Q}_{p}$-vector
space. Then $V_{0,\textup{dR}}=V_{\textup{dR}}\otimes_{\mathbb{Q}_{p}}K$ where

$V_{\textup{dR}}:=(V\otimes_{K}B)^{\Gamma}.$

The construction of $V_{\textup{dR}}$ is functorial with respect to $K$-linear
$\Gamma$-equivariant maps. This gives rise to a functor

$\left\\{\begin{array}[]{c}\textup{$K$-vector spaces with}\\\ \textup{a
cristalline $K$-linear action of
$\Gamma$}\end{array}\right\\}\longrightarrow\left\\{\begin{array}[]{c}\textup{$K$-vector
spaces}\end{array}\right\\}$

which is exact and compatible with tensor product. The Hodge filtration is
again exact in the above sense and compatible with tensor product. Similarly,
for an affine scheme $X=\operatorname{Spec}A$ over $K$ with a cristalline
action of $\Gamma$ by $K$-scheme isomorphisms we define

$X_{\textup{dR}}=\operatorname{Spec}{(A\otimes_{K}B)^{\Gamma}}.$

This construction is functorial with respect to $\Gamma$-equivariant morphisms
of $K$-schemes, hence gives rise to a functor

$\left\\{\begin{array}[]{c}\textup{$K$-schemes with}\\\ \textup{a cristalline
$K$-linear action of
$\Gamma$}\end{array}\right\\}\longrightarrow\left\\{\begin{array}[]{c}\textup{$K$-schemes}\end{array}\right\\}$

which is compatible with fibered products.

###  5.2. Affine group schemes with cristalline actions

Let $Z=\operatorname{Spec}R$ be an affine $\mathbb{Q}_{p}$-scheme endowed with
a cristalline action of $\Gamma$ and let $V$ be a vector bundle on $X$
together with a cristalline semilinear action on the $R$-module $\Gamma(Z,V)$.
In this section we write
$\mathbb{V}(V)=\operatorname{Spec}\operatorname{Sym}V^{\vee}$ for the total
space of $V$. Consider a subgroup $Z$-scheme $G\subset\operatorname{GL}(V)$
normalized by $\Gamma$. The action by conjugation of $\Gamma$ on $G$ is
cristalline. Indeed the closed embedding

$G\lhook\joinrel\longrightarrow\mathbb{V}(\operatorname{End}V\oplus\mathcal{O}_{Z}),\quad
g\longmapsto(g,\det(g)^{-1})$

is $\Gamma$-equivariant with respect to the action on
$\operatorname{End}(V\oplus\det V)$ given by

$\gamma.(f,\lambda)=(\rho(\gamma)f\rho(\gamma)^{-1},\lambda),\qquad\textup{for
}\gamma\in\Gamma,f\in\operatorname{End}(V),\textup{ and }\lambda\in\det V,$