# Shimura varieties (version of )

Sophie Morel  UMPA UMR CNRS 5669
ENS Lyon Site Monod
46 Allée d’Italie
69364 Lyon Cedex 07<EMAIL_ADDRESS>
(Date: today)

###### Abstract.

Lecture 1 gives an introduction to Shimura varieties over the complex numbers
(defined here as a special type of locally symmetric spaces) and to the
general theory of canonical models; it also discusses in more detail the
example of the Siegel modular varieties. Lecture 2 presents some families of
Shimura varieties (PEL type, Hodge type, abelian type) and the results that
are known about their canonical and integral models. Finally, lecture 3
discusses the cohomology of Shimura varieties, concentrating mostly on the
compact non-endoscopic case.

###### 2020 Mathematics Subject Classification:

Primary

I thank the organizers of the IHES summer school (Pierre-Henri Chaudouard, Wee
Teck Gan, Tasho Kaletha and Yannis Sakellaridis) for the enormous amount of
work they put into the school, and the audience for putting up with my
lectures. I also thank Mikhail Borovoi and Yujie Xu for remarks and
corrections on an earlier version of these notes, as well as the anonymous
referee for their careful reading of the texts and numerous comments and
corrections.

##  1\. Lecture 1: locally symmetric spaces and Shimura varieties

###  1.1. Locally symmetric spaces

Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, for example
$\mathrm{SL}_{n}$, $\mathrm{Sp}_{2n}$, or a special orthogonal or special
unitary group. Locally symmetric spaces for $G(\mathbb{R})$ are “nice enough”
spaces whose cohomology is related to automorphic representations of $G$. A
good reference for locally symmetric spaces is the introductory paper [51] by
Ji.

To simplify the presentation, we will assume here that $G(\mathbb{R})$ is
connected. Let $K_{\infty}$ be a maximal compact subgroup of $G(\mathbb{R})$,
and let $X=G(\mathbb{R})/K_{\infty}$. If $\Gamma$ is a discrete subgroup of
$G(\mathbb{R})$ such that $\Gamma\backslash G(\mathbb{R})$ (or equivalently
$\Gamma\backslash X$) is compact and that $\Gamma$ acts properly and freely on
$X$, 111This holds for example if $\Gamma$ is torsion free. then there is a
classical connection between the cohomology of $\Gamma\backslash X$ and
automorphic representations of $G(\mathbb{R})$, called _Matsushima’s formula_
(see Matsushima’s paper [81]). We will state a modern reformulation in Lecture
3 (see Theorem 3.1), but roughly it relates the Betti numbers of
$\Gamma\backslash X$ and the multiplicities of representations of
$G(\mathbb{R})$ in $L^{2}(\Gamma\backslash G(\mathbb{R}))$.

In fact, Matsushima’s paper deals with semi-simple real Lie groups. Here, we
have an algebraic group defined over $\mathbb{Q}$, so we have a particularly
nice way to produce discrete subgroups of $G(\mathbb{R})$. Remember that a
subgroup $\Gamma$ of $G(\mathbb{Q})$ is called an _arithmetic subgroup_ if
there exists a closed embedding of algebraic groups $G\subset\mathrm{GL}_{N}$
such that, setting
$G(\mathbb{Z})=G(\mathbb{Q})\cap\mathrm{GL}_{N}(\mathbb{Z})$, we have that
$\Gamma\cap G(\mathbb{Z})$ is of finite index in $\Gamma$ and in
$G(\mathbb{Z})$. 222We can check that this definition does not depend on the
embedding $G\subset\mathrm{GL}_{N}$, see [51, Proposition 4.2]. If $\Gamma$ is
small enough, then it acts properly and freely on $X$ ([51, Proposition 5.5]),
so the quotient $\Gamma\backslash X$ is a real analytic manifold. Also, the
quotient $\Gamma\backslash G(\mathbb{R})$ is compact if and only if $G$ is
anisotropic (over $\mathbb{Q}$), which means that $G$ has no nontrivial
parabolic subgroup defined over $\mathbb{Q}$ ([51, Theorem 5.10]). If
$\Gamma\backslash X$ is not compact but $G(\mathbb{R})$ has a discrete series,
then there is an extension of Matsushima’s formula, due to Borel and Casselman
in [19], that involves $L^{2}$ cohomology of $\Gamma\backslash X$; see 3.5.3.

We actually would like to see automorphic representations of $G(\mathbb{A})$
(not just $G(\mathbb{R})$) in the cohomology of our spaces, so we will use
adelic versions of $\Gamma\backslash X$. Let $K$ be an open compact subgroup
of $G(\mathbb{A}_{f})$, where $\mathbb{A}_{f}$ is the ring of finite adèles;
for example, if we have chosen an embedding $G\subset\mathrm{GL}_{N}$, then we
could take
$K=G(\mathbb{A}_{f})\cap\operatorname{Ker}(\mathrm{GL}_{N}(\widehat{\mathbb{Z}})\rightarrow\mathrm{GL}_{N}(\mathbb{Z}/n\mathbb{Z}))$,
for some positive integer $n$ (these are called _principal congruence
subgroups_). Let

$M_{K}=G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K,$

where the group $K$ acts by right translations on the factor
$G(\mathbb{A}_{f})$, and the group $G(\mathbb{Q})$ acts by left translations
on both factors simultaneously. Choose a system of representatives
$(x_{i})_{i\in I}$ of the finite 333See Lemmas 5.11 and 5.12 of [87]. quotient
$G(\mathbb{Q})\backslash G(\mathbb{A}_{f})/K$, and set
$\Gamma_{i}=G(\mathbb{Q})\cap x_{i}Kx_{i}^{-1}$ for every $i\in I$. Then the
$\Gamma_{i}$ are arithmetic subgroups of $G(\mathbb{Q})$, and we have

$M_{K}=\coprod_{i\in I}\Gamma_{i}\backslash X,$

so $M_{K}$ is a real analytic manifold if $K$ is small enough. But now we have
an action of $G(\mathbb{A}_{f})$ on the projective system $(M_{K})_{K\subset
G(\mathbb{A}_{f})}$, so we get an action on
$\varinjlim_{K}\mathrm{H}^{*}(M_{K})$, where $\mathrm{H}^{*}$ is any
“reasonable” cohomology theory, for example Betti cohomology. If $G$ is
anisotropic over $\mathbb{Q}$, then Matsushima’s result can be reformulated to
give a description of this action in terms of irreducible representations of
$G(\mathbb{A})$ appearing in $L^{2}(G(\mathbb{Q})\backslash G(\mathbb{A}))$;
this is the version we give in Theorem 3.1. There is also a version of the
Borel-Casselman generalization (see 3.5.3).

There is another way to think about the action of $G(\mathbb{A}_{f})$ on
$(M_{K})_{K\subset G(\mathbb{A}_{f})}$, which does not involve a limit on $K$.
Fix a Haar measure on $G(\mathbb{A}_{f})$ such that open compact subgroups of
$G(\mathbb{A}_{f})$ have rational volume (this is possible because these
groups are all commensurable); then every open subset of $G(\mathbb{A}_{f})$
has rational volume, because it is a finite union of translates of open
compact subgroups. The _Hecke algebra_ of $G$ is the space $\mathcal{H}_{G}$
of locally constant functions with compact support from $G(\mathbb{A}_{f})$ to
$\mathbb{Q}$; if $f,g\in\mathcal{H}_{G}$, then the convolution product $f*g$
still has rational values by the choice of Haar measure, so convolution
defines a multiplication on $\mathcal{H}_{G}$. For every open compact subgroup
$K$ of $G(\mathbb{A}_{f})$, the _Hecke algebra at level $K$_ is the subalgebra
$\mathcal{H}_{G,K}$ of bi-$K$-invariant functions in $\mathcal{H}_{G}$; we
have $\mathcal{H}_{G}=\bigcup_{K}\mathcal{H}_{G,K}$.

Fix $K$ small enough. Then $\mathrm{H}^{*}(M_{K})$ is the set of $K$-invariant
vectors in $\varinjlim_{K^{\prime}\subset
G(\mathbb{A}_{f})}\mathrm{H}^{*}(M_{K^{\prime}})$, so it has an action of
$\mathcal{H}_{G,K}$. 444In fact, we can recover the action of
$G(\mathbb{A}_{f})$ on $\varinjlim_{K\subset
G(\mathbb{A}_{f})}\mathrm{H}^{*}(M_{K})$ from the action of
$\mathcal{H}_{G,K}$ on $M_{K}$ for every $K$ small enough. We can describe
this action using Hecke correspondences: let $g\in G(\mathbb{A}_{f})$ and let
$K^{\prime}$ be an open compact subgroup of $G(\mathbb{A}_{f})$ such
$K^{\prime}\subset K\cap gKg^{-1}$. Then we have a _Hecke correspondence_
$(T_{1},T_{g}):M_{K^{\prime}}\rightarrow M_{K}^{2}$ sending the class of
$(x,h)$ in $M_{K^{\prime}}$ to that of $((x,h),(x,hg))$ in $M_{K}\times
M_{K}$, and $T_{1},T_{g}$ are both finite covering maps if $K$ is small
enough. Then the action of
$\frac{1}{\operatorname{vol}(K^{\prime})}1\mkern-5.0mu\mathrm{l}_{KgK}$ on
$\mathrm{H}^{*}(M_{K})$ is given by pulling back cohomology classes along
$T_{1}$, then pushing them forward along $T_{g}$ (see Section 16 of [64]).

We can also ask whether there is more structure on the spaces
$\Gamma\backslash X$ (or $M_{K}$). For example, suppose that
$G=\mathrm{SL}_{2}$ and $K_{\infty}=\mathrm{SO}(2)$. Then, for $\Gamma$ an
arithmetic subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$, the space
$\Gamma\backslash X$ is a modular curve, so it is the set of complex points of
an algebraic variety defined over a number field $F$, and we can use the
commuting actions of Hecke correspondences and of the absolute Galois group of
$F$ on the étale cohomology of this variety to construct some instance of the
global Langlands correspondence for $\mathrm{GL}_{2}$. See for example
Deligne’s Bourbaki seminar [34].

