#  A Note on Systems of Realizations on Shimura Varieties

Ziquan Yang

###### Abstract

Let $(G,\Omega)$ be a Shimura datum of abelian type. It is well known that the
corresponding Shimura variety $\mathrm{Sh}(G,\Omega)$ should be a moduli space
of abelian motives equipped with some additional structures. In this half-
expository note, we give under some simplifying assumptions a moduli
interpretation of $\mathrm{Sh}(G,\Omega)$ over the reflex field purely in
terms of systems of realizations. The main purpose is to introduce some
convenient formalism that can be used to avoid the technicalities about
dealing with various notions of families of motives.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Generalities on Galois Descent
    2. 2.2 Motives and Absolute Hodge Cycles
    3. 2.3 Tensors and Level Structures
  3. 3 Systems of Realizations
    1. 3.1 Basic Definitions
    2. 3.2 Characterizing “abelian-motivic” systems
    3. 3.3 Level Structures on Systems
    4. 3.4 Galois Descent Lemmas
  4. 4 Shimura Varieties
    1. 4.1 Automorphic VHS and étale local systems
    2. 4.2 Constructing Systems of Realizations
    3. 4.3 Moduli Interpretations over Reflex Fields

##  1 Introduction

It is well known that Shimura varieties of abelian type are supposed to be
moduli spaces of “abelian motives equipped with centain additional
structures”. The purpose of this note is to provide a reference for a precise
statement, under some simplifying assumptions. Let $(G,\Omega)$ be a Shimura
datum of abelian type with reflex field $E$, and for each compact open
subgroup $\mathsf{K}\subseteq G(\mathbb{A}_{f})$, denote the Shimura variety
$G(\mathbb{Q})\backslash\Omega\times G(\mathbb{A}_{f})/\mathsf{K}$ by
$\mathrm{Sh}_{\mathsf{K}}(G,\Omega)_{\mathbb{C}}$ and its canonical model over
$E$ by $\mathrm{Sh}_{\mathsf{K}}(G,\Omega)$. Let $\mathrm{Sh}(G,\Omega)$ be
the limit $\varprojlim_{\mathsf{K}}\mathrm{Sh}_{\mathsf{K}}(G,\Omega)$ as
$\mathsf{K}$ runs through compact open subgroups.

Before giving a moduli interpretation, we first show that:

######  Theorem A.

_(see also(4.2.5))_ Assume that $(G,\Omega)$ has $\mathbb{Q}$-rational weight,
the center of $G$ has no anisotropic torus split over $\mathbb{R}$, and
$(G,\Omega)$ has a nice Hodge-type cover (see (4.2.1)). Then there is a
natural functor from the category $\mathrm{Rep}(G)$ of finite-dimensional
$G$-representations over $\mathbb{Q}$ to the category of
$G(\mathbb{A}_{f})$-equivariant weakly abelian-motivic systems of realizations
over $\mathrm{Sh}(G,\Omega)$.

Roughly speaking, a system of realizations is a cohomological avatar of a
“genuine” family of motives. For a subfield $k\subseteq\mathbb{C}$ and a
smooth $k$-variety $S$, a system of realizations is given by a variation of
Hodge structures (VHS) over $S_{\mathbb{C}}$ whose de rham and étale
components descend to $S$ (see (3.1.1)). The advantage of working with systems
of realizations, as opposed to “genuine” families of motives, is that the
former is defined purely in terms of sheaves so that its pullback and descent
properties are well understood (see also (3.1.5)). The only price to pay is
that we need to explicitly keep track of a compatability condition between the
descent data of the étale and de Rham realizations of the VHS in a system.
This is why we introduce the notion of weakly abelian-motivic systems of
realizations (see (3.2.3)), which are those whose descent data are controlled
by abelian motives. It is a pointwise definition and may look a little odd at
first glance, but we hope the reader will find it actually very flexible to
work with once one gets used to it (see (3.2.5)).

######  Theorem B.

Let $(G,\Omega)$ be as in Theorem A and $\mathsf{K}$ be a neat compact open
subgroup. Then for every faithful $G$-representation $V$ and smooth
$E$-variety $S$, there is a natural bijection between
$\mathrm{Sh}_{\mathsf{K}}(G,\Omega)(S)$ and isomorphism classes of pairs
$(\mathsf{W},[\xi])$ such that $\mathsf{W}$ is a weakly abelian-motivic system
of realizations over $S$, and $[\xi]$ is a $\mathsf{K}$-level structure which
is $\mathsf{W}$-rational and is of type $\Omega$.

A more precise version is stated in (4.3.1). The meaning of
“$\mathsf{W}$-rational” and “of type $\Omega$” are explained in §3.3, and the
motivation for these definitions is explained in (4.1.4). The moduli
interpretation of the complex Shimura variety
$\mathrm{Sh}_{\mathsf{K}}(G,\Omega)_{\mathbb{C}}$ in terms of VHS equipped
with additional structures ([10, Prop. 3.10]) follows relatively easily from
the complex uniformization $G(\mathbb{Q})\backslash\Omega\times
G(\mathbb{A}_{f})/\mathsf{K}$ and standard results due to Baily and Borel.
This note is mainly focused on the canonical model
$\mathrm{Sh}_{\mathsf{K}}(G,\Omega)$ over the reflex field. Morally of course
Theorem B should follow from [10, Thm 3.31]. Unfortunately, some proofs in
loc. cit. seem to be only sketched, and there are some technical subtleties
about the notion of a family of motives used there that should be further
explained (see also (3.1.5)).

The author hopes that the notion of weakly abelian-motivic systems of
realizations set up in this note gives a convenient alternative formalism for
one to fill in the small gaps in literature. Our general approach here is
inspired by [4, §2.2] and [8, §5.24]. We expect that the simplifying
assumption of the existence of a “nice Hodge-type cover” (4.2.1) can be
removed by suitably adapting some constructions in [4], [6], and [9], if
needed.

##### Acknowledgment

We thank Ben Moonen for encouraging the author to write this note and Paul
Hamacher for his comments.

##### Notations

Let $K$ be a field and $L$ be an extension of $K$. For any $K$-scheme $X$, we
write $X(L)$ for the set of $K$-linear morphisms $\mathrm{Spec\,}(L)\to X$.

By a variety over a field, we shall always mean a geometrically reduced
separated scheme of finite type over the field. If $X$ is a smooth projective
variety over $\mathbb{C}$, we write $\mathrm{H}^{i}_{B}(X,\mathbb{Q})$ for the
$i$th Betti (i.e., singular) cohomology and $\mathrm{H}^{i}(X,\mathbb{Q})$ for
the associated Hodge structure.

Let $R$ be a commutative ring and $M$ be a finite free $R$-module. Let
$M^{\otimes}$ denote the direct sum of all $R$-modules that can be formed from
$M$ using the operations of taking duals, tensor products, symmetric powers
and exterior powers. Whenever applicable, we use these notations also for
sheaves of modules in some Grothendieck topology which is understood in the
context.

If $S$ is a smooth variety over $\mathbb{C}$ and $\mathbb{W}$ is a VHS
(variation of Hodge structures) over $S$, we often use $\mathbb{W}_{B}$ to
denote the underlying $\mathbb{Q}$-local system and $\mathbb{W}_{\mathrm{dR}}$
the filtered flat vector bundle over $S$. We will always assume that the VHS
is pure, i.e., the weight filtration is split.

Let $S$ be a scheme. We write $\mathsf{Loc}(S)$ for the category of étale
local systems of $\mathbb{A}_{f}$-modules and $\mathsf{Vect}(S)$ for the
category of vector bundles on $S$. If $S$ is a smooth $\mathbb{C}$-variety and
$\mathcal{V}\in\mathsf{Vect}(S)$, write $\mathcal{V}^{\mathrm{an}}$ for the
associated complex-analytic vector bundle. For a field-valued point
$s:\mathrm{Spec\,}(\kappa)\to S$, and $\mathcal{F}\in\mathsf{Loc}(S)$ (resp.
$\mathcal{V}\in\mathsf{Vect}(S)$), write $\mathcal{F}_{s}$ for
$s^{*}\mathcal{F}$ (resp. $\mathcal{V}_{s}$ for
$s^{*}\mathcal{V}=\mathcal{V}{\otimes}_{\mathcal{O}_{S}}\kappa$).

##  2 Preliminaries

### 2.1 Generalities on Galois Descent

We will make heavy use of Galois descent arguments, so we first spell out some
tautology and set up some notations to facilitate discussion. We do not intend
to state the results in utmost generality, but only the simple cases we need.

Let $k$ be a field of characteristic $0$ and $L$ is an algebraically closed
field over $k$. We allow $L$ to have infinite transcendence degree over $k$,
e.g., $L=\mathbb{C}$ and $k$ is a number field. Let $X_{0}$ be a smooth
$k$-variety. Let $X$ be the base change $X_{0}{\otimes}_{k}L$. Note that there
is a canonical identification $X(L)=X_{0}(L)$. Given a $k$-linear morphism
$s:\mathrm{Spec\,}(L)\to X_{0}$ (i.e., an element of $X_{0}(L)$) and
$\sigma\in\mathrm{Aut}(L/k)$, let $\sigma(s)$ be the composition
$\mathrm{Spec\,}(L)\stackrel{{\scriptstyle\mathrm{Spec\,}(\sigma)}}{{\to}}\mathrm{Spec\,}(L)\stackrel{{\scriptstyle
s}}{{\to}}X_{0}$. Then $s\mapsto\sigma(s)$ defines the
$\mathrm{Aut}(L/k)$-action on $X_{0}(L)$, and thus on $X(L)$ via the
identificaton $X(L)=X_{0}(L)$ above.

#### (2.1.1)

Suppose now that $\mathcal{F}\in\mathsf{Loc}(X)$. A descent of $\mathcal{F}$
to $X_{0}$ is a pair
$(\mathcal{F}_{0}\in\mathsf{Loc}(X_{0}),\alpha:\mathcal{F}_{0}|_{X}\,{\cong}\,\mathcal{F})$.
For every $x\in X(L)$ and $\sigma\in\mathrm{Aut}(L/k)$, the descent
$(\mathcal{F}_{0},\alpha)$ gives us an isomorphism of stalks
$\mathcal{F}_{x}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}_{\sigma(x)}$,
which we label as $\sigma_{\mathcal{F}_{0},x}$. Indeed, when we write
$\mathcal{F}_{x}$ and $\mathcal{F}_{\sigma(x)}$, we are viewing $x$ as an
element of $X(L)$, but via the identification $X(L)=X_{0}(L)$ we also have
identifications $\mathcal{F}_{x}=\mathcal{F}_{0,x}$ and
$\mathcal{F}_{\sigma(x)}=\mathcal{F}_{0,\sigma(x)}$. By applying [14, 04FN] to
$(X_{0},x,\sigma(x))$, we obtain a canonical isomorphism
$\mathcal{F}_{0,x}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}_{0,\sigma(x)}$.
This gives $\sigma_{\mathcal{F}_{0},x}$. If $x_{0}\in X_{0}$ is the image of
the geometric point $x$, then $x=\sigma(x)$ for
$\sigma\in\mathrm{Aut}(k(x)/k(x_{0}))=\mathrm{Aut}(L/k(x_{0}))$. The
collection $\\{\sigma_{\mathcal{F}_{0},x}\\}_{\mathrm{Aut}(L/k(x_{0}))}$ gives
precisely the $\mathrm{Aut}(L/k(x_{0}))$-action on the stalk
$\mathcal{F}_{0,x}$ ([14, 03QW]).

Similarly, for $\mathcal{V}\in\mathsf{Vect}(X)$, a descent of $\mathcal{V}$ to
$X_{0}$ is a pair
$(\mathcal{V}_{0}\in\mathsf{Vect}(X_{0}),\beta:\mathcal{V}_{0}|_{X}\,{\cong}\,\mathcal{V})$,
which gives at each $x\in X(L)$ a $\sigma$-linear isomorphism
$\sigma_{\mathcal{V}_{0},x}:\mathcal{V}_{x}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{V}_{\sigma(x)}$.
A global section $\mathbf{s}\in\mathrm{H}^{0}(X,\mathcal{F})$ (resp.
$\mathbf{t}\in\mathrm{H}^{0}(X,\mathcal{V})$) descends to
$\mathrm{H}^{0}(X_{0},\mathcal{F}_{0})$ (resp.
$\mathrm{H}^{0}(X_{0},\mathcal{V}_{0})$) if and only if for every
$\sigma\in\mathrm{Aut}(L/k)$ and $x\in X(L)$,
$\sigma_{\mathcal{F}_{0},x}(\mathbf{s}_{x})=\mathbf{s}_{\sigma(x)}$ (resp.
$\sigma_{\mathcal{V}_{0},x}(\mathbf{t}_{x})=\mathbf{t}_{\sigma(x)}$).

We often omit the identification $\alpha$ from the notation, and simply say
that $\mathcal{F}_{0}$ is a descent of $\mathcal{F}$. However, we emphasize
that as an object of $\mathsf{Loc}(X_{0})$, $\mathcal{F}_{0}$ may certainly
have nontrivial automorphisms, but there is no nontrivial automorphism, say
$g\in\mathrm{Aut}(\mathcal{F}_{0})$, such that
$\alpha\circ(g|_{X}:\mathcal{F}_{0}|_{X}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{F}_{0}|_{X})=\alpha$.
That is, as a descent of $\mathcal{F}$, $\mathcal{F}_{0}$ cannot have
nontrivial automorphisms. A similar remark applies to descent of vector
bundles. Therefore, if we say that two descents of a fixed object in
$\mathsf{Loc}(X)$ or $\mathsf{Vect}(X)$ to $X_{0}$ are isomorphic, then they
are uniquely isomorphic.