In order to generalize this picture, as outlined in Langlands’s papers [69]
and [70], we first need to know when the spaces $\Gamma\backslash X$ or
$M_{K}$ are the set of $\mathbb{C}$-points of an algebraic variety, and
whether this algebraic variety is defined over a number field. As we will see
in the section on canonical models (1.4), another advantage of $M_{K}$ over
$\Gamma\backslash X$ is that, when the answer to both questions above is
“yes”, then the $M_{K}$ for $K$ varying tend to all be defined over the same
field, while this is not the case for the $\Gamma\backslash X$.

######  Remark 1.1.

The first step is to check whether $\Gamma\backslash X$ has the structure of a
complex manifold, and there are obvious obstructions to that. For example, if
$G=\mathrm{SL}_{3}$ and $K_{\infty}=\mathrm{SO}(3)$, then $\Gamma\backslash X$
is $5$-dimensional as a real manifold, so it cannot have the structure of a
complex manifold.

Choose a $G(\mathbb{R})$-invariant Riemannian metric on
$X=G(\mathbb{R})/K_{\infty}$ (such a metric is unique up to rescaling on each
irreducible factor). Then $X$ is a _symmetric space_ , that is, a Riemannian
manifold such that:

  * (a)

The group of isometries of $X$ acts transitively on $X$;

  * (b)

For every $p\in X$, there exists a symmetry $s_{p}$ of $X$ (i.e. an involutive
isometry) such that $p$ is an isolated fixed point of $s_{p}$.

Moreover, the symmetric space $X$ is of _noncompact type_ , that is, it has
negative curvature. For $\Gamma$ a small enough arithmetic subgroup of
$G(\mathbb{Q})$, the Riemannian manifold $\Gamma\backslash X$ is a _locally
symmetric space_ ; in particular, it does not satisfy condition (a) anymore,
and it satisfies a variant of condition (b) where we only ask for the symmetry
to be defined in a neighborhood of the point. See Ji’s notes [51] for a review
of locally symmetric spaces.

We say that $X$ is a _Hermitian symmetric domain_ if it admits a
$G(\mathbb{R})$-invariant Hermitian metric. See Section 1 of Milne’s notes
[87] for a review of Hermitian symmetric domains.

######  Example 1.2.

Let $d$ be a positive integer. The _Siegel upper half space_
$\mathfrak{h}_{d}^{+}$ is the set of symmetric $d\times d$ complex matrices
$Y\in M_{d}(\mathbb{C})$ such $\operatorname{Im}(Y)$ is positive definite; if
$d=1$, then this is just the usual upper half plane. We claim that
$\mathfrak{h}_{d}^{+}$ is a Hermitian symmetric domain. For the proofs of the
basic properties of $\mathfrak{h}_{d}^{+}$ that we state below, a good
reference is Siegel’s paper [112].

We first need to see $\mathfrak{h}_{d}^{+}$ as a symmetric space. Let
$\mathrm{Sp}_{2d}$ be the symplectic group of the symplectic form with matrix
$\begin{pmatrix}0&I_{d}\\\ -I_{d}&0\end{pmatrix}$, where
$I_{d}\in\mathrm{GL}_{d}(\mathbb{Z})$ is the identity matrix. For every
commutative ring $R$, we have

$\mathrm{Sp}_{2d}(R)=\left\\{g\in\mathrm{GL}_{2d}(R)\mid{}^{t}g\begin{pmatrix}0&I_{d}\\\
-I_{d}&0\end{pmatrix}g=\begin{pmatrix}0&I_{d}\\\
-I_{d}&0\end{pmatrix}\right\\}.$

Note that $\mathrm{Sp}_{2}=\mathrm{SL}_{2}$. We make
$\mathrm{Sp}_{2d}(\mathbb{R})$ act on $\mathfrak{h}_{d}^{+}$ by the following
formula:

$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}\cdot Y=(AY+B)(CY+D)^{-1},$

where $A,B,C,D$ are $d\times d$ matrices such that $\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\mathrm{Sp}_{2d}(\mathbb{R})$ (see page 9 of [112]). Then
this action is transitive (see page 10 of [112]). Let $K_{\infty}$ be the
stabilizer in $\mathrm{Sp}_{2d}(\mathbb{R})$ of
$iI_{d}\in\mathfrak{h}_{d}^{+}$. Then
$K_{\infty}=O(2d)\cap\mathrm{Sp}_{2d}(\mathbb{R})$ (this is easy to check
directly), so it is a maximal compact subgroup of
$\mathrm{Sp}_{2d}(\mathbb{R})$, 555In fact, we have an isomorphism
$\mathrm{U}(d)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}K_{\infty}$ sending
$X+iY\in\mathrm{U}(d)$ (with $X,Y\in\mathrm{GL}_{d}(\mathbb{R})$) to
$\begin{pmatrix}X&Y\\\ -Y&X\end{pmatrix}$. and we have
$\mathfrak{h}_{d}^{+}\simeq\mathrm{Sp}_{2d}(\mathbb{R})/K_{\infty}$ as real
analytic manifolds.

Also, the space $\mathfrak{h}_{d}^{+}$ is an open subset of the complex vector
space of symmetric matrices in $M_{d}(\mathbb{C})$, so it has an obvious
structure of complex manifold. It remains to construct a
$\mathrm{Sp}_{2d}(\mathbb{R})$-invariant Hermitian metric on
$\mathfrak{h}_{d}^{+}$. Let $\mathcal{D}_{d}$ be the set of symmetric matrices
$A\in M_{d}(\mathbb{C})$ such that $I_{d}-A^{*}A$ is positive definite, this
is a bounded domain in the complex vector space of symmetric matrices in
$M_{d}(\mathbb{C})$, hence is equipped with a canonical Hermitian metric
called the _Bergman metric_ , which has negative curvature (see for example
[87, Theorem 1.3]); in particular, this metric is invariant by all holomorphic
automorphisms of $\mathcal{D}_{d}$. Now note that we have an isomorphism
$\mathfrak{h}_{d}^{+}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{D}_{d}$
sending $X\in\mathfrak{h}_{d}^{+}$ to $(iI_{d}-X)(iI_{d}+X)^{-1}$ (whose
inverse sends $A\in\mathcal{D}_{d}$ to $i(I_{d}-A)(I_{d}+A)^{-1}$), see pages
8-9 of [112]. We can give a formula for the resulting Hermitian metric on
$\mathfrak{h}_{d}^{+}$: up to a positive scalar, it is given by

$ds^{2}=\operatorname{Tr}(\operatorname{Im}(Y)^{-2}dY\operatorname{Im}(Y)^{-1}d\overline{Y})$

(see formula (28) on page 17 of [112]).

The isomorphism
$\mathfrak{h}_{d}^{+}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{D}_{d}$
is called a _bounded realization_ of $\mathfrak{h}_{d}^{+}$.

We now state the classification of Hermitian symmetric domains in terms of
real algebraic groups. Let $U(1)=\\{z\in\mathbb{C}\mid|z|=1\\}$.

######  Theorem 1.3.

[87, Theorem 1.21] Suppose that $G(\mathbb{R})$ is adjoint and connected. The
locally symmetric space $X$ is a Hermitian symmetric domain if and only if
there exists a morphism of real Lie groups $u:\mathrm{U}(1)\rightarrow
G(\mathbb{R})$ such that:

  * (a)

The only characters of $\mathrm{U}(1)$ that appear in its representation on
$\operatorname{Lie}(G(\mathbb{R}))$ via $\mathrm{Ad}\circ u$ are $1$, $z$ and
$z^{-1}$;

  * (b)

Conjugation by $u(-1)$ is a Cartan involution of $G(\mathbb{R})$, which means
that the real Lie subgroup $\\{g\in G(\mathbb{C})\mid
g=u(-1)\overline{g}u(-1)^{-1}\\}$ of $G(\mathbb{C})$ is compact;

  * (c)

The image of $u(-1)$ in every simple factor of $G(\mathbb{R})$ is nontrivial.

Moreover, we can choose $u$ such that $K_{\infty}$ is the centralizer of $u$
in $G(\mathbb{R})$, which means that $X$ is isomorphic to the set of
conjugates of $u$ by elements of $G(\mathbb{R})$.

We explain the construction of the morphism $u$. Suppose that $X$ is a
Hermitian symmetric domain, and let $p\in X$. For every $z\in\mathbb{C}$ with
$|z|=1$, multiplication by $z$ on $T_{p}X$ preserves the Hermitian metric and
sectional curvatures, so there exists a unique isometry $u_{p}(z)$ of $X$
fixing $p$ and such that $T_{p}u_{p}(z)$ is multiplication by $z$. The
uniqueness implies that $u_{p}(z)u_{p}(z^{\prime})=u_{p}(zz^{\prime})$ if
$|z|=|z^{\prime}|=1$, so we get a morphism of groups from $\mathrm{U}(1)$ to
the group of isometries of $X$, which is equal to
$G(\mathbb{R})^{0}_{\mathrm{ad}}=G(\mathbb{R})$.

######  Remark 1.4.

If we don’t assume that $G(\mathbb{R})$ is adjoint, then $u$ goes from
$\mathrm{U}(1)$ to $G(\mathbb{R})_{\mathrm{ad}}$, as we see in the sketch
above, and it will not necessarily lift to a morphism from $\mathrm{U}(1)$ to
$G(\mathbb{R})$.

######  Example 1.5.

  * (1)

If $G=\mathrm{Sp}_{2d}$, let $h:\mathbb{C}^{\times}\rightarrow G(\mathbb{R})$
be defined by

$h(a+ib)=\begin{pmatrix}aI_{d}&-bI_{d}\\\ bI_{d}&aI_{d}\end{pmatrix}.$

Then we can take $u:\mathrm{U}(1)\rightarrow\mathrm{PSp}_{2d}(\mathbb{R})$
given by $u(z)=h(\sqrt{z})$. Note that $u$ does not lift to a morphism from
$\mathrm{U}(1)$ into $G(\mathbb{R})$.

  * (2)

If $G=\mathrm{PGL}_{n}$ with $n\geq 3$, then the centralizer of a character
$u:\mathrm{U}(1)\rightarrow G(\mathbb{R})$ cannot be a maximal compact
subgroup of $G(\mathbb{R})$ (exercise), so the locally symmetric space of
maximal compact subgroups of $G(\mathbb{R})$ is not Hermitian.

Theorem 1.3 puts some pretty strong restrictions on the root systems of the
simple factors of $G(\mathbb{R})$, see Theorem 1.25 of [87] and the table
following it. In particular, the type $A$ simple factors of $G(\mathbb{R})$
must be of the form $\mathrm{PSU}(p,q)$, and $G(\mathbb{R})$ can have no
simple factor of type $E_{8}$, $F_{4}$ or $G_{2}$.

The natural next step would be to wonder for which Hermitian symmetric domains
$X$ the quotients $\Gamma\backslash X$ are algebraic varieties, but in fact it
turns out that the answer is “for all of them”, as was proved by Baily and
Borel. Borel later proved that this structure of algebraic variety is unique.

######  Theorem 1.6 (Baily-Borel and Borel, [12] and [17]).

Suppose that $X=G(\mathbb{R})/K_{\infty}$ is a Hermitian symmetric domain.
Then, for any torsion free arithmetic subgroup $\Gamma$ of $G(\mathbb{Q})$,
the quotient $\Gamma\backslash X$ has a unique structure of algebraic variety
over $\mathbb{C}$ compatible with its structure of complex analytic manifold.

The very rough idea is that the sheaf of automorphic forms on
$\Gamma\backslash X$ of sufficiently high weight will define an embedding of
$\Gamma\backslash X$ into a projective space. Borel’s uniqueness theorem
actually says that, if $V$ is a quasi-projective complex algebraic variety,
then any holomorphic map $V\rightarrow X$ is regular; see the discussion below
Theorem 3.14 of [87] for more details on its proof.

Remember that we did not just want the locally symmetric spaces
$\Gamma\backslash X$ to be algebraic varieties, we also wanted them to be
defined over a number field, and we would ideally like the number field in
question to only depend on $G$ and $K_{\infty}$. For this, it will actually be
easier to work with reductive groups instead of semi-simple groups. As a
motivation for this, and for the definition of Shimura varieties, we now spend
some more time on the case of the symplectic group.

###  1.2. The Siegel modular variety

:

See the end of this subsection (1.2.6 on page 1.2.6) for some background on
abelian schemes.

####  1.2.1. Symplectic groups

We use the notation of Example 1.2. It is well-known that
$\mathfrak{h}_{1}^{+}$ parametrizes elliptic curves over $\mathbb{C}$: an
element $\tau\in\mathfrak{h}_{1}^{+}$ is sent to the elliptic curve
$E_{\tau}=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$, and $E_{\tau}\simeq
E_{\tau^{\prime}}$ if and only if $\tau,\tau^{\prime}\in\mathfrak{h}_{1}^{+}$
in the same orbit for the action of
$\mathrm{SL}_{2}(\mathbb{Z})=\mathrm{Sp}_{2}(\mathbb{Z})$; so we can recover
$\tau$ from $E_{\tau}$ and the data of a symplectic isomorphism
$\mathrm{H}_{1}(E_{\tau},\mathbb{Z})\simeq\mathbb{Z}^{2}$ where
$\mathbb{Z}^{2}$ is equipped with the standard symplectic form. We want to
give a similar picture for higher-dimensional abelian varieties; in fact, the
analogy works best if we consider abelian varieties with a principal
polarization (Definition 1.30).

We first introduce some notation and recall the definition of the (general)
symplectic group as a group scheme over $\mathbb{Z}$. If $R$ is a commutative
ring, we denote by $\psi_{R}$ the perfect symplectic pairing on $R^{2d}$ with
matrix $\begin{pmatrix}0&I_{d}\\\ -I_{d}&0\end{pmatrix}$. So we have

$\psi_{R}((x_{1},\ldots,x_{d},y_{1},\ldots,y_{d}),(x^{\prime}_{1},\ldots,x^{\prime}_{d},y^{\prime}_{1},\ldots,y^{\prime}_{d}))=\sum_{i=1}^{d}x_{i}y^{\prime}_{i}-\sum_{i=1}^{d}x^{\prime}_{i}y_{i}.$

The _general symplectic group_ $\mathrm{GSp}_{2d}$ is the reductive group
scheme over $\mathbb{Z}$ whose points over a commutative ring $R$ are given
by:

$\mathrm{GSp}_{2d}(R)=\\{g\in\mathrm{GL}_{2d}(R)\mid\exists c(g)\in
R^{\times},\ \psi_{R}(g\cdot,g\cdot)=c(g)\psi_{R}\\}.$

The scalar $c(g)$ is called the _multiplier_ of $g\in\mathrm{GSp}_{2d}(R)$.
Sending $g$ to $c(g)$ defines a morphism of group schemes
$c:\mathrm{GSp}_{2d}\rightarrow\mathrm{GL}_{1}$, whose kernel
$\mathrm{Sp}_{2d}$ is called the _symplectic group_.

######  Example 1.7.

We have $\mathrm{GSp}_{2}=\mathrm{GL}_{2}$, $c=\det$ and
$\mathrm{Sp}_{2}=\mathrm{SL}_{2}$.

####  1.2.2. Complex abelian varieties

Let $A$ be complex abelian variety of dimension $d$; we identify $A$ and its
set of complex points. Then $A$ is a connected complex Lie group of dimension
$d$, so we have $A\simeq\operatorname{Lie}(A)/\Lambda$, with
$\operatorname{Lie}(A)\simeq\mathbb{C}^{d}$ the universal cover of $A$ and
$\Lambda=\pi_{1}(A)=\mathrm{H}_{1}(A,\mathbb{Z})\simeq\mathbb{Z}^{2d}$ a
lattice in the underlying $\mathbb{R}$-vector space. Let $A^{\vee}$ be the
dual abelian variety, i.e. the space of degree $0$ line bundles on $A$
(Definition 1.28). We can identify $\operatorname{Lie}(A^{\vee})$ with the
space of semi-linear forms on $\operatorname{Lie}(A)$ and
$\mathrm{H}_{1}(A^{\vee},\mathbb{Z})$ with the subspace $\Lambda^{\vee}$ of
forms whose imaginary part takes integer values on $\Lambda$ (see [92], §9).
For every positive integer $n$, we have $A[n]=\frac{1}{n}\Lambda/\Lambda$ and
$A^{\vee}[n]=\frac{1}{n}\Lambda^{\vee}/\Lambda^{\vee}$, and the canonical
pairing $A[n]\times A^{\vee}[n]\rightarrow\mu_{n}(\mathbb{C})$ is given by

$(v,u)\mapsto e^{-2i\pi n\operatorname{Im}(u(v))}$

(see [92], §24). We then have a bijection between the set of polarizations on
$A$ (Definition 1.30) and the set of positive definite Hermitian forms666We
take Hermitian forms to be semi-linear in the first variable and linear in the
second variable. $H$ on $\operatorname{Lie}(A)$ such that the symplectic form
$\operatorname{Im}(H)$ takes integer values on $\Lambda$; given such a form
$H$, the corresponding isogeny $\lambda_{H}$ from $A$ to $A^{\vee}$ is given
on $\mathbb{C}$-points by:

$\left\\{\begin{array}[]{rcl}\lambda_{H}:\operatorname{Lie}(A)/\Lambda&\rightarrow&\operatorname{Lie}(A^{\vee})/\Lambda^{\vee}\\\
w&\mapsto&(v\mapsto H(v,w))\end{array}\right.$

It follows that the Weil pairing (see Remark 1.31(2)) corresponding to
$\lambda_{H}$ is the map

(1.1) $\left\\{\begin{array}[]{rcl}A[n]\times
A[n]&\rightarrow&\mu_{n}(\mathbb{C})\\\ (v,w)&\mapsto&e^{-2i\pi
n\operatorname{Im}(H(v,w))}.\end{array}\right.$

As $v,w\in\frac{1}{n}\Lambda$, we have
$\operatorname{Im}(H(v,w))\in\frac{1}{n^{2}}\mathbb{Z}$, so the formula does
define a $n$th root of $1$ in $\mathbb{C}$.

In particular, the polarization $\lambda_{H}$ is principal if and only if
$\Lambda$ is self-dual with respect to the symplectic form
$\operatorname{Im}(H)$, that is,

$\Lambda=\\{w\in\operatorname{Lie}(A)\mid\forall v\in\Lambda,\
\operatorname{Im}(H(v,w))\in\mathbb{Z}\\}.$

In that case, the symplectic $\mathbb{Z}$-module
$(\Lambda,\operatorname{Im}(H))$ is isomorphic to $\mathbb{Z}^{2d}$ with the
form $\psi_{\mathbb{Z}}$.

Let $\widetilde{\mathcal{M}}_{d}$ be the set of isomorphism classes of triples
$(A,\lambda,\eta_{\mathbb{Z}})$, where $A$ is a $d$-dimensional complex
abelian variety, $\lambda$ is a principal polarization on $A$ and
$\eta_{\mathbb{Z}}$ is an isomorphism of symplectic spaces from
$\mathrm{H}_{1}(A,\mathbb{Z})$ to $(\mathbb{Z}^{2d},\psi_{\mathbb{Z}})$. We
have an action of $\mathrm{Sp}_{2d}(\mathbb{Z})$ on
$\widetilde{\mathcal{M}}_{d}$: if
$c=(A,\lambda,\eta_{\mathbb{Z}})\in\widetilde{\mathcal{M}}_{d}$ and
$x\in\mathrm{Sp}_{2g}(\mathbb{Z})$, set $x\cdot
c=(A,\lambda,x\circ\eta_{\mathbb{Z}})$.