#### (2.1.2)

Let
$X,X_{0},\mathcal{F},(\mathcal{F}_{0},\alpha:\mathcal{F}_{0}|_{X}\,{\cong}\,\mathcal{F})$
be as above. Suppose that $X_{0}$ is a finite étale cover of another smooth
$k$-variety $Y_{0}$ with $Y:=Y_{0}{\otimes}_{k}L$, and $\mathcal{F}$ admits a
descent
$(\mathcal{G}\in\mathsf{Loc}(Y),\gamma:\mathcal{G}|_{X}\,{\cong}\,\mathcal{F})$.
A simultaneous descent of $(\mathcal{F}_{0},\mathcal{G})$ to $Y_{0}$ is a
triple
$(\mathcal{G}_{0}\in\mathsf{Loc}(Y_{0}),\xi:\mathcal{G}_{0}|_{X_{0}}\,{\cong}\,\mathcal{F}_{0},\zeta:\mathcal{G}_{0}|_{Y}\,{\cong}\,\mathcal{G})$
such that the diagram on the right below commutes:

${X}$${{X_{0}}}$${\mathcal{F}}$${{\mathcal{F}_{0}|_{X}}}$${Y}$${{Y_{0}}}$${{\mathcal{G}|_{X}}}$${{\mathcal{G}_{0}|_{X}}}$$\scriptstyle{\lrcorner}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$$\scriptstyle{\zeta|_{X}}$$\scriptstyle{\xi|_{X}}$

It is not hard to see that such a triple $(\mathcal{G}_{0},\xi,\zeta)$ is
unique up to unique isomorphism provided that it exists. Similarly, we define
the notion of simultaneous descent of vector bundles.

### 2.2 Motives and Absolute Hodge Cycles

Let $k$ be a field of characteristic $0$. We denote by
$\mathsf{Mot}_{\mathrm{AH}}(k)$ the neutral $\mathbb{Q}$-linear Tannakian
category of motives over $k$ for absolute Hodge cycles (cf. [13, §2] where it
is denoted by $\mathscr{M}_{k}$). We have the Tate objects $\mathbf{1}(n)$ for
every $n\in\mathbb{Z}$ in this category. For any object
$M\in\mathsf{Mot}_{\mathrm{AH}}(k)$, we write $M(n)$ for
$M{\otimes}\mathbf{1}(n)$ and by an absolute Hodge class on $M$ we mean a
morphism $\mathbf{1}\to M$.

Following [7, §1], we denote by $\omega_{\ell}$ the $\ell$-adic realization
functor which sends $\mathsf{Mot}_{\mathrm{AH}}(k)$ to the category of finite
dimensional $\mathbb{Q}_{\ell}$-vector spaces with an action of
$\mathrm{Gal}_{k}:=\mathrm{Gal}(\bar{k}/k)$, where $\bar{k}$ is some chosen
algebraic closure of $k$. Putting $\omega_{\ell}$’s together we obtain
$\omega_{\mathrm{{\acute{e}}t}}$, which takes values in the category of finite
free $\mathbb{A}_{f}$-modules with a $\mathrm{Gal}_{k}$-action. Let
$\omega_{\mathrm{dR}}$ denote the de Rham realization functor, which takes
values in the category of filtered $k$-vector spaces. If $k=\mathbb{C}$, we
additionally consider the Betti realization $\omega_{B}$ (resp. the Hodge
realization $\omega_{\mathrm{Hdg}}$) which takes values in the category of
$\mathbb{Q}$-vector spaces (resp. Hodge structures).

Since the operations of taking duals, tensor products, and symmetric and
exterior powers make sense for the Tannakian category
$\mathsf{Mot}_{\mathrm{AH}}(k)$ and are compatible with the cohomological
realizations, it makes sense to say whether a tensor
$s\in(\omega_{\mathrm{{\acute{e}}t}}(M)\times\omega_{\mathrm{dR}}(M))^{\otimes}$
(resp. $s\in\omega_{B}(M)^{\otimes}$) is absolute Hodge for
$M\in\mathsf{Mot}_{\mathrm{AH}}(k)$ (resp.
$M\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$, when $k=\mathbb{C}$).

Let $\mathsf{Mot}_{\mathsf{Ab}}(k)\subseteq\mathsf{Mot}_{\mathrm{AH}}(k)$ be
the full Tannakian sub-category generated by the Artin motives and the motives
attached to abelian varieties. As in loc. cit., we will repeatedly make use of
the following fact:

######  Theorem (2.2.1).

_([2, Ch 1])_ The functor $\omega_{\mathrm{Hdg}}$ is fully faithful when
restricted to $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$. In particular, for
every $M\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$, every element
$s\in\omega_{B}(M)\cap\mathrm{Fil}^{0}\omega_{\mathrm{dR}}(M)$ is given by an
absolute Hodge cycle.

We often refer to objects in $\mathsf{Mot}_{\mathsf{Ab}}(k)$ as abelian
motives.

#### (2.2.2)

If $L$ is an extension of $k$, then there is a natural faithful functor of
Tannakian categories
$-{\otimes}_{k}L:\mathsf{Mot}_{\mathrm{AH}}(k)\to\mathsf{Mot}_{\mathrm{AH}}(L)$
compatible with fiber functors. A construction we shall often consider is the
following: Let $M$ be an object of $\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$
and $\sigma\in\mathrm{Aut}(\mathbb{C})$. We denote
$M{\otimes}_{\sigma}\mathbb{C}$ by $M^{\sigma}$. Base change properties of
étale and de Rham cohomology give us canonical isomorphisms
$\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$
and
$\omega_{\mathrm{dR}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{dR}}(M^{\sigma})$.
Note that the latter is $\sigma$-linear. Here the supscript “bc” stands for
“base change”, and we add this supscript when we need to distinguish these
isomorphisms from other potentially different isomorhisms (e.g., in most
Galois descent arguments). If $s\in\omega_{B}(M)^{\otimes}$ is absolute Hodge,
then we write $s^{\sigma}$ for the (necessarily unique) element in
$\omega_{B}(M^{\sigma})^{\otimes}$ such that $s$ and $s^{\sigma}$ have the
same étale and de Rham realizations via the isomorphisms
$\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$
and
$\omega_{\mathrm{dR}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{dR}}(M^{\sigma})$.

### 2.3 Tensors and Level Structures

#### (2.3.1)

Let $G$ be a connected reductive group over $\mathbb{Q}$, $V$ be a finite
dimensional $G$-representation over $\mathbb{Q}$ and $\mathsf{K}\subseteq
G(\mathbb{A}_{f})$ be a compact open subgroup. Let $T$ be a Noetherian
$\mathbb{Q}$-scheme and
$\mathsf{W}_{\mathrm{{\acute{e}}t}}\in\mathsf{Loc}(T)$. A
$(G,V,\mathsf{K})$-level structure, or simply a $\mathsf{K}$-level structure
when $(G,V)$ is understood, on $\mathsf{W}_{\mathrm{{\acute{e}}t}}$ is the
data $[\eta]$ that give each geometric point $t\to T$ a
$\pi_{1}^{\mathrm{{\acute{e}}t}}(T,t)$-invariant element
$[\eta]_{t}\in\mathsf{K}\backslash\mathrm{Isom}(V{\otimes}\mathbb{A}_{f},\mathsf{W}_{\mathrm{{\acute{e}}t},t})$,
such that if $t,t^{\prime}$ are two geometric points on the same connected
component of $T$, parallel transport along any étale path from $t$ to
$t^{\prime}$ sends $[\eta]_{t}$ to $[\eta]_{t^{\prime}}$. Alternatively, one
can also consider the pro-étale sheaf
$\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{W}_{\mathrm{{\acute{e}}t}})$
consisting of isomorphims
$V{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t}}$
over some pro-étale cover and define a $\mathsf{K}$-level structure to be a
global section of
$\mathsf{K}\backslash\underline{\mathrm{Isom}}(V{\otimes}\mathbb{A}_{f},\mathsf{W}_{\mathrm{{\acute{e}}t}})$.111When
the action of $\mathsf{K}$ on $V{\otimes}\mathbb{A}_{f}$ is not faithful, this
quotient is interpreted as taking the orbit of the action by the image of
$\mathsf{K}$ in $\mathrm{GL}(V{\otimes}\mathbb{A}_{f})$. We are not
considering a “stacky” quotient.

If $T^{\prime}\to T$ is a morphism of Noetherian $\mathbb{Q}$-schemes, we
naturally construct the pullback of $[\eta]$ to $T^{\prime}$, which is a
$\mathsf{K}$-level structure on
$\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{T^{\prime}}$. We denote it by
$[\eta]_{T^{\prime}}$. If $M\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$ is a
motive, and $[\eta_{M}]$ is a $\mathsf{K}$-level structure on
$\omega_{\mathrm{{\acute{e}}t}}(M)$, then $[\eta_{M}]^{\sigma}$ denotes
$\mathsf{K}$-level structure on $\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$
induced by composing (a representative of) $[\eta_{M}]$ with
$\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$.

#### (2.3.2)

We note that if a $\mathsf{K}$-level structure $[\eta]$ can be put on
$\mathsf{W}_{\mathrm{{\acute{e}}t}}$, then for each $v\in(V^{\otimes})^{G}$
(i.e., the $G$-invariants in $V^{\otimes}$), there exists a unique section
$\boldsymbol{v}_{\mathrm{{\acute{e}}t}}\in\mathrm{H}^{0}(\mathsf{W}_{\mathrm{{\acute{e}}t}}^{\otimes})$
such that for every geometric point $t\to T$, some (and hence every)
representative of $[\eta]_{t}$ sends $v{\otimes}1$ to
$\boldsymbol{v}_{\mathrm{{\acute{e}}t},t}$. We write this global section as
$\eta(v)_{\mathrm{{\acute{e}}t}}$, since it is completely determined by $\eta$
and $v$.

We will often make use of the following simple observation: Let $k$ and $L$ be
as in (2.1), and suppose that $T$ is a smooth $k$-variety. A
$\mathsf{K}$-level structure $[\eta]$ on
$\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{T_{L}}$ descends to a level structure on
$\mathsf{W}_{\mathrm{{\acute{e}}t}}$ if and only if for every $t\in
T(L)=T_{L}(L)$ and $\sigma\in\mathrm{Aut}(L/k)$,
$\sigma_{\mathsf{W}_{\mathrm{{\acute{e}}t}},t}$ sends $[\eta]_{t}$ to
$[\eta]_{\sigma(t)}$.

##  3 Systems of Realizations

### 3.1 Basic Definitions

######  Definition (3.1.1).

Let $k$ be a subfield of $\mathbb{C}$ and $S$ be a smooth $k$-variety. By a
system of realizations we mean a tuple
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}},i_{\mathrm{dR}},i_{\mathrm{{\acute{e}}t}})$
where

  * •

$\mathsf{V}_{B}$ is a $\mathbb{Q}$-local system over
$S_{\mathbb{C}}:=S{\otimes}_{k}\mathbb{C}$;

  * •

$\mathsf{V}_{\mathrm{dR}}$ is a filtered flat vector bundle over $S$;

  * •

$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ is an étale local system of
$\mathbb{A}_{f}$-coefficients over $S$;

  * •

$i_{\mathrm{dR}}$ is an isomorphism of flat vector bundles
$(\mathsf{V}_{B}{\otimes}\mathcal{O}^{\mathrm{an}}_{S_{\mathbb{C}}},\mathrm{id}{\otimes}d)\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}})^{\mathrm{an}}$
such that $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}})$ forms
a pure polarizable VHS;

  * •

$i_{\mathrm{{\acute{e}}t}}$ is an isomorphism
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$.

We may often omit $(i_{\mathrm{dR}},i_{\mathrm{{\acute{e}}t}})$ in the
notation and simply write
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}=\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$
and
$\mathsf{V}_{B}{\otimes}\mathcal{O}_{S_{\mathbb{C}}}^{\mathrm{an}}=\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$.
There are some changes of topology here: Technically,
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ is defined for the analytic topology,
but it has a canonically associated étale local system, which we also denote
by $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$. Similarly, we often do not
distinguish $\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$ with its
analytification
$\mathsf{V}_{B}{\otimes}\mathcal{O}^{\mathrm{an}}_{S_{\mathbb{C}}}$, because
over a smooth complex variety there is a natural equivalence of categories
between flat algebraic vector bundles with regular singularieties and flat
holomorphic vector bundles ([1, Thm II.5.9]). In particular, a flat global
section of $\mathsf{V}_{B}{\otimes}\mathcal{O}_{S_{\mathbb{C}}}^{\mathrm{an}}$
is also a flat global section of $\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$.

#### (3.1.2)

If $k^{\prime}$ is another subfield of $\mathbb{C}$ which contains $k$,
$S^{\prime}$ is a $k^{\prime}$-variety, and there is a $k$-linear morphism of
schemes $S^{\prime}\to S$, then we can naturally form the restriction
$\mathsf{V}|_{S^{\prime}}$ (or pullback) of $\mathsf{V}\in\mathsf{R}(S)$ to
$S^{\prime}$. Indeed, $S^{\prime}{\otimes}_{k^{\prime}}\mathbb{C}$ is the same
as $S{\otimes}_{k}\mathbb{C}$, so the Betti component $\mathsf{V}_{B}$ stays
the same. Then we simply restrict $\mathsf{V}_{\mathrm{dR}}$ and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ to $S^{\prime}$. If $k=\mathbb{C}$, we do
not distinguish $\mathsf{R}(S)$ with the category of polarizable VHS, as these
two categories the naturally equivalent.