Let $(A,\lambda,\eta_{\mathbb{Z}})\in\widetilde{\mathcal{M}}_{d}$. Then
$\Lambda=\mathrm{H}_{1}(A,\mathbb{Z})$ is a lattice in the real vector space
$\operatorname{Lie}(A)$, we have $A=\operatorname{Lie}(A)/\Lambda$ and, if
$H_{\lambda}$ is the Hermitian form associated to the polarization $\lambda$,
then the symplectic form $\operatorname{Im}(H_{\lambda})_{\mid\Lambda}$ is
sent to the form $\psi_{\mathbb{Z}}$ on $\mathbb{Z}^{2d}$ by the isomorphism
$\eta_{\mathbb{Z}}:\Lambda\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{Z}^{2d}$.
So, if we see $\mathbb{R}^{2d}$ as a complex vector space via the isomorphisms
(of real vector spaces)
$\operatorname{Lie}(A)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\Lambda\otimes_{\mathbb{Z}}\mathbb{R}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{R}^{2d}$,
then the Hermitian form on $\mathbb{R}^{2d}$ corresponding to $H_{\lambda}$ is
$(v,w)\mapsto\psi_{\mathbb{R}}(iv,w)+i\psi_{\mathbb{R}}(v,w)$.

This shows $\eta_{\mathbb{Z}}$ determines all the data of the isomorphism
class of $(A,\lambda,\eta_{\mathbb{Z}})$, except for the structure of complex
vector space on $\mathbb{R}^{2d}\simeq\operatorname{Lie}(A)$. This structure
of complex vector space is equivalent to the data of an $\mathbb{R}$-linear
endomorphism $J$ of $\mathbb{R}^{2d}$ such that $J^{2}=-1$ (the endomorphism
$J$ corresponds to multiplication by $i$). We also need the
$\mathbb{R}$-bilinear map
$\mathbb{R}^{2d}\times\mathbb{R}^{2d}\rightarrow\mathbb{C}$ defined by
$(v,w)\mapsto\psi_{\mathbb{R}}(J(v),w)+i\psi_{\mathbb{R}}(v,w)$ to be a
positive definite Hermitian form on $\mathbb{R}^{2d}$, which is equivalent to
the following conditions:

  * (a)

$\psi_{\mathbb{R}}(J(v),J(w))=\psi_{\mathbb{R}}(v,w)$ for all
$v,w\in\mathbb{R}^{2d}$;

  * (b)

the $\mathbb{R}$-bilinear form $(v,w)\mapsto\psi_{\mathbb{R}}(J(v),w)$ on
$\mathbb{R}^{2d}$ (which is symmetric by (a)) is positive definite.

Conversely, if we have a complex structure $J$ on $\mathbb{R}^{2d}$ satisfying
(a) and (b), then we get a positive definite Hermitian form $H$ on
$\mathbb{R}^{2d}$ whose imaginary part takes integer values on the lattice
$\mathbb{Z}^{2d}$, so the complex torus $A=\mathbb{R}^{2d}/\mathbb{Z}^{2d}$
has a polarization $\lambda$ induced by $H$, hence is an abelian variety (for
example by the Kodaira embedding theorem), and we get an element
$(A,\lambda,\mathbf{id}_{\mathbb{Z}^{2d}})$ of $\widetilde{\mathcal{M}}_{d}$.

To sum up, we have defined a bijection from $\widetilde{\mathcal{M}}_{d}$ to
the set $X^{\prime}$ of endomorphisms $J$ of $\mathbb{R}^{2d}$ such that
$J^{2}=-1$ and that $J$ satisfies conditions (a) and (b).

Now observe that, if $W$ is a $\mathbb{R}$-vector space, then the data of an
endomorphism $J$ of $W$ such that $J^{2}=-1$ (i.e. of the structure of a
$\mathbb{C}$-vector space on $W$) is equivalent to the data of a
$\mathbb{C}$-linear endomorphism $J_{\mathbb{C}}$ of
$W\otimes_{\mathbb{R}}\mathbb{C}$ such that
$W\otimes_{\mathbb{R}}\mathbb{C}=\operatorname{Ker}(J_{\mathbb{C}}-i\cdot\mathbf{id}_{W})\oplus\operatorname{Ker}(J_{\mathbb{C}}+i\cdot\mathbf{id}_{W})$
and
$\operatorname{Ker}(J_{\mathbb{C}}+i\cdot\mathbf{id}_{W})=\overline{\operatorname{Ker}(J_{\mathbb{C}}-i\cdot\mathbf{id}_{W})}$,
where $v\mapsto\overline{v}$ is the involution of
$W\otimes_{\mathbb{R}}\mathbb{C}$ induced by complex conjugation on
$\mathbb{C}$. This is equivalent to giving a $\mathbb{C}$-vector subspace $E$
of $W\otimes_{\mathbb{R}}\mathbb{C}$ such that
$W\otimes_{\mathbb{R}}\mathbb{C}=E\oplus\overline{E}$, i.e. a $d$-dimensional
complex subspace $E$ of $W\otimes_{\mathbb{R}}\mathbb{C}$ such that
$E\cap\overline{E}=\\{0\\}$ (take
$E=\operatorname{Ker}(J_{\mathbb{C}}-i\cdot\mathbf{id}_{W})$). 777In fancier
terms, we are saying that putting a structure of complex vector space on $W$
is the same as putting a pure Hodge structure of type $\\{(-1,0),(0,-1)\\}$ on
it (or of type $\\{(1,0),(0,1)\\}$, depending on your normalization). When
$W=\mathbb{R}^{2d}$ and the complex structure comes from an element
$(A,\lambda,\eta_{\mathbb{Z}})$ of $\widetilde{\mathcal{M}}_{d}$, then this
Hodge structure is the one coming from the Hodge structure on
$\mathrm{H}_{1}(A,\mathbb{R})$ via the isomorphism
$\mathrm{H}_{1}(A,\mathbb{R})\stackrel{{\scriptstyle\eta_{\mathbb{Z}}\otimes\mathbb{R}}}{{\rightarrow}}\mathbb{Z}^{2d}\otimes_{\mathbb{Z}}\mathbb{R}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{R}^{2d}$.

We apply this to $W=\mathbb{R}^{2d}$. Let $J$ be a complex structure on
$\mathbb{R}^{2d}$, and let $E$ be the corresponding $\mathbb{C}$-vector
subspace of $W\otimes_{\mathbb{R}}\mathbb{C}=\mathbb{C}^{2d}$. Then condition
(a) on $J$ is equivalent to the fact that:

  * (a’)

$\psi_{\mathbb{C}}(v,w)=0$ for all $v,w\in E$

(i.e. to the fact that $E$ is a Lagrangian subspace888A maximal isotropic
subspace. of $\mathbb{C}^{2d}$), and condition (b) on $J$ is equivalent to the
fact that

  * (b’)

$-i\psi_{\mathbb{C}}(v,\overline{v})\in\mathbb{R}_{>0}$ for all $v\in
E\setminus\\{0\\}$.

Note that these two conditions on a $\mathbb{C}$-vector subspace $E$ of
$\mathbb{C}^{2d}$ imply that $E\cap\overline{E}=\\{0\\}$. So we get a
bijection from $X^{\prime}$ to the set of Lagrangian subspaces $E$ of
$\mathbb{C}^{2d}$ satisfying (b’).

If we represent Lagrangian subspaces of $\mathbb{C}^{2d}$ by their bases, seen
as complex matrices of size $2d\times d$, then the action of
$\mathrm{Sp}_{2d}(\mathbb{R})$ is just left multiplication. For example, if we
see $\mathbb{R}^{2d}$ as a complex vector space via the isomorphism
$\mathbb{R}^{2d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C}^{d}$
sending $(x_{1},\ldots,x_{d},y_{1},\ldots,y_{d})$ to
$(x_{1}+iy_{1},\ldots,x_{d}+iy_{d})$, then the corresponding endomorphism of
$\mathbb{R}^{2d}$ is $\begin{pmatrix}0&-I_{d}\\\ I_{d}&0\end{pmatrix}\in
X^{\prime}$, and the corresponding subspace of $\mathbb{C}^{2d}$ is the one
with basis $\begin{pmatrix}iI_{d}\\\ I_{d}\end{pmatrix}$.

More generally, if $Y\in\mathfrak{h}_{d}^{+}$, the subspace of
$\mathbb{C}^{2d}$ with basis $\begin{pmatrix}Y\\\ I_{d}\end{pmatrix}$ is a
Lagrangian subspace satisfying condition (b’), and every such Lagrangian
subspace is of that form. So we get bijections
$\widetilde{\mathcal{M}}_{d}\simeq X^{\prime}\simeq\mathfrak{h}_{d}^{+}$, and
we can check that the second bijection is
$\mathrm{Sp}_{2d}(\mathbb{R})$-equivariant. Unraveling the definitions, we see
that $Y\in\mathfrak{h}_{d}^{+}$ corresponds to the element
$(A_{Y},\lambda_{Y},\eta_{\mathbb{Z},Y})$ of $\widetilde{\mathcal{M}}_{d}$
such that $A_{Y}=\mathbb{C}^{d}/(\mathbb{Z}^{d}+Y\mathbb{Z}^{d})$,
$\lambda_{Y}$ is the principal polarization given by the Hermitian form with
matrix $\operatorname{Im}(Y)^{-1}$ on $\mathbb{C}^{d}$, and
$\eta_{\mathbb{Z},Y}:\mathbb{Z}^{d}+Y\mathbb{Z}^{d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{Z}^{2d}$
is the isomorphism sending $a\in\mathbb{Z}^{d}$ to
$(a,0)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}=\mathbb{Z}^{2d}$ and $Ya\in
Y\mathbb{Z}^{d}$ to
$(0,a)\in\mathbb{Z}^{d}\times\mathbb{Z}^{d}=\mathbb{Z}^{2d}$.