#### (3.1.3)

Our definition looks slightly different from [3, Def. 6.1] (see also [10,
§3.41]). However, if $K$ is a number field (which is no longer equipped with a
chosen embedding into $\mathbb{C}$) and $S$ is a smooth $K$-variety, then a
system of realizations over $S$ as defined in [3, Def. 6.1] is equivalent to a
system of realizations over $S_{(\mathbb{Q})}$ in our definition, where
$S_{(\mathbb{Q})}$ is the $\mathbb{Q}$-variety obtained by composing the
structural morphism of $S$ with
$\mathrm{Spec\,}(K)\to\mathrm{Spec\,}(\mathbb{Q})$.

#### (3.1.4)

With the obvious definitions of morphisms, tensor products and duals, systems
of realizations over a smooth variety $S$ over $k\subseteq\mathbb{C}$ form a
$\mathbb{Q}$-linear Tannakian category $\mathsf{R}(S)$. We have Tate objects
$\mathbf{1}(n)$ and denote $\mathsf{V}(n)$ by
$\mathsf{V}{\otimes}\mathbf{1}(n)$ (cf. the discussion below loc. cit.). We
write $\mathrm{H}^{0}(\mathsf{V})$ for the $\mathbb{Q}$-module
$\mathrm{Hom}(\mathbf{1},\mathsf{V})$. Concretely, an element in
$\mathrm{H}^{0}(\mathsf{V})$ is given by a triple
$(\mathbf{s}_{B},\mathbf{s}_{\mathrm{dR}},\mathbf{s}_{\mathrm{{\acute{e}}t}})$
where $\mathbf{s}_{B}\in\mathrm{H}^{0}(\mathsf{V}_{B})$ is a global section
which is everywhere of Hodge type $(0,0)$, and $\mathbf{s}_{\mathrm{dR}}$ and
$\mathbf{s}_{\mathrm{{\acute{e}}t}}$ are respectively global sections in
$\mathrm{H}^{0}(\mathsf{V}_{\mathrm{dR}})$ and
$\mathrm{H}^{0}(\mathsf{V}_{\mathrm{{\acute{e}}t}})$ whose restrictions over
$S_{\mathbb{C}}$ are the de Rham and étale realizations of $\mathbf{s}_{B}$.

######  Remark (3.1.5).

We introduce systems of realizations mainly to circumvent the problem of not
having a definition of a family of motives in literature that has been
documented to have satisfactory pullback and descent properties. For example,
if one uses [10, Def. 2.37], then it is not a priori clear how to perform
pullback along non-dominant morphisms, though when we restrict to considering
abelian motives over smooth $\mathbb{C}$-varieties this definition has nice
descent properties (e.g., combine Lem. 2.33 and Prop. 2.42 in loc. cit.). On
the other hand, [12, Def. 4.3.3] is amenable to defining pullback, but its
descent properties have not been discussed.

### 3.2 Characterizing “abelian-motivic” systems

######  Definition (3.2.1).

Let $S$ be a smooth variety over $k\subseteq\mathbb{C}$. We say that a system
of realizations $\mathsf{V}$ over $S$ is strongly abelian motivic (strongly
AM) if there exists another system $\mathsf{H}$ which is given by the first
cohomology of an abelian scheme $\mathcal{A}/S$, a realization
$\mathsf{H}^{\natural}$ that is a direct sum of those constructed out of
$\mathsf{H}$ by taking Tate twists, duals, tensor products, exterior and
symmetric powers, as well as an idempotent projector
$\boldsymbol{\pi}=(\boldsymbol{\pi}_{B},\boldsymbol{\pi}_{\mathrm{dR}},\boldsymbol{\pi}_{\mathrm{{\acute{e}}t}}):\mathsf{H}^{\natural}\to\mathsf{H}^{\natural}$
such that $\mathsf{V}$ is isomorphic to the image of $\boldsymbol{\pi}$. We
write the full subcategory of $\mathsf{R}(S)$ consisting of such objects as
$\mathsf{R}^{\circ}_{\mathrm{am}}(S)$.

We say that $\mathsf{V}$ is étale-locally strongly AM, if for some finite
étale cover $S^{\prime}\to S$ of $k$-varieties, the restriction
$\mathsf{V}|_{S^{\prime}}$ is strongly AM. We write the full subcategory of
$\mathsf{R}(S)$ consisting of such objects as $\mathsf{R}_{\mathrm{am}}(S)$.

######  Remark (3.2.2).

In [10], there is also a similar notion of an abelian motivic VHS (Def. 2.30),
and a notion of an abelian motive on a smooth connected variety over a field
of characteristic $0$ (Def. 2.37). To distinguish with Milne’s definitions, we
added “strongly” to indicate that we ask the relevant abelian scheme
$\mathcal{A}$ to be defined over the entire $S$, as opposed to just an open
dense subset of $S$. One reason for doing so is that being strongly AM is
clearly stable under base change along any morphisms between smooth
$k$-varieties, rather than just the dominant ones (see also (3.1.5)).

The following definition is an extension of (3.2.1).

######  Definition (3.2.3).

We say that $\mathsf{V}$ is weakly abelian-motivic (weakly AM) if for every
$s\in S(\mathbb{C})$, there exists
$M\in\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$ such that
$\omega_{\mathrm{Hdg}}(M)\,{\cong}\,(\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s})$;
moreover, for any such isomorphism
$\gamma:\omega_{\mathrm{Hdg}}(M)\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s})$
and $\sigma\in\mathrm{Aut}(\mathbb{C}/k)$, there exists an isomorphism
$\gamma^{\sigma}:\omega_{\mathrm{Hdg}}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{B,\sigma(s)},\mathsf{V}_{\mathrm{dR},\sigma(s)})$
such that the Betti components $\gamma_{B},\gamma^{\sigma}_{B}$ of $\gamma$
and $\gamma^{\sigma}$ fit into commutative diagrams:

${{\omega_{\mathrm{{\acute{e}}t}}(M)}}$${{\mathsf{V}_{B,s}{\otimes}\mathbb{A}_{f}}}$${{\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})}}$${{\mathsf{V}_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}}$$\scriptstyle{\gamma^{\sigma}_{B}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\gamma_{B}{\otimes}\mathbb{A}_{f}}$
and
${{\omega_{\mathrm{dR}}(M)}}$${{\mathsf{V}_{B,s}{\otimes}\mathbb{C}}}$${{\omega_{\mathrm{dR}}(M^{\sigma})}}$${{\mathsf{V}_{B,\sigma(s)}{\otimes}\mathbb{C}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{\mathsf{V}_{\mathrm{dR}},s}}$$\scriptstyle{\gamma^{\sigma}_{B}{\otimes}\mathbb{C}}$$\scriptstyle{\gamma_{B}{\otimes}\mathbb{C}}$
(1)

We note that $\gamma^{\sigma}$ is uniquely determined by $\gamma$ provided
that it exists. For the notations
$\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}$,
$\sigma_{\mathsf{V}_{\mathrm{dR}},s}$ and “$\,{\cong}\,_{\mathrm{bc}}$” in the
above diagrams, see (2.1.1) and (2.2.2). When we write $\mathsf{V}_{B,s}$ and
$\mathsf{V}_{B,\sigma(s)}$, we are viewing $s$ and $\sigma(s)$ as (usually
different) closed points on $S_{\mathbb{C}}$, and to apply the notations
$\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}$ and
$\sigma_{\mathsf{V}_{\mathrm{dR}},s}$ we are viewing
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ and $\mathsf{V}_{\mathrm{dR}}$ as
descents of $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ and
$\mathsf{V}_{B}{\otimes}\mathcal{O}^{\mathrm{an}}_{S}$ over $S$ respectively.

Denote the full subcategory of $\mathsf{R}(S)$ given by these objects by
$\mathsf{R}^{*}_{\mathrm{am}}(S)$.

In the above definition, we say that the pair $(M,\gamma)$ is a “test object”
at $s$ for $\mathsf{V}$ to be weakly AM, but its choice does not matter:

######  Lemma (3.2.4).

For a $\mathsf{V}\in\mathsf{R}(S)$ as in (3.2.3), if the existence of
$\gamma^{\sigma}$ is satisfied by one $(M,\gamma)$, then it is satisfied by
any other pair $(M^{\prime},\gamma^{\prime})$.

###### Proof.

Consider the isomorphism
$\psi:=(\gamma^{\prime})^{-1}\circ\gamma:\omega_{\mathrm{Hdg}}(M)\stackrel{{\scriptstyle\sim}}{{\to}}\omega_{\mathrm{Hdg}}(M^{\prime})$.
Since $M$ and $M^{\prime}$ are both abelian motives (i.e., objects on
$\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$), $\psi$ is absolute Hodge. Then for
any $\sigma\in\mathrm{Aut}(\mathbb{C}/k)$, $\psi$ induces an isomorphism
$\psi^{\sigma}:\omega_{B}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}\omega_{B}((M^{\prime})^{\sigma})$
such that for the Betti components $\psi_{B}$ and $\psi^{\sigma}_{B}$ the
diagrams

${{\omega_{\mathrm{{\acute{e}}t}}(M)}}$${{\omega_{\mathrm{{\acute{e}}t}}(M^{\prime})}}$${{\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})}}$${{\omega_{\mathrm{{\acute{e}}t}}((M^{\prime})^{\sigma})}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\psi_{B}^{\sigma}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\psi_{B}{\otimes}\mathbb{A}_{f}}$
and
${{\omega_{\mathrm{dR}}(M)}}$${{\omega_{\mathrm{dR}}(M^{\prime})}}$${{\omega_{\mathrm{dR}}(M^{\sigma})}}$${{\omega_{\mathrm{dR}}((M^{\prime})^{\sigma})}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\psi_{B}^{\sigma}{\otimes}\mathbb{C}}$$\scriptstyle{\psi_{B}{\otimes}\mathbb{C}}$
(2)

commute. Therefore, the diagrams in (1) commute with $\gamma^{\sigma}$
replaced by $\gamma^{\sigma}\circ(\psi^{\sigma})^{-1}$. ∎

#### (3.2.5)

Our motivation for defining $\mathsf{R}^{*}_{\mathrm{am}}(S)$ is that it is
very flexible to work with. We list some simple observations which we will use
repeatedly later:

  1. (i)

$\mathsf{R}^{*}_{\mathrm{am}}(S)$ is a Tannakian subcategory of
$\mathsf{R}(S)$. In particular, it is stable under taking Tate twists, tensor
products, duals, symmetric and exterior products.

  2. (ii)

Let $S^{\prime}$ be another smooth $k$-variety admitting a morphism to $S$. If
$\mathsf{V}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$, then
$\mathsf{V}|_{S^{\prime}}\in\mathsf{R}^{*}_{\mathrm{am}}(S^{\prime})$; if
$S^{\prime}(\mathbb{C})\to S(\mathbb{C})$ is surjective, then the converse is
also true.

  3. (iii)

$\mathsf{R}_{\mathrm{am}}(S)$ is contained in
$\mathsf{R}^{*}_{\mathrm{am}}(S)$ (see (3.2.7) below).

For readers’ convenience, we spell out a tautology on how the condition of
being weakly abelian motivic interacts with global tensors and level
structures:

#### (3.2.6)

Let $S$ be as above and take
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}_{\mathrm{am}}^{*}(S)$.
Let $\boldsymbol{v}_{B}\in\mathrm{H}^{0}(\mathsf{V}^{\otimes}_{B})$ be a
global section which is everywhere of Hodge type $(0,0)$. Let
$\boldsymbol{v}_{\mathrm{{\acute{e}}t},\mathbb{C}}\in\mathrm{H}^{0}(\mathsf{V}^{\otimes}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}})$
and
$\boldsymbol{v}_{\mathrm{dR},\mathbb{C}}\in\mathrm{H}^{0}(\mathsf{V}^{\otimes}_{\mathrm{dR}}|_{S_{\mathbb{C}}})$
be the étale and de Rham realizations of $\boldsymbol{v}_{B}$ respectively.
Let $s\in S(\mathbb{C})$, $\sigma\in\mathrm{Aut}(\mathbb{C}/k)$ and
$(M,\gamma)$ be a test object at $s$ for $\mathsf{V}$ to be weakly AM and
remember the resulting $\gamma^{\sigma}$. Let
$v_{M}:=\gamma_{B}^{-1}(\boldsymbol{v}_{B,s})\in\omega_{B}(M)^{\otimes}$. Then
since $M$ is an abelian motive, $v_{M}$ is absolute Hodge. Therefore, it makes
sense to form $v_{M}^{\sigma}$ (see (2.2.2)). By (1) the following are
equivalent:

  1. (i)

$\gamma^{\sigma}$ sends $v_{M}^{\sigma}$ to $\boldsymbol{v}_{B,\sigma(s)}$.

  2. (ii)

$\gamma_{B}^{\sigma}{\otimes}\mathbb{A}_{f}$ sends $v_{M}^{\sigma}{\otimes}1$
to
$\boldsymbol{v}_{B,\sigma(s)}{\otimes}1\in\mathsf{V}^{\otimes}_{\mathrm{{\acute{e}}t},\sigma(s)}=\mathsf{V}_{B,\sigma(s)}^{\otimes}{\otimes}\mathbb{A}_{f}$.

  3. (iii)

$\gamma_{B}^{\sigma}{\otimes}\mathbb{C}$ sends $v_{M}^{\sigma}{\otimes}1$ to
$\boldsymbol{v}_{B,\sigma(s)}{\otimes}1\in\mathsf{V}^{\otimes}_{\mathrm{dR},\sigma(s)}=\mathsf{V}^{\otimes}_{B,\sigma(s)}{\otimes}\mathbb{C}$.

Using the tautological remarks in (2.1.1), one easily checks that (ii) (resp.
(iii)) holds for every $s$ and $\sigma$ if and only if
$\boldsymbol{v}_{\mathrm{{\acute{e}}t},\mathbb{C}}$ (resp.
$\boldsymbol{v}_{\mathrm{dR},\mathbb{C}}$) descends to $S$. Therefore, using
the equivalence of (ii) and (iii) with (i), we see that if one of
$\boldsymbol{v}_{\mathrm{{\acute{e}}t},\mathbb{C}}$ or
$\boldsymbol{v}_{\mathrm{dR},\mathbb{C}}$ descends to $S$, then so does the
other.

Similarly, suppose that for some $(G,V,\mathsf{K})$ in (2.3.1) we can put a
$\mathsf{K}$-level structure $[\eta]$ on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$.
Again consider the $(M,\gamma,\gamma^{\sigma})$ above. Let $[\eta_{M}]$ be the
$\mathsf{K}$-level structure on $\omega_{\mathrm{{\acute{e}}t}}(M)$ obtained
by pulling back of $[\eta]_{s}$ along $\gamma_{B}{\otimes}\mathbb{A}_{f}$.
Then (2.3.2) and the commutativity of the first diagram in (3.2.3) implies
that $\gamma_{B}^{\sigma}{\otimes}\mathbb{A}_{f}$ sends $[\eta_{M}]^{\sigma}$
to $[\eta]_{\sigma(s)}$, where $[\eta_{M}]^{\sigma}$ is the composition
$V{\otimes}\mathbb{A}_{f}\to\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$.

Now we return to show (3.2.5)(iii):

######  Proposition (3.2.7).

Let $S$ be a smooth variety over a field $k\subseteq\mathbb{C}$ and
$\mathsf{V}$ be an object of $\mathsf{R}(S)$. If
$\mathsf{V}\in\mathsf{R}_{\mathrm{am}}(S)$, then
$\mathsf{V}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$.

###### Proof.

First, we are allowed to assume that $\mathsf{V}$ is strongly abelian motivic
(not just étale locally) thanks to (3.2.5)(ii). Let
$\mathcal{A}/S,\mathsf{H},\mathsf{H}^{\natural},\boldsymbol{\pi}$ be as in
(3.2.1). Fix an arbitrary $s\in S(\mathbb{C})$ and
$\sigma\in\mathrm{Aut}(\mathbb{C}/k)$. Then the descent of the family
$\mathcal{A}_{\mathbb{C}}/S_{\mathbb{C}}$ to $\mathcal{A}/S$ provides us with
a canonical isomorphism
$(\mathcal{A}_{s})^{\sigma}:=\mathcal{A}_{s}{\otimes}_{\sigma}\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{A}_{\sigma(s)}$
of $\mathbb{C}$-varieties. In analogy to (2.1.1), we denote it by
$\sigma_{\mathcal{A},s}$. It tautologically fits into commutative diagrams

${{\mathrm{H}^{1}_{\mathrm{{\acute{e}}t}}(\mathcal{A}_{s},\mathbb{A}_{f})}}$${{\mathsf{H}_{\mathrm{{\acute{e}}t},s}}}$${{\mathrm{H}^{1}_{\mathrm{{\acute{e}}t}}((\mathcal{A}_{s})^{\sigma},\mathbb{A}_{f})}}$${{\mathrm{H}^{1}_{\mathrm{{\acute{e}}t}}(\mathcal{A}_{\sigma(s)},\mathbb{A}_{f})=\mathsf{H}_{\mathrm{{\acute{e}}t},\sigma(s)}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{\mathcal{A},s}}$$\scriptstyle{\sigma_{\mathsf{H}_{\mathrm{{\acute{e}}t}},s}}$${{\mathrm{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{s})}}$${{\mathsf{H}_{\mathrm{dR},s}}}$${{\mathrm{H}^{1}_{\mathrm{dR}}((\mathcal{A}_{s})^{\sigma}/\mathbb{C})}}$${{\mathrm{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{\sigma(s)}/\mathbb{C})=\mathsf{H}_{\mathrm{dR},\sigma(s)}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{\mathcal{A},s}}$$\scriptstyle{\sigma_{\mathsf{H}_{\mathrm{dR}},s}}$
(3)

We caution the reader that for the diagram on the right, the vertical arrows
are $\sigma$-linear, whereas the horizontal ones are $\mathbb{C}$-linear.

Fix an identification of $\mathsf{V}$ with $\mathrm{im}(\boldsymbol{\pi})$ and
denote by $T$ the tensorial construction such that
$\mathsf{H}^{\natural}=T(\mathsf{H})$. For every $x\in S(\mathbb{C})$, let
$\mathfrak{h}^{1}(\mathcal{A}_{x})\in\mathsf{Mot}_{\mathrm{AH}}(\mathbb{C})$
be the motive whose Betti, étale and de Rham realisations are equal to the
first Betti, étale and de Rham cohomology of $\mathcal{A}_{x}$, respectively,
and let $\mathfrak{h}^{1}(\mathcal{A}_{x})^{\natural}\coloneqq
T(\mathfrak{h}^{1}(\mathcal{A}_{x}))$. We denote by $M_{x}$ the submotive of
$\mathfrak{h}^{1}(\mathcal{A}_{x})^{\natural}$ cut out by the absolute Hodge
cycle $\boldsymbol{\pi}_{B,x}$, which is an idempotent projector on
$\mathfrak{h}^{1}(\mathcal{A}_{x})^{\natural}$. Then we may identify
$\omega_{\mathrm{Hdg}}(M_{x})$ with
$(\mathsf{V}_{B,x},\mathsf{V}_{\mathrm{dR},x})$.

We take $M=M_{s}$ and denote the identification
$\omega_{\mathrm{Hdg}}(M)=(\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s})$ by
$\gamma$. Then we use $(M,\gamma)$ as a test object at $s$ for $\mathsf{V}$ to
be weakly AM. Note that $M^{\sigma}$ is canonically identified with the
submotive of
$T(\mathfrak{h}^{1}((\mathcal{A}_{s})^{\sigma}))=T(\mathfrak{h}^{1}(\mathcal{A}_{x}))^{\sigma}$
cut out by $\boldsymbol{\pi}^{\sigma}_{B,s}$. Now, since
$\boldsymbol{\pi}_{\mathrm{{\acute{e}}t}}$ and
$\boldsymbol{\pi}_{\mathrm{dR}}$ are global sections of
$\mathrm{End}(\mathsf{H}^{\natural}_{\mathrm{{\acute{e}}t}})$ and
$\mathrm{End}(\mathsf{H}^{\natural}_{\mathrm{dR}})$ respectively,
$\sigma_{\mathsf{H}_{\mathrm{{\acute{e}}t}},s}$ sends
$\boldsymbol{\pi}_{\mathrm{{\acute{e}}t},s}$ to
$\boldsymbol{\pi}_{\mathrm{{\acute{e}}t},\sigma(s)}$ and
$\sigma_{\mathsf{H}_{\mathrm{dR}},s}$ sends $\boldsymbol{\pi}_{\mathrm{dR},s}$
to $\boldsymbol{\pi}_{\mathrm{dR},\sigma(s)}$ (cf. (2.1.1)). The diagrams in
(3) hence tell us that $\sigma_{\mathcal{A},s}$ sends
$\boldsymbol{\pi}^{\sigma}_{B,s}$ to $\boldsymbol{\pi}_{B,\sigma(s)}$.
Therefore, we conclude that $\sigma_{\mathcal{A},s}$ induces an isomorphism
$M^{\sigma}=(M_{s})^{\sigma}\stackrel{{\scriptstyle\sim}}{{\to}}M_{\sigma(s)}$
of objects in $\mathsf{Mot}_{\mathsf{Ab}}(\mathbb{C})$. Denote this
isomorphism by $\sigma_{M,s}$. Then the diagrams (3) induce commutative
diagrams

${{\omega_{\mathrm{{\acute{e}}t}}(M)}}$${{\mathsf{V}_{\mathrm{{\acute{e}}t},s}}}$${{\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})}}$${{\omega_{\mathrm{{\acute{e}}t}}(M_{\sigma(s)})=\mathsf{V}_{\mathrm{{\acute{e}}t},\sigma(s)}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{M,s}}$$\scriptstyle{\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}}$${{\omega_{\mathrm{dR}}(M)}}$${{\mathsf{V}_{\mathrm{dR},s}}}$${{\omega_{\mathrm{dR}}(M^{\sigma})}}$${{\omega_{\mathrm{dR}}(M_{\sigma(s)})=\mathsf{V}_{\mathrm{dR},\sigma(s)}}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$$\scriptstyle{\sigma_{M,s}}$$\scriptstyle{\sigma_{\mathsf{V}_{\mathrm{dR}},s}}$
(4)

In orther words, the Hodge realization of $\sigma_{M,s}$ provides the
isomorphism $\gamma^{\sigma}$ we are seeking for the test object
$(M=M_{s},\gamma)$ at $s$. ∎

### 3.3 Level Structures on Systems

Finally, we make some definitions regarding level structures on systems of
realizations. The first two are not used until section 4. Let
$(G,V,\mathsf{K})$ be as in (2.3.1), and we recall the notations introduced
there. Let $S$ be a smooth variety over $k\subseteq\mathbb{C}$ and
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}(S)$.
Let $[\eta]$ be a $\mathsf{K}$-level structure on
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$.

######  Definition (3.3.1).

We say that $[\eta]$ on $\mathsf{V}_{\mathrm{{\acute{e}}t}}$ is
$\mathsf{V}$-rational if it satisfies the following property: For every $v\in
I:=(V^{\otimes})^{G}$, the global section
$\eta(v)_{\mathrm{{\acute{e}}t}}\in\mathrm{H}^{0}(\mathsf{V}_{\mathrm{{\acute{e}}t}}^{\otimes})$
(see (2.3.2)) is the étale component of a global section
$\boldsymbol{v}=(\boldsymbol{v}_{B},\boldsymbol{v}_{\mathrm{dR}},\boldsymbol{v}_{\mathrm{{\acute{e}}t}})\in\mathrm{H}^{0}(\mathsf{V}^{\otimes})$
(i.e.,
$\eta(v)_{\mathrm{{\acute{e}}t}}=\boldsymbol{v}_{\mathrm{{\acute{e}}t}}$). We
denote $\boldsymbol{v}$ by $\eta(v)$. Moreover, for every $s\in
S(\mathbb{C})$, $(\mathsf{V}_{B,s},\\{\eta(v)_{B,s}\\}_{v\in
I})\,{\cong}\,(V,\\{v\\}_{v\in I})$.

######  Definition (3.3.2).

If $V$ is a faithful representation, $\Omega$ is a $G(\mathbb{R})$-conjugacy
class of morphisms $\mathbb{S}\to G_{\mathbb{R}}$, and $[\eta]$ is
$\mathsf{V}$-rational, we say that $(\mathsf{V},[\eta])$ is of type $\Omega$
if for every $s\in S(\mathbb{C})$, under some (and hence every) isomorphism
$(\mathsf{V}_{B,s},\\{\eta(v)_{B,s}\\}_{v\in I})\,{\cong}\,(V,\\{v\\}_{v\in
I})$ the Hodge structure on $\mathsf{V}_{B,s}$ is defined by an element of
$\Omega$.

######  Definition (3.3.3).

A $\mathsf{K}$-level structure $[\eta]$ on
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ is said to be rigid on $\mathsf{V}$ if
for any $s\in S(\mathbb{C})$, there is no nontrivial automorphism of
$\mathsf{V}_{B,s}$ which preserves the Hodge filtration on
$\mathsf{V}_{\mathrm{dR},s}$ and the level structure $[\eta]_{s}$.

### 3.4 Galois Descent Lemmas

######  Lemma (3.4.1).

Let $S$ be a smooth variety over a field $k\subseteq\mathbb{C}$ and take
$\mathsf{V},\mathsf{W}\in\mathsf{R}_{\mathrm{am}}^{*}(S)$. Let
$\varphi_{\mathbb{C}}$ be a morphism
$\mathsf{V}|_{S_{\mathbb{C}}}\to\mathsf{W}|_{S_{\mathbb{C}}}$. Then
$\varphi_{\mathbb{C}}$ descends to a morphism $\mathsf{V}\to\mathsf{W}$ if and
only if either the étale or the de Rham component of $\varphi_{\mathbb{C}}$
descends to $S$.

###### Proof.

Considering $\varphi_{\mathbb{C}}$ as a global section of
$(\mathsf{V}^{\vee}{\otimes}\mathsf{W})|_{S_{\mathbb{C}}}$, we know that
either its étale or de Rham component descent to $S$. By (3.2.6) this implies
that $\varphi_{\mathbb{C}}$ itself descends to $S$. ∎

######  Corollary (3.4.2).

Let $S$ be a smooth variety over a field $k\subseteq\mathbb{C}$. Suppose that
$(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$ is a VHS over
$S_{\mathbb{C}}$ equipped with a descent $\mathsf{V}_{\mathrm{{\acute{e}}t}}$
of $\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ to $S$. If there exists a descent
$(\mathsf{V}_{\mathrm{dR}},\varphi:\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}\,{\cong}\,\mathsf{V}_{\mathrm{dR},\mathbb{C}})$
of $\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ as a filtered flat vector bundle over
$S$ such that
$\mathsf{V}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}(S)$
belongs to the subcategory $\mathsf{R}^{*}_{\mathrm{am}}(S)$, then the descent
$(\mathsf{V}_{\mathrm{dR}},\varphi)$ is unique up to unique isomorphism.