Now we want an interpretation of the quotients
$\Gamma\backslash\mathfrak{h}_{d}^{+}$, for $\Gamma$ an arithmetic subgroup of
$\mathrm{Sp}_{2d}(\mathbb{Q})$. We will do this for the groups
$\Gamma(n)=\operatorname{Ker}(\mathrm{Sp}_{2d}(\mathbb{Z})\rightarrow\mathrm{Sp}_{2d}(\mathbb{Z}/n\mathbb{Z}))$,
where $n$ is a positive integer ($\Gamma(n)$ is called the _principal
congruence subgroup_ at level $n$). An arithmetic subgroup of
$\mathrm{Sp}_{2d}(\mathbb{Q})$ that contains some $\Gamma(n)$ is called a
_congruence subgroup_. For $d=1$, there exist arithmetic subgroups that are
not congruence subgroups (see the introduction of Raghunathan’s paper [101]
for a counterexample, attributed to Fricke-Klein); for $d\geq 2$, it is known
that every arithmetic subgroup is a congruence subgroup (this was proved
independently by Bass-Lazard-Serre in [13] and Mennicke in [82]).

We will need the notion of a level structure; we give the general definition
here.

######  Definition 1.8.

Let $S$ be a scheme, $(A,\lambda)$ be a principally polarized abelian scheme
of relative dimension $d$ over $S$, and $n$ be a positive integer. A _level
$n$ structure_ on $(A,\lambda)$ is a couple $(\eta,\varphi)$, where
$\eta:A[n]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\underline{\mathbb{Z}/n\mathbb{Z}}^{2d}_{S}$
and
$\varphi:\underline{\mathbb{Z}/n\mathbb{Z}}_{S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mu_{n,S}$
are isomorphisms of group schemes such that
$\varphi\circ\psi_{\mathbb{Z}/n\mathbb{Z}}\circ(\eta\times\eta)$ is the Weil
pairing associated to $\lambda$ on $A[n]$.

######  Remark 1.9.

A level $n$ structure on $(A,\lambda)$ can only exist if $n$ is invertible on
$S$ and $\mu_{n,S}$ is a constant group scheme.

Note that isomorphisms
$\varphi:\underline{\mathbb{Z}/n\mathbb{Z}}_{S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mu_{n,S}$
correspond to sections $\zeta\in\mu_{n}(S)$ generating $\mu_{n,S}$ (i.e. to
primitive $n$th roots of $1$ over $S$), by sending $\varphi$ to
$\zeta=\varphi(1)$. So we will also see level structures as couples
$(\eta,\zeta)$, with $\zeta\in\mu_{n}(S)$ primitive.

Let $\zeta_{n}=e^{-2i\pi/n}\in\mu_{n}(\mathbb{C})$. If
$Y\in\mathfrak{h}_{d}^{+}$, then $\frac{1}{n}\eta_{\mathbb{Z},Y}$ defines an
isomorphism of groups
$\eta_{Y}:A_{Y}[n]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathbb{Z}/n\mathbb{Z})^{2d}$,
and it follows from formula (1.1) that $(\eta,\zeta_{n})$ is a level $n$
structure on $(A_{Y},\lambda_{Y})$.

Using the fact that
$\mathrm{Sp}_{2d}(\mathbb{Z})\rightarrow\mathrm{Sp}_{2d}(\mathbb{Z}/n\mathbb{Z})$
is surjective for every $n\in\mathbb{N}$, which follows from strong
approximation for $\mathrm{Sp}_{2d}$, 999See the papers [99] and [100] of
Platonov. we finally get:

######  Proposition 1.10.

Let $n$ be a positive integer. The map $Y\mapsto(A_{Y},\lambda_{Y},\eta_{Y})$
induces a bijection from $\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ to the set
of isomorphism classes of triples $(A,\lambda,\eta)$, where $(A,\lambda)$ is a
principally polarized complex abelian variety of dimension $d$ and
$\eta:A[n]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\mathbb{Z}/n\mathbb{Z})^{2d}$
is an isomorphism of groups such that $(\eta,\zeta_{n})$ is a level $n$
structure on $(A,\lambda)$.

So we have reinterpreted $\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ as the set
of $\mathbb{C}$-points of a certain moduli problem. Considering this moduli
problem over more general bases, we have an “obvious” way to make
$\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ into an algebraic variety, which we
now review.

####  1.2.3. The connected Siegel modular variety

Let $\mathcal{O}_{n}=\mathbb{Z}[1/n][T]/(T^{n}-1)$. If $S$ is a scheme over
$\mathcal{O}_{n}$, we denote by
$\varphi_{0}:\underline{\mathbb{Z}/n\mathbb{Z}}_{S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mu_{n,S}$
the isomorphism sending $1$ to the class of $T$.

######  Definition 1.11.

Let $\mathcal{M}^{\prime}_{d,n}$ be the contravariant functor from the
category of $\mathcal{O}_{n}$-schemes to the category of sets sending $S$ to
the set of isomorphisms classes of triples $(A,\lambda,\eta)$, where
$(A,\lambda)$ is a principally polarized abelian scheme of relative dimension
$d$ over $S$ and
$\eta:A[n]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\underline{\mathbb{Z}/n\mathbb{Z}}^{2d}_{S}$
is an isomorphism of group schemes such that $(\eta,\varphi_{0})$ is a level
$n$ structure on $(A,\lambda)$.

An isomorphism from $(A,\lambda,\eta)$ to
$(A^{\prime},\lambda^{\prime},\eta^{\prime})$ is an isomorphism of abelian
varieties $u:A\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A^{\prime}$ such
that $\lambda^{\prime}\circ u=u^{\vee}\circ\lambda$ and
$\eta^{\prime}=\eta\circ(u,u)$, and, for $f:T\rightarrow S$ a morphism of
$\mathcal{O}_{n}$-schemes, the map
$\mathcal{M}^{\prime}_{d,n}(S)\rightarrow\mathcal{M}^{\prime}_{d,n}(T)$ is
base change along $f$.

######  Theorem 1.12 (Mumford, cf. [41]).

Suppose that $n\geq 3$. Then the functor $\mathcal{M}^{\prime}_{d,n}$ is
representable by a smooth quasi-projective $\mathcal{O}_{n}$-scheme purely of
dimension $d(d+1)/2$ and with connected geometric fibers, which we still
denote by $\mathcal{M}^{\prime}_{d,n}$ and call the _connected Siegel modular
variety of level $n$_.

######  Remark 1.13.

If $n\in\\{1,2\\}$, then triples $(A,\lambda,\eta)$ as in Definition 1.11 may
have automorphisms, so we should see $\mathcal{M}^{\prime}_{d,n}$ as a stack.
It will then be representable by a smooth Deligne-Mumford stack over
$\mathcal{O}_{n}$ that is a finite étale quotient of the scheme
$\mathcal{M}^{\prime}_{d,3n}$.

We can now strengthen Proposition 1.10 to the following result.

######  Proposition 1.14.

Let $n\geq 3$ be an integer. Then the map
$Y\mapsto(A_{Y},\lambda_{Y},\eta_{Y})$ induces an isomorphism of complex
algebraic varieties from $\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ to
$\mathcal{M}^{\prime}_{d,n}(\mathbb{C})$.

Let $\alpha$ be the map of the proposition. To show that $\alpha$ is regular,
we use the fact that the triples $(A_{Y},\lambda_{Y},\eta_{Y})$ are obtained
as the fibers of an abelian scheme
$\mathcal{A}\rightarrow\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ given together
with a polarization $\lambda$ and a level $n$ structure $\eta$. The abelian
scheme $\mathcal{A}$ is a locally symmetric space for the non-reductive group
$\mathbb{Q}^{2d}\rtimes\mathrm{Sp}_{2d}$, where $\mathrm{Sp}_{2d}$ acts on
$\mathbb{Q}^{2d}$ via its standard representation, and is an example of a
_mixed Shimura variety_ (see for example Pink’s thesis [96] for the theory of
mixed Shimura varieties and their compactifications), and $\lambda$ and $\eta$
can also be given by explicit formulas interpolating those for $\lambda_{Y}$
and $\eta_{Y}$. Once we have this fact, the map $\alpha$ corresponds to the
$\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$-point of
$\mathcal{M}^{\prime}_{d,n}$ defined by $(\mathcal{A},\lambda,\eta)$, hence it
is a morphism of algebraic varieties.

Let us explain why $\alpha^{-1}$ is holomorphic (for more details, see the
proof of Proposition 1.1 in Milne’s [88]); by Borel’s uniqueness theorem (see
Theorem 1.6), this will imply that it is an isomorphism of algebraic
varieties. We don’t know anything explicit about the algebraic variety
structure on $\mathcal{M}:=\mathcal{M}^{\prime}_{d,n}(\mathbb{C})$, but we do
know that we have a universal abelian scheme
$f:\mathcal{A}\rightarrow\mathcal{M}$ of relative dimension $d$, with a
principal polarization $\lambda$ and a level $n$ structure $\eta$. By
definition, the map $\alpha^{-1}$ sends $x\in\mathcal{M}$ to the element of
$\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ corresponding to the triple
$(\mathcal{A}_{x},\lambda_{x},\eta_{x})$, so we need to understand how to
construct this. The exponential map gives a surjective morphism
$\exp:\operatorname{Lie}(\mathcal{A}/\mathcal{M})\rightarrow\mathcal{A}$ of
abelian groups schemes over $\mathcal{M}$ (where the source is the relative
Lie algebra of $\mathcal{A}\rightarrow\mathcal{M}$), whose kernel is the
relative homology of $\mathcal{A}\rightarrow\mathcal{M}$ with coefficients in
$\mathbb{Z}$. So we get an exact sequence of abelian sheaves over
$\mathcal{M}$:

$0\rightarrow\mathcal{L}:=(R^{1}f_{*}\underline{\mathbb{Z}})^{\vee}\rightarrow\operatorname{Lie}(\mathcal{A}/\mathcal{M})\stackrel{{\scriptstyle\exp}}{{\rightarrow}}\mathcal{A}\rightarrow
0$