Recall that an automorphism of $(\mathsf{V}_{\mathrm{dR}},\varphi)$ is an
automorphism of $\mathsf{V}_{\mathrm{dR}}$ which fixes $\varphi$, so it has to
be the identity. Therefore, it suffices to prove uniqueness up to isomorphism.

###### Proof.

Assume that we have
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}}),\mathsf{V}^{\prime}=(\mathsf{V}_{B},\mathsf{V}^{\prime}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}^{\ast}_{\mathrm{am}}(S)$
such that there exist isomorphisms
$\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\varphi}}{{\cong}}\mathsf{V}_{\mathrm{dR},\mathbb{C}}\stackrel{{\scriptstyle\varphi^{\prime}}}{{\cong}}\mathsf{V}^{\prime}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$.
By Lemma (3.4.1) the isomorphism
$(\mathrm{id}_{\mathsf{V}_{B}},\varphi^{\prime}\circ\varphi,\mathrm{id}_{\mathsf{V}_{\mathrm{{\acute{e}}t}}}|_{S_{\mathbb{C}}})\colon\mathsf{V}|_{S_{\mathbb{C}}}\to\mathsf{V}^{\prime}|_{S_{\mathbb{C}}}$
descends to $S$ since its étale component does, proving the claim.∎

######  Lemma (3.4.3).

Let $S$ be a smooth variety over a field $k\subseteq\mathbb{C}$ and
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$
be an object of $\mathsf{R}^{*}_{\mathrm{am}}(S)$. Suppose that $S$ is an
étale cover of another smooth $k$-variety $T$, and the restriction
$\mathsf{V}|_{S_{\mathbb{C}}}$ descends to
$\mathsf{W}_{\mathbb{C}}=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR},\mathbb{C}},\mathsf{W}_{\mathrm{{\acute{e}}t},\mathbb{C}})\in\mathsf{R}(T_{\mathbb{C}})$.
Consider the diagram

${{S_{\mathbb{C}}}}$${S}$${{T_{\mathbb{C}}}}$${T}$$\scriptstyle{\lrcorner}$

If
$(\mathsf{V}_{\mathrm{{\acute{e}}t}},\mathsf{W}_{\mathrm{{\acute{e}}t},\mathbb{C}})$
admits a simultaneous descent to a local system
$\mathsf{W}_{\mathrm{{\acute{e}}t}}$ over $T$, then
$(\mathsf{V}_{\mathrm{dR}},\mathsf{W}_{\mathrm{dR},\mathbb{C}})$ also admits a
simultaneous descent to some filtered flat vector bundle
$\mathsf{W}_{\mathrm{dR}}$ over $T$, such that
$(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}},\mathsf{W}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}^{*}_{\mathrm{am}}(T)$.
Moreover, if $\mathsf{V}\in\mathsf{R}_{\mathrm{am}}(S)$, then
$\mathsf{W}\in\mathsf{R}_{\mathrm{am}}(T)$.

We remark that the identifications
$\mathsf{W}_{B}{\otimes}\mathcal{O}^{\mathrm{an}}_{T_{\mathbb{C}}}=\mathsf{W}_{\mathrm{dR}}|_{T_{\mathbb{C}}}$
and
$\mathsf{W}_{B}{\otimes}\mathbb{A}_{f}=\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{T_{\mathbb{C}}}$
are encoded in the definition of simultaneous descent (see (2.1.2)).

###### Proof.

Let $p_{1},p_{2}$ be the two projections of $U:=S\times_{T}S$ to $S$. The
descent of $\mathsf{V}_{\mathbb{C}}:=\mathsf{V}|_{S_{\mathbb{C}}}$ to
$\mathsf{W}_{\mathbb{C}}$ is provided by an isomorphism
$\varphi_{\mathbb{C}}:(p_{1}^{*}\mathsf{V})|_{U_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}(p_{2}^{*}\mathsf{V})|_{U_{\mathbb{C}}}$
that satisfies the cocycle condition. That
$(\mathsf{V}_{\mathrm{{\acute{e}}t}},\mathsf{W}_{\mathrm{{\acute{e}}t},\mathbb{C}})$
admits a simultanenous descent to a local system
$\mathsf{W}_{\mathrm{{\acute{e}}t}}$ over $T$ implies that the étale component
of $\varphi_{\mathbb{C}}$ descends to $U$, so that (3.4.1) implies that
$\varphi_{\mathbb{C}}$ descends to an isomorphism
$\varphi:p_{1}^{*}\mathsf{V}\stackrel{{\scriptstyle\sim}}{{\to}}p_{2}^{*}\mathsf{V}$
over $U$. Note that $\varphi$ still satisfies the cocycle condition, because
it is a condition that can be checked after we base change to
$U_{\mathbb{C}}$. The de Rham component of $\varphi$, which respects the
connections and filtrations, provides the descent datum to construct the
desired descent of filtered flat vector bundle $\mathsf{W}_{\mathrm{dR}}$, so
that
$\mathsf{W}:=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}},\mathsf{W}_{\mathrm{{\acute{e}}t}})$
is a system of realizations over $T$. By (3.2.5)(ii), that
$\mathsf{V}\in\mathsf{R}^{*}_{\mathrm{am}}(S)$ implies
$\mathsf{W}\in\mathsf{R}^{*}_{\mathrm{am}}(T)$. If moreover
$\mathsf{V}\in\mathsf{R}_{\mathrm{am}}(S)$, then
$\mathsf{W}\in\mathsf{R}_{\mathrm{am}}(T)$ by the very definition of
$\mathsf{R}_{\mathrm{am}}(-)$. ∎

######  Lemma (3.4.4).

_(Rigidity Lemma)_ Let $S$ be a smooth variety over a field
$k\subseteq\mathbb{C}$ and $(G,V,\mathsf{K})$ be as in (2.3.1). Let
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$
and
$\mathsf{W}=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}},\mathsf{W}_{\mathrm{{\acute{e}}t}})$
be objects of $\mathsf{R}^{*}_{\mathrm{am}}(S)$ equipped with rigid
$\mathsf{K}$-level structures $[\eta]$ and $[\xi]$ respectively. Then every
isomorphism
$\lambda:(\mathsf{V},[\eta])|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W},[\xi])|_{S_{\mathbb{C}}}$
over $S_{\mathbb{C}}$ descends to an isomorphism
$(\mathsf{V},[\eta])\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W},[\xi])$
over $S$.

###### Proof.

The conclusion amounts to saying that the isomorphisms
$\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$
and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$
over $S_{\mathbb{C}}$ given by $\lambda$ descend to isomorphisms
$\mathsf{V}_{\mathrm{dR}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{dR}}$
and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t}}$
over $S$ respectively, and the latter moreover sends $[\eta]$ to $[\xi]$. In
fact, it suffices to show that
$\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{S_{\mathbb{C}}}$
descends. Indeed, it is then automatic that $[\eta]$ is sent to $[\xi]$ (cf.
(2.3.2)), and one applies (3.4.1) to conclude that
$\mathsf{V}_{\mathrm{dR}}|_{S_{\mathbb{C}}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{dR}}|_{S_{\mathbb{C}}}$
descends as well.

We are given that $\lambda$ sends
$[\eta^{\mathrm{an}}]:=[\eta]_{S_{\mathbb{C}}}$ to
$[\xi^{\mathrm{an}}]:=[\xi]_{S_{\mathbb{C}}}$. Write $\lambda_{x}$ for the
fiber of $\lambda$ at a point $x\in S(\mathbb{C})$ and $\lambda_{B,x}$ for its
Betti component. Note that $\lambda_{B,x}{\otimes}\mathbb{A}_{f}$ sends
$[\eta^{\mathrm{an}}]_{x}$ to $[\xi^{\mathrm{an}}]_{x}$.

Fix an arbitrary $s\in S(\mathbb{C})$ and
$\sigma\in\mathrm{Aut}(\mathbb{C}/k)$. Let $(M,\gamma)$ be a test object at
$s$ for $\mathsf{V}$ to be weakly AM and remember the resulting
$\gamma^{\sigma}:\omega_{\mathrm{Hdg}}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{V}_{B,\sigma(s)},\mathsf{V}_{\mathrm{dR},\sigma(s)})$
(see (3.2.3)). Set $\nu=\lambda_{s}\circ\gamma$. Then $(M,\nu)$ becomes a test
object at $s$ for $\mathsf{W}$, and take the resulting
$\nu^{\sigma}:\omega_{\mathrm{Hdg}}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W}_{B,\sigma(s)},\mathsf{W}_{\mathrm{dR},\sigma(s)})$
such that the diagrams in (3.2.3) commute with $\gamma^{\sigma}$ replaced by
$\nu^{\sigma}$. Let $[\eta_{M}]$ be the level structure on
$\omega_{\mathrm{{\acute{e}}t}}(M)$ given by pulling back
$[\eta^{\mathrm{an}}]_{s}$ along $\gamma_{B}{\otimes}\mathbb{A}_{f}$. Then
since $\lambda_{B,s}{\otimes}\mathbb{A}_{f}$ sends $[\eta^{\mathrm{an}}]_{s}$
to $[\xi^{\mathrm{an}}]_{s}$, $[\eta_{M}]$ is also the pullback of
$[\xi^{\mathrm{an}}]_{s}$ along $\nu_{B}{\otimes}\mathbb{A}_{f}$. By (3.2.6),
$\gamma^{\sigma}_{B}{\otimes}\mathbb{A}_{f}$ sends $[\eta_{M}]^{\sigma}$ to
$[\eta^{\mathrm{an}}]_{\sigma(s)}$ and
$\nu^{\sigma}_{B}{\otimes}\mathbb{A}_{f}$ sends $[\eta_{M}]^{\sigma}$ to
$[\xi^{\mathrm{an}}]_{\sigma(s)}$.

Here comes the key observation: Since
$\lambda_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}$ sends
$[\eta^{\mathrm{an}}]_{\sigma(s)}$ to $[\xi^{\mathrm{an}}]_{\sigma(s)}$,
$\lambda_{\sigma(s)}\circ\gamma^{\sigma}$ is another isomorphism
$\omega_{\mathrm{Hdg}}(M^{\sigma})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W}_{B,\sigma(s)},\mathsf{W}_{\mathrm{dR},\sigma(s)})$
whose étale realization sends $[\eta_{M}]^{\sigma}$ to
$[\xi^{\mathrm{an}}]_{\sigma(s)}$. By the rigidity assumption, there is no
nontrivial automorphism of the Hodge structure
$(\mathsf{W}_{B,\sigma(s)},\mathsf{W}_{\mathrm{dR},\sigma(s)})$ whose étale
realization fixes $[\xi^{\mathrm{an}}]_{\sigma(s)}$. Therefore, we must have
$\nu^{\sigma}=\lambda_{\sigma(s)}\circ\gamma^{\sigma}$.

Now one readily checks that the following diagram commutes:

${{\omega_{\mathrm{{\acute{e}}t}}(M)}}$${{\mathsf{V}_{B,s}{\otimes}\mathbb{A}_{f}}}$${{\mathsf{W}_{B,s}{\otimes}\mathbb{A}_{f}}}$${{\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})}}$${{\mathsf{V}_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}}}$${{\mathsf{W}_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}}}$$\scriptstyle{\lambda_{B,s}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\gamma_{B}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\nu_{B}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\sigma_{\mathsf{V}_{\mathrm{{\acute{e}}t}},s}}$$\scriptstyle{\lambda_{B,\sigma(s)}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\sigma_{\mathsf{W}_{\mathrm{{\acute{e}}t}},s}}$$\scriptstyle{\gamma^{\sigma}_{B}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\nu_{B}^{\sigma}{\otimes}\mathbb{A}_{f}}$$\scriptstyle{\,{\cong}\,_{\mathrm{bc}}}$

Indeed, by the construction of $\gamma^{\sigma}$ and $\nu^{\sigma}$, the two
squares which contain
$\omega_{\mathrm{{\acute{e}}t}}(M)\,{\cong}\,_{\mathrm{bc}}\omega_{\mathrm{{\acute{e}}t}}(M^{\sigma})$
commute. The upper triangle commutes by definition of $\nu$, and the lower
triangle commutes by the preceeding paragraph. Therefore, the square at the
front has to commute as well. As $s\in S(\mathbb{C})$ and
$\sigma\in\mathrm{Aut}(\mathbb{C}/k)$ were arbitrary, this implies that
$\lambda_{B}{\otimes}\mathbb{A}_{f}$ descends to an isomorphism
$\mathsf{V}_{\mathrm{{\acute{e}}t}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{W}_{\mathrm{{\acute{e}}t}}$
as desired. ∎

######  Remark (3.4.5).

The above rigidity lemma roughly says that Galois descent of isomorphisms
between weakly AM systems of realizations is forced upon us by the existence
of certain rigid level structures. The following well known example can be
seen as a special case of the above lemma, in which the role played by the
rigid level structure is made more explicit: Suppose for $i=1,2$ and $N\geq
3$, $(E_{i},\alpha_{i})$ is an elliptic curve over a smooth variety $S$ over
$k\subseteq\mathbb{C}$ with a (full) $N$-level structure
$\alpha_{i}:\underline{(\mathbb{Z}/N\mathbb{Z})}_{S}^{\oplus
2}\stackrel{{\scriptstyle\sim}}{{\to}}(E_{i}/S)[N]$. Set
$\mathsf{K}:=\ker(\mathrm{GL}_{2}(\widehat{\mathbb{Z}})\to\mathrm{GL}_{2}(\mathbb{Z}/N\mathbb{Z}))$.
Then the first cohomology of $E_{i}$ defines an object
$\mathsf{H}_{i}\in\mathsf{R}_{\mathrm{am}}^{\circ}(S)$ and $\alpha_{i}$
defines a $\mathsf{K}$-level structure on $\mathsf{H}_{i}$.

If
$(E_{1},\alpha_{1})|_{S_{\mathbb{C}}}\,{\cong}\,(E_{2},\alpha_{2})|_{S_{\mathbb{C}}}$
(or equivalently
$(\mathsf{H}_{1},\alpha_{1})|_{S_{\mathbb{C}}}\,{\cong}\,(\mathsf{H}_{2},\alpha_{2})|_{S_{\mathbb{C}}}$),
then in fact $(E_{1},\alpha_{1})\,{\cong}\,(E_{2},\alpha_{2})$ over $S$
already (and hence
$(\mathsf{H}_{1},\alpha_{1})\,{\cong}\,(\mathsf{H}_{2},\alpha_{2})$). This
follows a fortiori from the existence of a fine moduli space of elliptic
curves with level $N$-structure over $k$. Indeed, two $S$-valued points on
this moduli space must be equal if they are equal after tensoring with
$\mathbb{C}$. But without using the moduli theory this can be seen as follows:
Without loss of generality, suppose that $S$ is connected and $K$ is the
fraction field of $S$. Then $K$-forms of $(E_{i},\alpha_{i})|_{K}$ are
classified by
$\mathrm{H}^{1}(\mathrm{Gal}_{K},\mathrm{Aut}((E_{i},\alpha_{i})|_{\bar{K}}))$,
which is trivial as $\mathrm{Aut}((E_{i},\alpha_{i})|_{\bar{K}})$ is trivial.
The resulting isomorphism
$(E_{1},\alpha_{1})|_{K}\,{\cong}\,(E_{2},\alpha_{2})|_{K}$ (which has to be
unique) extends over $S$.

##  4 Shimura Varieties

Let $(G,\Omega)$ be a Shimura datum which satisfies the axioms in [9, Ch II,
(2.1)]. Let $E(G,\Omega)$ be the reflex field, $Z$ be the center of $G$ and
$Z_{s}$ be the maximal anisotropic subtorus of $Z$ that is split over
$\mathbb{R}$. In this note, we always assume that

$\text{the weight is defined over $\mathbb{Q}$ and $Z_{s}$ is trivial}.$ (5)

Note that in particular the latter condition ensures that $Z(\mathbb{Q})$ is
discrete in $Z(\mathbb{A}_{f})$ ([11, Rmk 5.27]). We will often drop the
Hermitian symmetric domain $\Omega$ from the notation of Shimura varieties
when no confusion would arise.

For any neat compact open subgroup $\mathsf{K}\subseteq G(\mathbb{A}_{f})$,
let $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ denote the resulting Shimura
variety with a complex uniformization $G(\mathbb{Q})\backslash\Omega\times
G(\mathbb{A}_{f})/\mathsf{K}$ and let $\mathrm{Sh}_{\mathsf{K}}(G)$ denote the
canonical model over $E(G,\Omega)$. Let $\mathrm{Sh}(G)$ denote the inverse
limit $\varprojlim_{\mathsf{K}}\mathrm{Sh}_{\mathsf{K}}(G)$ as $\mathsf{K}$
runs through all compact open subgroups. Under our assumptions,
$\mathrm{Sh}(G)(\mathbb{C})$ is described by
$G(\mathbb{Q})\backslash\Omega\times G(\mathbb{A}_{f})$ ([11, (5.28)]). Note
that $\mathrm{Sh}_{\mathsf{K}}(G)=\mathrm{Sh}(G)/\mathsf{K}$.

### 4.1 Automorphic VHS and étale local systems

Let $G\to\mathrm{GL}(V)$ be a representation. For any neat compact open
subgroup $\mathsf{K}\subseteq G(\mathbb{A}_{f})$, we can attach to
$\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ a polarizbale VHS
$\mathsf{V}_{\mathrm{Hdg}}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$
(cf. [9, Ch. II 3.3], [15, §2.2]). In particular, $\mathsf{V}_{B}$ is defined
to be the contraction product $V\times^{G(\mathbb{Q})}[\Omega\times
G(\mathbb{A}_{f})/\mathsf{K}]$. The étale local system defined by
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ can be alternatively constructed as
$(V{\otimes}\mathbb{A}_{f})\times^{\mathsf{K}}[G(\mathbb{Q})\backslash\Omega\times
G(\mathbb{A}_{f})]$. This allows us to descend it to
$\mathrm{Sh}_{\mathsf{K}}(G)$. To be pedantic, choose an
$\widehat{\mathbb{Z}}$-lattice $\Lambda$ which is stabilized by $\mathsf{K}$
and let $\rho:\mathsf{K}\to\mathrm{GL}(\Lambda)$ be the representation. Set
$\mathsf{K}_{n}=\mathsf{K}\cap\rho^{-1}\ker(\mathrm{GL}(\Lambda)\to\mathrm{GL}(\Lambda/n))$.
The natural map
$\mathrm{Sh}_{\mathsf{K}_{n}}(G)_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$
is a finite étale map with deck transformation group
$\mathsf{K}/\mathsf{K}_{n}$. Then we define a $\Lambda/n$-local system on
$\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ by

$L_{n}:=(\Lambda/n)\times^{\mathsf{K}/\mathsf{K}_{n}}\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}.$

Then
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}=(\varprojlim_{n}L_{n}){\otimes}\mathbb{Q}$,
where $n$ goes to $\infty$ multiplicatively. Since
$\mathrm{Sh}_{\mathsf{K}_{n}}(G)_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$
is defined over $E(G,\Omega)$, the étale sheaf
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}$ descends to
$\mathrm{Sh}_{\mathsf{K}}(G)$. The resulting etale sheaf
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ over $\mathrm{Sh}_{\mathsf{K}}(G)$ is
well defined and is independent of the choice of $\Lambda$, so we obtain a
well defined tensor functor from the category $\mathrm{Rep}(G)$ of
$\mathbb{Q}$-representations of $G$ to
$\mathsf{Loc}(\mathrm{Sh}_{\mathsf{K}}(G))$ (cf. the description in [5,
340-341]). We say that $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$
is the automorphic VHS on $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ is the automorphic étale local system on
$\mathrm{Sh}_{\mathsf{K}}(G)$ associated to $V$.

We use a superscript $\mathsf{K}$ (e.g.,
$\mathsf{V}_{\mathrm{Hdg}}^{\mathsf{K}}$) when we need to emphasize that we
are considering the objects at level $\mathsf{K}$. We also remark that
$\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}}$ admits a tautological
$\mathsf{K}$-level structure $[\eta^{\mathsf{K}}_{V}]$, since by construction
its restriction to $\mathrm{Sh}(G)$ comes equipped with a trivialization
$V{\otimes}\underline{\mathbb{A}_{f}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}}|_{\mathrm{Sh}(G)}$,
and $\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}}|_{\mathrm{Sh}(G)}$ admits
a $G(\mathbb{A}_{f})$-equivariant structure. Conversely, the trivialization
over $\mathrm{Sh}(G)$ is recovered by
$\varprojlim_{\mathsf{K}^{\prime}\subseteq\mathsf{K}}[\eta_{V}^{\mathsf{K}^{\prime}}]$.

The reader may easily check that the above constructions are suitably
functorial in $V$:

######  Theorem (4.1.1).

Let the objects
$\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}},\mathsf{V}_{\mathrm{{\acute{e}}t}}$
and $[\eta_{V}]$ be as defined above for a neat $\mathsf{K}$. Let $W$ be
another representation of $G$ and construct the corresponding
$\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR},\mathbb{C}},\mathsf{W}_{\mathrm{{\acute{e}}t}}$,
and $[\eta_{W}]$. Every morphism $V\to W$ induces a morphism
$(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})\to(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR},\mathbb{C}})$
of VHS over $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ and a morphism
$\mathsf{V}_{\mathrm{{\acute{e}}t}}\to\mathsf{W}_{\mathrm{{\acute{e}}t}}$ of
étale local systems over $\mathrm{Sh}_{\mathsf{K}}(G)$. These induced
morphisms are uniquely characterized by the following properties:

  1. (a)

The restriction of
$\mathsf{V}_{\mathrm{{\acute{e}}t}}\to\mathsf{W}_{\mathrm{{\acute{e}}t}}$ to
$\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ is equal to
$\mathsf{V}_{B}{\otimes}\mathbb{A}_{f}\to\mathsf{W}_{B}{\otimes}\mathbb{A}_{f}$
under the canonical identifications.

  2. (b)

The diagram

${{V{\otimes}\underline{\mathbb{A}_{f}}}}$${{\mathsf{V}_{\mathrm{{\acute{e}}t}}|_{\mathrm{Sh}(G)}}}$${{W{\otimes}\underline{\mathbb{A}_{f}}}}$${{\mathsf{W}_{\mathrm{{\acute{e}}t}}|_{\mathrm{Sh}(G)}}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

commutes, where the horizontal arrows are the trivializations provided by
$\varprojlim_{\mathsf{K}^{\prime}\subseteq\mathsf{K}}[\eta^{\mathsf{K}^{\prime}}_{V}]$
and
$\varprojlim_{\mathsf{K}^{\prime}\subseteq\mathsf{K}}[\eta^{\mathsf{K}^{\prime}}_{W}]$
respectively.

Define
$[\eta^{\mathrm{an}}_{V}]:=[\eta_{V}]_{\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}}$.
Using the automorphic VHS
$(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$, we can already give a
moduli interpretation of $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$:

######  Theorem (4.1.2).

Assume that $V$ is faithful. For every smooth $\mathbb{C}$-variety $T$, let
$\mathcal{M}_{V}(T)$ be the groupoid of pairs
$(\mathbb{W},[\xi^{\mathrm{an}}])$ where
$\mathbb{W}=(\mathbb{W}_{B},\mathbb{W}_{\mathrm{dR}})$ is a VHS over $T$ and
$[\xi^{\mathrm{an}}]$ is a $\mathsf{K}$-level structure on
$\mathbb{W}_{B}{\otimes}\mathbb{A}_{f}$ which is $\mathbb{W}$-rational, and
$(\mathbb{W},[\xi^{\mathrm{an}}])$ is of type $\Omega$.

Then
$(\mathsf{V}_{\mathbb{C}}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}}),[\eta^{\mathrm{an}}_{V}])$
is an object of $\mathcal{M}_{V}(\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}})$
and for every object $(\mathbb{W},[\xi^{\mathrm{an}}])\in\mathcal{M}_{V}(T)$
there exists a unique morphism
$\rho:T\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ such that
$\rho^{*}(\mathsf{V}_{\mathbb{C}},[\eta^{\mathrm{an}}_{V}])\,{\cong}\,(\mathbb{W},[\xi^{\mathrm{an}}])$.

###### Proof.

This follows from [10, Prop. 3.10]. Let $I$ be an indexing set of
$(V^{\otimes})^{G}$ and write the elements in $(V^{\otimes})^{G}$ as
$\\{s_{\alpha}\\}_{\alpha\in I}$. That $[\xi^{\mathrm{an}}]$ is
$\mathbb{W}$-rational implies that for every $\alpha$,
$\xi^{\mathrm{an}}(s_{\alpha})_{\mathrm{{\acute{e}}t}}$ comes from a global
section in $\mathrm{H}^{0}(\mathbb{W}_{B}^{\otimes})$ which is everywhere of
Hodge type $(0,0)$ (see (3.3.1)) and let us denote it by
$\mathbf{t}_{\alpha,B}$. It is not hard to check that
$[\eta^{\mathrm{an}}_{V}]$ is $\mathsf{V}_{\mathbb{C}}$-rational, and we let
$\mathbf{s}_{\alpha,B}\in\mathrm{H}^{0}(\mathsf{V}_{B})$ be the global section
defined by $\eta_{V}(s_{\alpha})_{\mathrm{{\acute{e}}t}}$.

Here is a sketch of the argument in loc. cit. in our notations: Using the
complex uniformization of $\mathrm{Sh}_{\mathsf{K}}(G)(\mathbb{C})$ and the
fact that both $[\eta^{\mathrm{an}}_{V}]$ and $[\xi^{\mathrm{an}}]$ are of
$\Omega$-type, one checks that for each $t\in T(\mathbb{C})$, there is a
unique $s\in\mathrm{Sh}_{\mathsf{K}}(G)(\mathbb{C})$ such that
$(\mathbb{W}_{t},\\{\mathbf{t}_{\alpha,B,t}\\},[\xi^{\mathrm{an}}]_{t})\,{\cong}\,((\mathsf{V}_{B,s},\mathsf{V}_{\mathrm{dR},s}),\\{\mathbf{s}_{\alpha,B,s}\\},[\eta^{\mathrm{an}}_{V}]_{s})$,
so that $t\mapsto s$ defines a map of sets
$T(\mathbb{C})\to\mathrm{Sh}_{\mathsf{K}}(G)(\mathbb{C})$. Then one uses the
assumption that $\mathbb{W}$ is a VHS (so that the Hodge filtration on
$\mathbb{W}_{t}$ varies holomorphically with $t$) to show that the map is
holomorphic. Finally, a theorem of Baily and Borel (recalled in Thm 2.24 in
loc. cit.) says that any such holomorphic map is algebraic. ∎

######  Remark (4.1.3).

We remark that the assumption (5) together with the neatness of $\mathsf{K}$
implies that for every $(\mathbb{W},[\xi^{\mathrm{an}}])\in\mathcal{M}_{V}(T)$
as above, the $\mathsf{K}$-level structure $[\xi^{\mathrm{an}}]$ is rigid, as
defined in (3.3.3) (cf. [15, §2.1], [10, Rmk 3.11]). In particular, the
groupoid $\mathcal{M}_{V}(T)$ is in fact a set. The above theorem implies that
$\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ represents the functor
$\mathcal{M}_{V}(-)$ from the category of smooth $\mathbb{C}$-varieties to
sets.