Let $x\in\mathcal{M}$. The level structure $\eta_{x}$ defines an isomorphism
$(\mathbb{Z}/n\mathbb{Z})^{2d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{A}_{x}[n]$,
that is, a basis of $\mathcal{A}_{x}[n]$; we can see the entries
$e_{1},\ldots,e_{2d}$ of this basis as elements of
$\frac{1}{n}\mathcal{L}_{x}/\mathcal{L}_{x}\subset\operatorname{Lie}(\mathcal{A}_{x}/\mathcal{M}_{x})/\mathcal{L}_{x}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{A}_{x}$,
and lift them to a basis $\widetilde{e}_{1},\ldots,\widetilde{e}_{2d}$ of
$\frac{1}{n}\mathcal{L}_{x}$, which gives an isomorphism
$\mathbb{Z}^{2d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}_{x}$,
hence an isomorphism
$\mathbb{R}^{2d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Lie}(\mathcal{A}_{x}/\mathcal{M}_{x})$.
Now we go back to the discussion in Subsection 1.2.2 and we see that the
element $Y_{x}$ of $\mathfrak{h}_{d}^{+}$ whose class is equal to
$\alpha^{-1}(x)$ is basically a basis of the Lagrangian subspace corresponding
to the complex structure on $\mathbb{R}^{2d}$ coming from the isomorphism
$\mathbb{R}^{2d}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Lie}(\mathcal{A}_{x}/\mathcal{M}_{x})$
that we just defined. So we need to show that we can find a neighborhood $U$
of $x$ such that $\widetilde{e}_{1},\ldots,\widetilde{e}_{2d}$ extend to
holomorphic sections of $\frac{1}{n}\mathcal{L}_{\mid U}\rightarrow U$ that
define a level $n$ structure on $\mathcal{A}_{y}$ at each point $y$ of $U$.
The extensions exist because the morphism
$\frac{1}{n}\mathcal{L}\rightarrow\mathcal{M}$ is étale, and the condition on
the extensions is open because it only involves finite groupe schemes over
$\mathcal{M}$.

In particular, we showed that $\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ is the
set of complex points of an algebraic variety defined over the number field
$\mathbb{Q}(\zeta_{n})$. Unfortunately, this number field depends on the level
$n$. The issue is that we need a fixed primitive $n$th root of $1$ in order to
define the moduli problem $\mathcal{M}^{\prime}_{d,n}$, so we need to be over
a basis where such a primitive $n$th root exists. To fix this problem, we will
allow the primitive $n$th root of $1$ to vary.

####  1.2.4. The Siegel modular variety

######  Definition 1.15.

Let $n$ be a positive integer. The _Siegel modular variety_
$\mathcal{M}_{d,n}$ is the contravariant functor from the category of
$\mathbb{Z}[1/n]$-schemes to the category of sets sending a scheme $S$ to the
set of isomorphism classes of triples $(A,\lambda,\eta,\varphi)$, where
$(A,\lambda)$ is a principally polarized abelian scheme of relative dimension
$d$ over $S$ and $(\eta,\varphi)$ is a level $n$ structure on $(A,\lambda)$.

An isomorphism from $(A,\lambda,\eta,\varphi)$ to
$(A^{\prime},\lambda^{\prime},\eta^{\prime},\varphi^{\prime})$ is an
isomorphism of abelian varieties
$u:A\stackrel{{\scriptstyle\sim}}{{\rightarrow}}A^{\prime}$ such that
$\lambda^{\prime}\circ u=u^{\vee}\circ\lambda$ and
$\eta^{\prime}=\eta\circ(u,u)$.

The group $\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})$ acts on
$\mathcal{M}_{d,n}$: if $g\in\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})$ and
$(A,\lambda,\eta,\varphi)\in\mathcal{M}_{d,n}(S)$, then

$g\cdot(A,\lambda,\eta,\varphi)=(A,\lambda,g\circ\eta,c(g)^{-1}\varphi).$

The kernel of this action is the group
$K(n)=\operatorname{Ker}(\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})\rightarrow\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z}))$.
If $n$ divides $m$, then we have a morphism
$\mathcal{M}_{d,m}\rightarrow\mathcal{M}_{d,n}$ that forgets part of the level
$m$ structure; this morphism is representable finite étale, and in fact it is
a torsor under the finite group $K(n)/K(m)$.

We have the following variant of Theorem 1.12.

######  Theorem 1.16 (Mumford, cf. [41]).

Suppose that $n\geq 3$. Then the functor $\mathcal{M}_{d,n}$ is representable
by a smooth quasi-projective $\mathbb{Z}[1/n]$-scheme purely of dimension
$d(d+1)/2$, which we still denote by $\mathcal{M}_{d,n}$ and call the _Siegel
modular variety of level $n$_.

######  Remark 1.17.

The scheme $\mathcal{M}_{d,n}$ is the Shimura variety for $\mathrm{GSp}_{2d}$
with level
$K(n)=\operatorname{Ker}(\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})\rightarrow\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z}))$,
or rather its integral model. If $K$ is an open compact subgroup of
$\mathrm{GSp}_{2d}(\mathbb{A}_{f})$ that is small enough, 101010For example,
$K\subset K(N)$ with $N\geq 3$. then we can also define the Shimura variety
$\mathcal{M}_{d,K}$ with level $K$, and even its integral model: choose $n$
such that $K(n)\subset K$. Then $K(n)$ is a normal subgroup of $K$, so the
group $K/K(n)\subset\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z})$ acts on
$\mathcal{M}_{d,n}$, and we set
$\mathcal{M}_{d,K}=\mathcal{M}_{d,n}/(K/K(n))$. It is easy to check that
$\mathcal{M}_{d,K,\mathbb{Q}}$ does not depend on the choice of $n$; however,
with our definition, the localization of $\mathbb{Z}$ over which
$\mathcal{M}_{d,K}$ is defined depends on the choice of $n$, so this is not
ideal.

In fact, for $K$ an open compact subgroup of
$\mathrm{GSp}_{2d}(\mathbb{A}_{f})$, we have a direct definition of a level
$K$ structure on a principally polarized abelian scheme (see Section 5 of
Kottwitz’s paper [64]). For $K$ small enough, the scheme
$\mathcal{M}_{d,K,\mathbb{Q}}$ is the moduli space of principally polarized
abelian schemes with level $K$ structure. In general, this moduli space is
representable by a Deligne-Mumford stack. We can also define the moduli space
over a localization of $\mathbb{Z}$, but the primes that we invert depend on
$K$; see the discussion in Subsubsection 2.1.4.

Let us explain the relationship between $\mathcal{M}_{d,n}$ and
$\mathcal{M}^{\prime}_{d,n}$. We define a map
$s:(\mathbb{Z}/n\mathbb{Z})^{\times}\rightarrow\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z})$
by $s(\alpha)=\begin{pmatrix}0&\alpha I_{d}\\\ I_{d}&0\end{pmatrix}$; note
that $s$ is a section of the multiplier
$c:\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z})\rightarrow(\mathbb{Z}/n\mathbb{Z})^{\times}$,
and that it is not a morphism of groups.

######  Proposition 1.18.

The morphism
$\mathcal{M}^{\prime}_{d,n}\times(\mathbb{Z}/n\mathbb{Z})^{\times}\rightarrow\mathcal{M}_{d,n,\mathcal{O}_{n}}$
sending $((A,\lambda,\eta),\alpha)$ to
$(A,\lambda,s(\alpha)\circ\eta,\varphi_{0}\circ\alpha)$ (where we see $\alpha$
as an automorphism of $\underline{\mathbb{Z}/n\mathbb{Z}}_{S}$ for any scheme
$S$) is an isomorphism.

As a corollary, we get a description of the complex points of
$\mathcal{M}_{d,n}$. Let
$\mathfrak{h}_{d}=\mathfrak{h}_{d}^{+}\cup(-\mathfrak{h}_{d}^{+})$ be the set
of symmetric matrices $Y\in M_{d}(\mathbb{C})$ such that
$\operatorname{Im}(Y)$ is positive definite or negative definite. The action
of $\mathrm{Sp}_{2d}(\mathbb{R})$ on $\mathfrak{h}_{d}$ extends to a
transitive action of $\mathrm{GSp}_{2d}(\mathbb{R})$, given by the same
formula. The stabilizer of $iI_{d}\in\mathfrak{h}_{d}$ in
$\mathrm{GSp}_{2d}(\mathbb{R})$ is $\mathbb{R}_{>0}K_{\infty}$, where
$K_{\infty}$ is as before the stabilizer of $iI_{d}$ in
$\mathrm{Sp}_{2d}(\mathbb{R})$, so
$\mathfrak{h}_{d}\simeq\mathrm{GSp}_{2d}(\mathbb{R})/\mathbb{R}_{>0}K_{\infty}$
as real analytic manifolds.

######  Corollary 1.19.

We have an isomorphism of complex algebraic varieties

$\mathcal{M}_{d,n}(\mathbb{C})\simeq\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)$

extending the isomorphism of Proposition 1.14, where
$K(n)=\operatorname{Ker}(\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})\rightarrow\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z}))$
and $\mathrm{GSp}_{2g}(\mathbb{Q})$ acts diagonally on
$\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})$.

This follows from the fact that

$\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)\simeq\mathrm{GSp}_{2d}(\mathbb{Q})^{+}\backslash(\mathfrak{h}_{d}^{+}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)),$

where
$\mathrm{GSp}_{2d}(\mathbb{Q})^{+}=\\{g\in\mathrm{GSp}_{2d}(\mathbb{Q})\mid
c(g)>0\\}$, and from strong approximation for $\mathrm{Sp}_{2d}$, 111111See
the papers [99] and [100] of Platonov. which implies that $c$ induces a
bijection

$\mathrm{GSp}_{2d}(\mathbb{Q})_{+}\backslash\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{Q}_{>0}\backslash\mathbb{A}_{f}^{\times}/c(K(n))\simeq\widehat{\mathbb{Z}}^{\times}/(1+n\widehat{\mathbb{Z}})\simeq(\mathbb{Z}/n\mathbb{Z})^{\times}.$

For every $i\in(\mathbb{Z}/n\mathbb{Z})^{\times}$, we choose
$x_{i}\in\mathrm{GSp}_{2d}(\mathbb{A}_{f})$ lifting $i$ and we set

$\Gamma(n)_{i}=\mathrm{GSp}_{2d}(\mathbb{Q})^{+}\cap
x_{i}K(n)x_{i}^{-1}=\mathrm{Sp}_{2d}(\mathbb{Q})\cap x_{i}K(n)x_{i}^{-1}.$

Then the $\Gamma(n)_{i}$ are arithmetic subgroups of
$\mathrm{Sp}_{2d}(\mathbb{Q})$, and we have

$\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)\simeq\coprod_{i\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\Gamma(n)_{i}\backslash\mathfrak{h}_{d}^{+}$

as complex manifolds.