######  Remark (4.1.4).

Unlike the set up in [10, Prop. 3.10], we do not keep a collection of global
Hodge tensors ($\mathfrak{s}$ in Milne’s notation, which plays the role of
$\\{\mathbf{t}_{\alpha,B}\\}$ in ours) of $\mathbb{W}$ as part of the data for
objects parametrized by $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$. Instead,
we impose the condition that the $\mathsf{K}$-level structure
$[\xi^{\mathrm{an}}]$ needs to be $\mathbb{W}$-rational. Doing so does not
change the mathematical content because $\mathbf{t}_{\alpha,B}$’s are uniquely
determined by the level structure $[\xi^{\mathrm{an}}]$ provided that they
exist.

### 4.2 Constructing Systems of Realizations

We shall work under the following assumption:

######  Definition (4.2.1).

We say that a Shimura datum $(G,\Omega)$ of abelian type which satisfies (5)
has a nice Hodge-type cover if there exists a morphism of Shimura data
$(\widetilde{G},\widetilde{\Omega})\to(G,\Omega)$ such that

  1. (i)

$(\widetilde{G},\widetilde{\Omega})$ is of Hodge type and also satisfies
assumption (5);

  2. (ii)

$\widetilde{G}\to G$ is surjective and the kernel lies in the center of
$\widetilde{G}$;

  3. (iii)

the induced embedding on reflex fields $E(G,\Omega)\subseteq
E(\widetilde{G},\widetilde{\Omega})$ is an equality.

In the above, by “cover” we just mean that $\widetilde{G}$ can be viewed as a
cover of $G$. The map between Hermitian symmetric domains
$\widetilde{\Omega}\to\Omega$ is usually not surjective, but only a cover over
some connected components of $\Omega$. In practice,
$\mathrm{Sh}(\widetilde{G},\widetilde{\Omega})$ does function as an “étale
cover” of $\mathrm{Sh}(G,\Omega)$ up to taking Hecke translates (e.g., in the
proof of (4.2.2) below).

In general, a Shimura datum $(G,\Omega)$ is said to be of abelian type if
there exists a Shimura datum $(\widetilde{G},\widetilde{\Omega})$ of Hodge
type and a central isogeny between derived subgroups
$\widetilde{G}^{\mathrm{der}}\to G^{\mathrm{der}}$ which induces an
isomorphism between the adjoint Shimura data
$(\widetilde{G}^{\mathrm{ad}},\widetilde{\Omega}^{\mathrm{ad}})\stackrel{{\scriptstyle\sim}}{{\to}}(G^{\mathrm{ad}},\Omega^{\mathrm{ad}})$,
so that the relationship between $(G,\Omega)$ and
$(\widetilde{G},\widetilde{\Omega})$ is more indirect than the one in (4.2.1);
in particular, $E(G,\Omega)$ could be strictly smaller than
$E(\widetilde{G},\widetilde{\Omega})$.

######  Theorem (4.2.2).

Assume that $(G,\Omega)$ is a Shimura datum of abelian type which satisfies
(5) and has a nice Hodge-type cover. Let $\mathsf{K}\subseteq
G(\mathbb{A}_{f})$ be a neat compact open subgroup. Let $G\to\mathrm{GL}(V)$
be a representation and $(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$
(resp. $\mathsf{V}_{\mathrm{{\acute{e}}t}}$) be the associated automorphic VHS
over $\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$ (resp. automorphic étale local
system on $\mathrm{Sh}_{\mathsf{K}}(G)$).

Then there exists a unique descent $\mathsf{V}_{\mathrm{dR}}$ of
$\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ to $\mathrm{Sh}_{\mathsf{K}}(G)$ such
that
$\mathsf{V}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$
is an object of $\mathsf{R}^{*}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$.
Moreover, $\mathsf{V}$ lies in the subcategory
$\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$.

We call
$\mathsf{V}:=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$
as above the automorphic system of realizations associated to $V$. Note that
by (3.4.2), the descent $\mathsf{V}_{\mathrm{dR}}$ of
$\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ that makes $\mathsf{V}$ an object of
$\mathsf{R}^{*}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$ is automatically
unique provided that it exists. We write $\mathsf{V}$ as
$\mathsf{V}^{\mathsf{K}}=(\mathsf{V}^{\mathsf{K}}_{B},\mathsf{V}^{\mathsf{K}}_{\mathrm{dR}},\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}})$
when we need to emphasize that we are considering objects at level
$\mathsf{K}$. We start with a preliminary reduction.

######  Lemma (4.2.3).

The truth of to (4.2.2) is independent of $\mathsf{K}$ (as long as
$\mathsf{K}$ is neat).

###### Proof.

Suppose that $\mathsf{K}^{\prime}\subseteq\mathsf{K}$. If we have already
constructed the desired $\mathsf{V}_{\mathrm{dR}}^{\mathsf{K}}$ for
$\mathsf{K}$ over $S:=\mathrm{Sh}_{\mathsf{K}}(G)$, then we can simply
pullback $\mathsf{V}_{\mathrm{dR}}^{\mathsf{K}}$ to
$S^{\prime}:=\mathrm{Sh}_{\mathsf{K}^{\prime}}(G)$, since the pullback of
$(\mathsf{V}_{B}^{\mathsf{K}},\mathsf{V}^{\mathsf{K}}_{\mathrm{dR},\mathbb{C}})$
to $S^{\prime}_{\mathbb{C}}$ and that of
$\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}}$ to $S^{\prime}$ are
identified with
$(\mathsf{V}_{B}^{\mathsf{K}^{\prime}},\mathsf{V}_{\mathrm{dR},\mathbb{C}}^{\mathsf{K}^{\prime}})$
and $\mathsf{V}_{\mathrm{{\acute{e}}t}}^{\mathsf{K}^{\prime}}$ respectively.
It is clear that the pullback of an object in $\mathsf{R}_{\mathrm{am}}(S)$ is
an object in $\mathsf{R}_{\mathrm{am}}(S^{\prime})$. Conversely, suppose that
we have constructed $\mathsf{V}_{\mathrm{dR}}^{\mathsf{K}^{\prime}}$ over
$S^{\prime}$ so that
$(\mathsf{V}^{\mathsf{K}^{\prime}}_{B},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{dR}},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}^{\prime}}(G))$.
Note that
$(\mathsf{V}^{\mathsf{K}^{\prime}}_{B},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{dR},\mathbb{C}})$
descends to $S^{\prime}_{\mathbb{C}}$ and
$\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{{\acute{e}}t}}$ descends to $S$
because
$(\mathsf{V}^{\mathsf{K}}_{B},\mathsf{V}^{\mathsf{K}}_{\mathrm{dR},\mathbb{C}})$
and $\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}}$ already exist. We
conclude by (3.4.3) that the object
$(\mathsf{V}^{\mathsf{K}^{\prime}}_{B},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{dR}},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}_{\mathrm{am}}(S^{\prime})$
descends to $S$. In particular, this gives the desired
$\mathsf{V}_{\mathrm{dR}}^{\mathsf{K}}$. ∎

######  Lemma (4.2.4).

Theorem (4.2.2) holds if the Shimura datum $(G,\Omega)$ in question is of
Hodge type.

###### Proof.

By (4.2.3) it suffices to consider the case when $\mathsf{K}$ is sufficiently
small. By assumption, there exists an embedding of Shimura data
$(G,\Omega)\hookrightarrow(\mathrm{GSp},\mathcal{H}^{\pm})$ where
$\mathrm{GSp}=\mathrm{GSp}(H)$ for some symplectic space $H$ and
$\mathcal{H}^{\pm}$ is the Siegel half planes. Let
$K\subseteq\mathrm{GSp}(\mathbb{A}_{f})$ be some sufficiently small compact
open subgroup and assume $\mathsf{K}\subseteq K$ up to shrinking $\mathsf{K}$,
so that we obtain a morphism
$\mathrm{Sh}_{\mathsf{K}}(G)\to\mathrm{Sh}_{K}(\mathrm{GSp})_{E}$, where
$E:=E(G,\Omega)$ is the reflex field. Let $\mathcal{A}$ be the universal
abelian scheme over $\mathrm{Sh}_{K}(\mathrm{GSp})$. Let
$\mathsf{H}=(\mathsf{H}_{B},\mathsf{H}_{\mathrm{dR}},\mathsf{H}_{\mathrm{{\acute{e}}t}})$
be the system of realizations over $S=\mathrm{Sh}_{\mathsf{K}}(G)$ given by
the first cohomology of $\mathcal{A}_{S}$.

Using that $H$ is a faithful representation of $G$, we may find a $N\gg 0$
with the property that $V$ is isomorphic to a subquotient of
$T_{N}(H):=\oplus_{a,b\leq N}H^{{\otimes}a}{\otimes}(H^{\vee})^{{\otimes}b}$
and define $T_{N}(\mathsf{H})$ similarly. Since $G$ is reductive, there exists
a $G$-invariant idempotent projector $\pi\in\mathrm{End}(T_{N}(H))$ such that
$V\,{\cong}\,\mathrm{im}(\pi)$ as $G$-representations. Fix such an
isomorphism. Note that we may view $\pi$ as a morphism
$\mathbb{Q}\to\mathrm{End}(T_{N}(H))$ (here $\mathbb{Q}$ stands for the
trivial representation of $G$). Therefore, by (4.1.1) we obtain a
$\boldsymbol{\pi}_{B}\in\mathrm{H}^{0}(\mathrm{End}(T_{N}(\mathsf{H}_{B})))$
which is everywhere of Hodge type $(0,0)$ (i.e., $\boldsymbol{\pi}_{B}$
defines an endomorphism of $T_{N}(\mathsf{H}|_{S_{\mathbb{C}}})$ as a VHS) and
whose étale realization descends to a global section
$\boldsymbol{\pi}_{\mathrm{{\acute{e}}t}}\in
H^{0}(\mathrm{End}(T_{N}(\mathsf{H}_{\mathrm{{\acute{e}}t}})))$. By (3.2.6),
the de Rham realization of $\boldsymbol{\pi}_{B}$ also descends to some
$\boldsymbol{\pi}_{\mathrm{dR}}\in\mathrm{H}^{0}(\mathrm{End}(T_{N}(\mathsf{H}_{\mathrm{dR}})))$,
so that
$\boldsymbol{\pi}:=(\boldsymbol{\pi}_{B},\boldsymbol{\pi}_{\mathrm{dR}},\boldsymbol{\pi}_{\mathrm{{\acute{e}}t}})\in\mathrm{H}^{0}(\mathrm{End}(T_{N}(\mathsf{H})))$.
Note that by (4.1.1) the fixed identification $V=\mathrm{im}(\pi)$ gives rise
to identifications
$\mathsf{V}|_{S_{\mathbb{C}}}=\mathrm{im}(\boldsymbol{\pi})|_{S_{\mathbb{C}}}$
and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}=\mathrm{im}(\boldsymbol{\pi}_{\mathrm{{\acute{e}}t}})$.
Therefore, we may construct $\mathsf{V}_{\mathrm{dR}}$ by taking
$\mathrm{im}(\boldsymbol{\pi}_{\mathrm{dR}})$. Then by construction
$\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}})$
is an object of $\mathsf{R}_{\mathrm{am}}(S)$. ∎

Now we prove (4.2.2) in full:

###### Proof.

Again by (4.2.3) we may assume that $\mathsf{K}$ is sufficiently small. Then
we choose a sufficiently small
$\widetilde{\mathsf{K}}\subseteq\widetilde{G}(\mathbb{A}_{f})$ whose image
lies in $\mathsf{K}$, so that we obtain a morphism
$p_{\widetilde{\mathsf{K}},\mathsf{K}}:\mathrm{Sh}_{\widetilde{\mathsf{K}}}(\widetilde{G})\to\mathrm{Sh}_{\mathsf{K}}(G)$
from the morphism of Shimura data. Below we write $\widetilde{S}$ for
$\mathrm{Sh}_{\widetilde{\mathsf{K}}}(\widetilde{G})$ and $S$ for
$\mathrm{Sh}_{\mathsf{K}}(G)$. We note that (ii) implies that every connected
component of $\widetilde{\Omega}$ is mapped isomorphically onto some connected
component of $\Omega$, even though $\widetilde{\Omega}\to\Omega$ may not be
surjective. Therefore, $p_{\widetilde{\mathsf{K}},\mathsf{K}}$ is a finite
étale cover of its image $\mathrm{im}(p_{\widetilde{\mathsf{K}},\mathsf{K}})$,
which is a union of connected components of $\mathrm{Sh}_{\mathsf{K}}(G)$.

Viewing $V$ as a $\widetilde{G}$-representation via the projection
$\widetilde{G}\to G$, we obtain the associated automorphic VHS on
$\widetilde{S}_{\mathbb{C}}$ and automorphic étale local system on
$\widetilde{S}$. Denote them by
$(\widetilde{\mathsf{V}}_{B},\widetilde{\mathsf{V}}_{\mathrm{dR},\mathbb{C}})$
and $\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}$ respectively. It is not
hard to see that these are naturally identified with the pullback of
$(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})$ and
$\mathsf{V}_{\mathrm{{\acute{e}}t}}$ along
$p_{\widetilde{\mathsf{K}},\mathsf{K}}$. By (4.2.4) there exists a descent
$\widetilde{\mathsf{V}}_{\mathrm{dR}}$ to $\widetilde{S}$ of
$\widetilde{\mathsf{V}}_{\mathrm{dR},\mathbb{C}}$ such that
$\widetilde{\mathsf{V}}:=(\widetilde{\mathsf{V}}_{B},\widetilde{\mathsf{V}}_{\mathrm{dR}},\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}})$
is an object of $\mathsf{R}_{\mathrm{am}}(\widetilde{S})$. Then we may
construct the desired descent $\mathsf{V}_{\mathrm{dR}}$ over
$\mathrm{im}(p_{\widetilde{\mathsf{K}},\mathsf{K}})$ by applying (3.4.3)
again.

In order to construct $\mathsf{V}_{\mathrm{dR}}$ on connected components not
contained in $\mathrm{im}(p_{\widetilde{\mathsf{K}},\mathsf{K}})$, we consider
Hecke translates. Let $g\in G(\mathbb{A}_{f})$ be any element and set
$\mathsf{K}_{g}:=\mathsf{K}\cap g\mathsf{K}g^{-1}$ and
$S_{g}:=\mathrm{Sh}_{\mathsf{K}_{g}}(G)$. Then $S_{g}$ is a finite étale cover
over $S$, and we denote the natural projection to $S$ by $p_{g}$. Note that
$S_{g,\mathbb{C}}$ has complex uniformization
$G(\mathbb{Q})\backslash\Omega\times G(\mathbb{A}_{f})/\mathsf{K}_{g}$. The
automorphism $(x,g^{\prime})\mapsto(x,g^{\prime}g)$ on $\Omega\times
G(\mathbb{A}_{f})$ descends to a morphism $S_{g,\mathbb{C}}\to
S_{\mathbb{C}}$, which we denote by $T_{g}$. It is well known that $T_{g}$ is
defined over the reflex field, i.e., descends to $S_{g}\to S$.

Now we recall that for varying $\mathsf{K}^{\prime}$, the automorphic VHS
$\\{(\mathsf{V}^{\mathsf{K}^{\prime}}_{B},\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{dR},\mathbb{C}})\\}_{\mathsf{K}^{\prime}}$
and the étale local systems
$\\{\mathsf{V}^{\mathsf{K}^{\prime}}_{\mathrm{{\acute{e}}t}}\\}_{\mathsf{K}^{\prime}}$
on $\mathrm{Sh}(G)$ (or its $\mathbb{C}$-fiber) are
$G(\mathbb{A}_{f})$-equivariant. In particular, there are natural
identifications
$T_{g,\mathbb{C}}^{*}(\mathsf{V}^{\mathsf{K}}_{B},\mathsf{V}^{\mathsf{K}}_{\mathrm{dR},\mathbb{C}})=(\mathsf{V}^{\mathsf{K}_{g}}_{B},\mathsf{V}^{\mathsf{K}_{g}}_{\mathrm{dR},\mathbb{C}})$
and
$T_{g}^{*}(\mathsf{V}^{\mathsf{K}}_{\mathrm{{\acute{e}}t}})=\mathsf{V}^{\mathsf{K}_{g}}_{\mathrm{{\acute{e}}t}}$.
Without loss of generality we may assume that the image of
$\widetilde{\mathsf{K}}\subseteq\widetilde{G}(\mathbb{A}_{f})$ in
$G(\mathbb{A}_{f})$ lies in $\mathsf{K}_{g}$. Then
$p_{\widetilde{\mathsf{K}},\mathsf{K}}$ factors through
$p_{\widetilde{\mathsf{K}},\mathsf{K}_{g}}$. Consider now the composition

${{\widetilde{S}}}$${{S_{g}}}$${S}$$\scriptstyle{p_{\widetilde{\mathsf{K}},\mathsf{K}_{g}}}$$\scriptstyle{T_{g}}$

which we shall denote by $\mathfrak{p}_{g}$. Then we have identifications
$\mathfrak{p}_{g,\mathbb{C}}^{*}(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR},\mathbb{C}})=(\widetilde{\mathsf{V}}_{B},\widetilde{\mathsf{V}}_{\mathrm{dR},\mathbb{C}})$
and
$\mathfrak{p}_{g}^{*}\mathsf{V}_{\mathrm{{\acute{e}}t}}=\widetilde{\mathsf{V}}_{\mathrm{{\acute{e}}t}}$.
We may now apply (3.4.3) again to descend
$\widetilde{\mathsf{V}}\in\mathsf{R}_{\mathrm{am}}(\widetilde{S})$ to
$\mathrm{im}(\mathfrak{p}_{g})$, which in particular defines a descent of
$\mathsf{V}_{\mathrm{dR},\mathbb{C}}$ when restricted to
$\mathrm{im}(\mathfrak{p}_{g})_{\mathbb{C}}$. For each connected component
$S^{\circ}$ of $S$, and each $g\in G(\mathbb{A}_{f})$ such that
$S^{\circ}\subseteq\mathrm{im}(\mathfrak{p}_{g})$, the resulting descent of
$\mathsf{V}_{\mathrm{dR},\mathbb{C}}|_{S^{\circ}_{\mathbb{C}}}$ does not
depend on $g$, by (3.4.2). We may now conclude using the fact that there is a
finite subset $J\subseteq G(\mathbb{A}_{f})$ such that $S=\cup_{g\in
J}\mathrm{im}(\mathfrak{p}_{g})$. ∎

######  Theorem (4.2.5).

For each neat $\mathsf{K}$, the association $V\mapsto\mathsf{V}^{\mathsf{K}}$
given by (4.2.2) defines a functor
$\mathcal{F}_{\mathsf{K}}:\mathrm{Rep}(G)\to\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$.
Moreover, if $\mathsf{K}^{\prime}\subseteq\mathsf{K}$,
$\mathcal{F}_{\mathsf{K}^{\prime}}$ is the composition of
$\mathcal{F}_{\mathsf{K}}$ and the pullback functor
$\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}^{\prime}}(G))\to\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}_{\mathsf{K}}(G))$;
in other words, $\varprojlim_{\mathsf{K}}\mathcal{F}_{\mathsf{K}}$ defines a
functor from $\mathrm{Rep}(G)$ to the category of
$G(\mathbb{A}_{f})$-equivariant objects in
$\mathsf{R}_{\mathrm{am}}(\mathrm{Sh}(G))$.

###### Proof.

This follows readily from (4.1.1) and (3.4.1). ∎

### 4.3 Moduli Interpretations over Reflex Fields

We are now ready to give the moduli interpretations of the Shimura varieties
over the reflex fields, under our simplifying assumptions.

######  Theorem (4.3.1).

Let $(G,\Omega)$ be a Shimura datum of abelian type which satisfies (5) and
has a nice Hodge-type cover. Assume that $V\in\mathrm{Rep}(G)$ is faithful.
Let $E^{\prime}\subseteq\mathbb{C}$ be a subfield which contains
$E=E(G,\Omega)$ and $T$ be a smooth $E^{\prime}$-variety. Let
$\mathcal{M}_{V,E^{\prime}}(T)$ be the groupoid of pairs $(\mathsf{W},[\xi])$
where
$\mathsf{W}=(\mathsf{W}_{B},\mathsf{W}_{\mathrm{dR}},\mathsf{W}_{\mathrm{{\acute{e}}t}})\in\mathsf{R}^{*}_{\mathrm{am}}(T)$,
and $[\xi]$ is a $\mathsf{K}$-level structure on
$\mathsf{W}_{\mathrm{{\acute{e}}t}}$ such that $[\xi]$ is
$\mathsf{W}$-rational and $(\mathsf{W},[\xi])$ is of type $\Omega$.

Then
$(\mathsf{V}=(\mathsf{V}_{B},\mathsf{V}_{\mathrm{dR}},\mathsf{V}_{\mathrm{{\acute{e}}t}}),[\eta_{V}])$
defined by (4.2.2) with $[\eta_{V}]$ be the tautological $\mathsf{K}$-level
structure is an object of $\mathcal{M}_{V,E}(\mathrm{Sh}_{\mathsf{K}}(G))$,
and for each $(\mathsf{W},[\xi])\in\mathcal{M}_{V,E^{\prime}}(T)$, there
exists a unique $\rho:T\to\mathrm{Sh}_{\mathsf{K}}(G)_{E^{\prime}}$ such that
$(\mathsf{W},[\xi])\,{\cong}\,\rho^{*}(\mathsf{V},[\eta_{V}])$.

Note that the objects in $\mathcal{M}_{V,E^{\prime}}(T)$ above are rigid, so
$\mathcal{M}_{V,E^{\prime}}(T)$ is a set (cf. (4.1.3)).

###### Proof.

By applying (4.1.2), we first construct a morphism
$\rho_{\mathbb{C}}:T_{\mathbb{C}}\to\mathrm{Sh}_{\mathsf{K}}(G)_{\mathbb{C}}$
such that there is an isomorphism

$\nu:\rho_{\mathbb{C}}^{*}(\mathsf{V},[\eta_{V}])\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W},[\xi])|_{T_{\mathbb{C}}}.$

Next, we argue that $\rho_{\mathbb{C}}$ descends over $E^{\prime}$. Let $t\in
T(\mathbb{C})$ be a point and let
$x\in\mathrm{Sh}_{\mathsf{K}}(G)(\mathbb{C})$ be its image under
$\rho_{\mathbb{C}}$. It suffices to show that for any
$\sigma\in\mathrm{Aut}(\mathbb{C}/E^{\prime})$,
$\rho_{\mathbb{C}}(\sigma(t))=\sigma(x)$. Let $(M,\gamma)$ be a test object at
$t$ for $\mathsf{W}$ to be weakly AM (see (3.2.3)). Let $[\xi_{M}]$ be the
$\mathsf{K}$-level structure on $\omega_{\mathrm{{\acute{e}}t}}(M)$ obtained
by pulling back $[\xi]_{t}$ along $\gamma_{B}{\otimes}\mathbb{A}_{f}$. Then by
(3.2.6), there is an isomorphism
$(\mathsf{W}_{\sigma(t)},[\xi]_{\sigma(t)})\,{\cong}\,(\omega_{\mathrm{Hdg}}(M^{\sigma}),[\xi_{M}]^{\sigma})$.
Note that the fiber $\nu_{t}$ of $\nu$ at $t$ is an isomorphism
$(\mathsf{V}_{x},[\eta_{V}]_{x})\stackrel{{\scriptstyle\sim}}{{\to}}(\mathsf{W}_{t},[\xi]_{t})$.
Therefore, $(M,\nu_{t}^{-1}\circ\gamma)$ is a test object at $x$ for
$\mathsf{V}$ to be weakly AM. Moreover, $[\xi_{M}]$ is the pullback of
$[\eta_{V}]_{x}$ along $(\nu_{t}^{-1}\circ\gamma)_{B}{\otimes}\mathbb{A}_{f}$.
By (3.2.6) again, there is an isomorphism
$(\mathsf{V}_{\sigma(x)},[\eta_{V}]_{\sigma(x)})\,{\cong}\,(\omega_{\mathrm{Hdg}}(M^{\sigma}),[\xi_{M}]^{\sigma})$.
Therefore, we have shown that
$(\mathsf{V}_{\sigma(x)},[\eta_{V}]_{\sigma(x)})\,{\cong}\,(\mathsf{W}_{\sigma(t)},[\xi]_{\sigma(t)})$
as they are both isomorphic to
$(\omega_{\mathrm{Hdg}}(M^{\sigma}),[\xi_{M}]^{\sigma})$. This implies by
(4.1.2) that $\rho_{\mathbb{C}}(\sigma(t))=\sigma(x)$.

Now we have shown that $\rho_{\mathbb{C}}$ descends to a morphism
$\rho:T\to\mathrm{Sh}_{\mathsf{K}}(G)_{E^{\prime}}$, and there is an
isomorphism between $(\mathsf{W},[\xi])$ and $\rho^{*}(\mathsf{V},[\eta_{V}])$
when restricted to $T_{\mathbb{C}}$. We conclude by the rigidity lemma (3.4.4)
that in fact we must have
$(\mathsf{W},[\xi])\,{\cong}\,\rho^{*}(\mathsf{V},[\eta_{V}])$. ∎

######  Remark (4.3.2).

The above proof is certainly inspired by that of [7, Cor. 5.4], which deals
with the special case when $(G,\Omega)$ is an orthogonal type Shimura datum,
$V$ is the standard representation of $G$, $T$ is the moduli space of quasi-
polarized K3 surfaces of a certain degree, and $(\mathsf{W},[\xi])$ is given
by the primitive cohomology of the universal family. We remark though that
after Madapusi-Pera proved the rationality of the peirod map, he proved in [7,
Prop. 5.6] that
$\rho^{*}\mathsf{V}_{\mathrm{{\acute{e}}t}}\,{\cong}\,\mathsf{W}_{\mathrm{{\acute{e}}t}}$
by resorting to Deligne’s big monodromy trick. It is in fact unnecessary if we
only work with neat level structures, by the rigidity lemma (3.4.4).

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