In fact, for $i\in(\mathbb{Z}/n\mathbb{Z})^{\times}$, we can take
$x_{i}=\begin{pmatrix}0&a_{i}I_{d}\\\ I_{d}&0\end{pmatrix}$ with
$a_{i}\in\widehat{\mathbb{Z}}^{\times}$ lifting $i$. In particular, we have
$x_{i}\in\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})$; as $K(n)$ is a normal
subgroup of $\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})$, we get
$x_{i}K(n)x_{i}^{-1}=K(n)$, hence $\Gamma(n)_{i}=\Gamma(n)$, and finally

$\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)\simeq\coprod_{i\in(\mathbb{Z}/n\mathbb{Z})^{\times}}\Gamma(n)\backslash\mathfrak{h}_{d}^{+}.$

######  Remark 1.20.

If $K$ is a small enough open compact subgroup of
$\mathrm{GSp}_{2d}(\mathbb{A}_{f})$, then we get an isomorphism of complex
manifolds:

$\mathcal{M}_{d,K}(\mathbb{C})\simeq\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K).$

####  1.2.5. Hecke correspondences

We can also descend the Hecke correspondences of Subsection 1.1 to morphisms
of schemes.

We proceed as in Section 3 of Laumon’s paper [74]. Let
$g\in\mathrm{GSp}_{2d}(\mathbb{A}_{f})$, and let $K,K^{\prime}$ be small
enough open compact subgroups of $\mathrm{GSp}_{2d}(\mathbb{A}_{f})$ such that
$K^{\prime}\subset K\cap gKg^{-1}$. We want to define finite étale morphisms
$T_{1},T_{g}:\mathcal{M}_{d,K^{\prime}}\rightarrow\mathcal{M}_{d,K}$, and the
Hecke correspondence associated to $(g,K,K^{\prime})$ is the couple
$(T_{1},T_{g})$.

Choose $n\geq 3$ such that $K(n)\subset K^{\prime}$; then
$\mathcal{M}_{d,K^{\prime}}=\mathcal{M}_{d,n}/(K^{\prime}/K(n))$ and
$\mathcal{M}_{d,K}=\mathcal{M}_{d,n}/(K/K(n))$. The morphism $T_{1}$ should
just forget part of the level structure: as $K^{\prime}/K(n)\subset K/K(n)$,
we have an obvious choice for
$T_{1}:\mathcal{M}_{d,K^{\prime}}\rightarrow\mathcal{M}_{d,K}$: we take the
morphism induced by the identity of $\mathcal{M}_{d,n}$.

To define $T_{g}$, we first consider the following special case: if $g\in
M_{2d}(\widehat{\mathbb{Z}})\cap\mathrm{GSp}_{2d}(\mathbb{A}_{f})$, let
$x=(A,\lambda,\eta,\varphi)\in\mathcal{M}_{d,n}(S)$. Let $u$ be the
endomorphism of $(\mathbb{Z}/n\mathbb{Z})^{2d}$ with matrix $g$. Then $T_{g}$
sends the class of $x$ in $\mathcal{M}_{d,K^{\prime}}(S)$ to the class of
$(A^{\prime},\lambda^{\prime},\eta^{\prime},\varphi)\in\mathcal{M}_{d,n}(S)$,
where $A^{\prime}$ is the quotient of $A$ by the finite flat subgroup scheme
$\eta^{-1}(\underline{\operatorname{Ker}u})$ of $A[n]$ and
$\lambda^{\prime},\eta^{\prime}$ are the morphisms deduced from
$\lambda,g\circ\eta$.

Note that, if $g=a\cdot I_{2d}$ with
$a\in\widehat{\mathbb{Z}}\cap\mathbb{A}_{f}^{\times}$ and $K^{\prime}=K$, then
the morphism $T_{g}:\mathcal{M}_{d,K}\rightarrow\mathcal{M}_{d,K}$ is an
isomorphism.

Finally, for a general $g\in\mathrm{GSp}_{2d}(\mathbb{A}_{f})$, we write
$g=g_{0}^{-1}g_{1}$ with
$g_{0}\in(\widehat{\mathbb{Z}}\cap\mathbb{A}_{f}^{\times})I_{2d}$ and
$g_{1}\in M_{2d}(\widehat{\mathbb{Z}})\cap\mathrm{GSp}_{2d}(\mathbb{A}_{f})$,
and we set $T_{g}=T_{g_{0}}^{-1}\circ T_{g_{1}}$.

######  Remark 1.21.

If we use instead the general definition of a level $K$ structure from Section
5 of [64], then it becomes much easier to define the Hecke correspondences on
the moduli problems; see Section 6 of loc. cit.

######  Remark 1.22.

We have two ways to think of $\mathcal{M}_{d,n}(\mathbb{C})$: as an adelic
double quotient or as finite disjoint union of spaces
$\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$, which are locally symmetric spaces
associated to the semi-simple group $\mathrm{Sp}_{2d}$. The first description
is more convenient to see the action of adelic Hecke operators, and the second
description is a bit more concrete and has simpler combinatorics. Note also
that the complex manifold $\Gamma(n)\backslash\mathfrak{h}_{d}^{+}$ is
isomorphic to the set of $\mathbb{C}$-points of the algebraic variety
$\mathcal{M}^{\prime}_{d,n}$, but this algebraic variety is defined over the
field $\mathbb{Q}[T]/(T^{n}-1)$, which depends on $n$. On the other hand, the
adelic double quotient
$\mathrm{GSp}_{2d}(\mathbb{Q})\backslash(\mathfrak{h}_{d}\times\mathrm{GSp}_{2d}(\mathbb{A}_{f})/K(n)$
is isomorphic to the set of $\mathbb{C}$-points of the algebraic variety
$\mathcal{M}_{d,n}$, which is defined over $\mathbb{Q}$ (and even
$\mathbb{Z}[1/n]$). So if we want to consider Shimura varieties as a
projective system of algebraic varieties over a number field, then it makes
sense to use the adelic double quotients, because they are all defined over
the same field.

####  1.2.6. Background on abelian schemes

Let $S$ be a scheme. We denote by $\mathrm{Sch}/S$ the category of
$S$-schemes.

######  Definition 1.23.

An _abelian scheme_ over $S$ is an $S$-group scheme $A\rightarrow S$ which is
smooth and proper with geometrically connected fibers. If $S$ is the spectrum
of a field $k$, an abelian scheme over $S$ is also called an _abelian variety_
over $k$.

A _morphism of abelian schemes_ over $S$ is a morphism of $S$-group schemes
between abelian schemes over $S$.

######  Proposition 1.24.

Let $A$ be an abelian scheme over $S$. Then the $S$-group scheme $A\rightarrow
S$ is commutative.

######  Definition 1.25.

Let $A$ be an abelian scheme over $S$, and let $e:S\rightarrow A$ be its zero
section. We consider the following two functors from
$(\mathrm{Sch}/S)^{\mathrm{op}}$ to the category of sets:

  * (a)

The functor $\mathbf{Pic}_{A/S,e}$ sending an $S$-scheme $T\rightarrow S$ to
the set of isomorphism classes of couples $(\mathcal{L},\varphi)$, where
$\mathcal{L}$ is an invertible sheaf on $A\times_{S}T$ and
$\varphi:\mathcal{O}_{T}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}e_{T}^{*}\mathcal{L}$
is an isomorphism, with $e_{T}=e\times_{S}T:T\rightarrow A\times_{S}T$. An
isomorphism from $(\mathcal{L},\varphi)$ to
$(\mathcal{L}^{\prime},\varphi^{\prime})$ is an isomorphism of
$\mathcal{O}_{T}$-modules
$\alpha:\mathcal{L}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathcal{L}^{\prime}$
such that $(e_{T}^{*}\alpha)\circ\varphi=\varphi^{\prime}$;

  * (b)

The subfunctor $\mathbf{Pic}_{A/S,e}^{0}$ of $\mathbf{Pic}_{A/S,e}$ sending an
$S$-scheme $T\rightarrow S$ to the set of isomorphism classes of couples
$(\mathcal{L},\varphi)$ as in (a) such that, for every point $t$ of $T$, every
smooth projective curve over the residue field $\kappa(t)$ of $t$ and every
morphism of $\kappa(t)$-schemes $f:C\rightarrow A\times_{S}t$, the line bundle
$f^{*}(\mathcal{L}_{\mid A\times_{S}t})$ is of degree $0$ on $C$.

######  Remark 1.26.

  * (1)

The functor $\mathbf{Pic}_{A/S,e}$ can be made into a functor into the
category of abelian groups: if $T$ is an $S$-scheme and
$(\mathcal{L},\varphi)$, $(\mathcal{L}^{\prime},\varphi^{\prime})$ represent
elements of $\mathbf{Pic}_{A/S,e}(T)$, their product is represented by
$(\mathcal{L}\otimes_{\mathcal{O}_{T}}\mathcal{L}^{\prime},\varphi\otimes\varphi^{\prime})$,
where $\varphi\otimes\varphi^{\prime}$ is the isomorphism

$\textstyle{\mathcal{O}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{T}\otimes_{\mathcal{O}_{T}}\mathcal{O}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\scriptstyle{\varphi\otimes\varphi^{\prime}}$$\textstyle{(e_{T}^{*}\mathcal{L})\otimes_{\mathcal{O}_{T}}(e_{T}^{*}\mathcal{L}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{e_{T}^{*}(\mathcal{L}\otimes_{\mathcal{O}_{T}}\mathcal{L}^{\prime}).}$

Moreover, for every $S$-scheme $T$, the set $\mathbf{Pic}_{A/S,e}^{0}(T)$ is a
subgroup of $\mathbf{Pic}_{A/S,e}(T)$.

  * (2)

If $X\rightarrow S$ is a scheme over $S$, then the relative Picard functor
$\mathbf{Pic}_{X/S}$ on $\mathrm{Sch}/S$ is the fppf sheafification of the
functor $T\mapsto\mathbf{Pic}(X\times_{S}T)$, where, for $Y$ a scheme, we
denote by $\mathbf{Pic}(Y)$ the set of isomorphism classes of line bundles on
$Y$ (that is an abelian group for the tensor product); see [114, Situation
0D25]. We can also define a subfonctor $\mathbf{Pic}_{X/S}^{0}$ of
$\mathbf{Pic}_{X/S}$ as in Definition 1.25. By [114, Lemma 0D28], if $A$ is an
abelian scheme over $S$, then there is an isomorphism of functors in abelian
groups $\mathbf{Pic}_{A/S}\simeq\mathbf{Pic}_{A/S,e}$, inducing an isomorphism
$\mathbf{Pic}_{A/s}^{0}\simeq\mathbf{Pic}_{A/S,e}^{0}$.

  * (3)

We can upgrade $A\mapsto\mathbf{Pic}_{A/S,e}$ and
$A\mapsto\mathbf{Pic}_{A/S,e}^{0}$ to contravariant functors in $A$: if
$f:A\rightarrow B$ is a morphism of abelian schemes over $S$, then it induces
a natural transformation
$f^{*}:\mathbf{Pic}_{B/S,e}\rightarrow\mathbf{Pic}_{A/S,e}$ sending
$(\mathcal{L},\varphi)$ to $(f^{*}(\mathcal{L}),f^{*}(\varphi))$, and $f^{*}$
sends $\mathbf{Pic}_{B/S,e}^{0}$ to $\mathbf{Pic}_{A/S,e}^{0}$.

######  Theorem 1.27.

Let $A$ be an abelian scheme over $S$. Then $\mathbf{Pic}_{A/S,e}^{0}$ is
representable by an abelian scheme over $S$.

###### Proof.

We know that $\mathbf{Pic}_{A/S,e}^{0}$ is representable by an algebraic space
over $S$ by a result of M. Artin (see [10] or [114, Proposition 0D2C]). We can
check on the moduli problem that this algebraic space is proper and smooth,
and its fibers over points of $S$ are abelian varieties by the classical
theory of the dual abelian variety (see sections II.8 and III.13 of Mumford’s
book [92]). It remains to prove that the algebraic space representing
$\mathbf{Pic}_{A/S,e}^{0}$ is a scheme; this is due to Raynaud, and a proof is
given in Theorem 1.9 of [41].

∎

######  Definition 1.28.

Let $A$ be an abelian scheme over $S$. The abelian scheme over $S$
representing $\mathbf{Pic}_{A/S,e}^{0}$ is called the _dual abelian scheme_ of
$A$ and denoted by $A^{\vee}$. In particular, we get a couple
$(\mathcal{P}_{A},\varphi_{A})$ representing the element of
$\mathbf{Pic}_{A/S,e}^{0}(A^{\vee})$ corresponding to
$\mathbf{id}_{A^{\vee}}$, with $\mathcal{P}_{A}$ a line bundle on
$A\times_{S}A^{\vee}$, called the _Poincaré line bundle_.

If $f:A\rightarrow B$ is a morphism of abelian schemes over $S$, we denote by
$f^{\vee}:B^{\vee}\rightarrow A^{\vee}$ the morphism corresponding to the
natural transformation
$f^{*}:\mathbf{Pic}_{B/S,e}^{0}\rightarrow\mathbf{Pic}_{A/S,e}^{0}$ of Remark
1.26.

######  Remark 1.29.

Let $e:S\rightarrow A^{\vee}$ be the unit section. Then the pullback of
$\mathcal{P}_{A}$ by $A\times_{S}e:A=A\times_{S}S\rightarrow
A\times_{S}A^{\vee}$ is the line bundle on $A$ corresponding to the element
$e$ of $A^{\vee}(S)=\mathbf{Pic}_{A/S,e}^{0}(S)$; in other words, it is
isomorphic to the trivial line bundle $\mathcal{O}_{A}$. So $\mathcal{P}_{A}$
defines an element of $\mathbf{Pic}_{A^{\vee}/S,e}^{0}(A)$, that is, a
morphism of $S$-schemes $A\rightarrow A^{\vee\vee}$, called the _biduality
morphism_. The _biduality theorem_ says that the biduality morphism is an
isomorphism. For $S$ the spectrum of a field, this is proved in Section III.13
of [92], and the general case reduces to this by looking at the fibers over
points of $S$.

Let $A$ be an abelian scheme over $S$ and let $\mathcal{L}$ be a line bundle
on $A$. We denote by $\mu,p_{1},p_{2},\varepsilon:A\times_{S}A\rightarrow A$
the addition morphism, the first projection, the second projection and the
zero morphism respectively. Then the line bundle
$(\mu^{*}\mathcal{L})\otimes(p_{1}^{*}\mathcal{L}^{\otimes-1})\otimes(p_{2}^{*}\mathcal{L}^{\otimes-1})\otimes(\varepsilon^{*}\mathcal{L})$
on $A\times_{S}A$ is trivial when restricted to $S\times_{S}A$ via the zero
section of $A$, hence it defines an element of $\mathbf{Pic}_{A/S,e}^{0}(A)$,
corresponding to a morphism of $S$-schemes $\lambda(\mathcal{L}):A\rightarrow
A^{\vee}$, which is a morphism of abelian varieties by the theorem of the cube
(see for example Section III.10 of [92]).

######  Definition 1.30.

Let $A$ be an abelian scheme over $S$. A _polarization_ on $A$ is an isogeny
$\lambda:A\rightarrow A^{\vee}$ (i.e. a finite faithfully flat morphism of
abelian schemes) such that, for every algebraically closed field $k$ and every
morphism $\operatorname{Spec}k\rightarrow S$, the morphism
$\lambda\times_{S}\operatorname{Spec}k:A\times_{S}\operatorname{Spec}k\rightarrow
A^{\vee}\times_{S}\operatorname{Spec}k=(A\times_{S}\operatorname{Spec}k)^{\vee}$
is of the form $\lambda(\mathcal{L})$, for $\mathcal{L}$ an ample line bundle
on $A\times_{S}\operatorname{Spec}k$. We say that a polarization is
_principal_ if it is an isomorphism.

A _principally polarized abelian scheme_ over $S$ is a pair $(A,\lambda)$,
where $A$ is an abelian scheme over $S$ and $\lambda$ is a principal
polarization on $A$.

######  Remark 1.31.

Let $\lambda$ be a polarization on $A$, and let $n$ be a positive integer.
Then, composing $\lambda:A[n]\rightarrow A^{\vee}[n]$ with the canonical
Cartier pairing $A[n]\times A^{\vee}[n]\rightarrow\mu_{n,S}$, 121212See page
183 of Mumford’s book [92] for the case of abelian varieties over a field. For
the case of general abelian schemes, see the beginning of Section 1 of Oda’s
paper [95], applied to the isogeny given by multiplication by $n$. we get a
pairing $A[n]\times A[n]\rightarrow\mu_{n,S}$, called the _Weil pairing_
associated to $\lambda$. If $\lambda$ is principal. this is a perfect pairing.

###  1.3. Shimura varieties over $\mathbb{C}$

Remember the upshot of Subsection 1.2: if we want algebraic varieties that are
all defined over the same number field, and Hecke correspondences that are
also defined on this number field, it is better to work with adelic double
quotients for a reductive group such as $\mathrm{GSp}_{2d}$ rather than with
locally symmetric spaces for a semi-simple group such as $\mathrm{Sp}_{2d}$.
This (and Theorem 1.3) motivates the definition of Shimura data, due to
Deligne in [35].

####  1.3.1. The Serre torus

Let $\mathbb{S}$ be $\mathbb{C}^{\times}$ seen as an algebraic group over
$\mathbb{R}$; this is called the _Serre torus_. In other words, the group
$\mathbb{S}$ is the Weil restriction of scalars from $\mathbb{C}$ to
$\mathbb{R}$ of $\mathrm{GL}_{1}$, so that
$\mathbb{S}(R)=(R\otimes_{\mathbb{R}}\mathbb{C})^{\times}$ for every
$\mathbb{R}$-algebra $R$. We denote by $w$ the injective morphism
$\mathrm{GL}_{1,\mathbb{R}}\rightarrow\mathbb{S}$ corresponding to the
inclusion $\mathbb{R}^{\times}\subset\mathbb{C}^{\times}$.

We have
$\mathbb{S}(\mathbb{C})=(\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C})^{\times}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C}^{\times}\times\mathbb{C}^{\times}$,
where the isomorphism sends $a\otimes 1+b\otimes i$ to $(a+ib,a-ib)$. So the
abelian group
$\mathrm{Hom}(\mathbb{S}_{\mathbb{C}},\mathrm{GL}_{1,\mathbb{C}})$ of
characters of $\mathbb{S}$ is free of rank $2$ and generated by the characters
$z$ and $\overline{z}$ corresponding to the two projections of
$\mathbb{C}^{\times}\times\mathbb{C}^{\times}$ on $\mathbb{C}^{\times}$. We
denote by $r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow\mathbb{S}_{\mathbb{C}}$ the
injective morphism corresponding to the injection of the first factor in
$\mathbb{C}^{\times}\times\mathbb{C}^{\times}$.

If $V$ is a real vector space and $\rho:\mathbb{S}\rightarrow\mathrm{GL}(V)$
is a morphism of algebraic groups (i.e. a representation of $\mathbb{S}$ on
$V$), then we have
$V_{\mathbb{C}}:=V\otimes_{\mathbb{R}}\mathbb{C}=\bigoplus_{p,q\in\mathbb{Z}}V^{p,q}$,
where $V^{p,q}$ is the subspace of $V_{\mathbb{C}}$ on which
$\mathbb{S}_{\mathbb{C}}$ acts by the character $z^{-p}\overline{z}^{-q}$;
moreover, as $\rho$ is defined over $\mathbb{R}$, we have
$\overline{V^{p,q}}=V^{q,p}$ for all $p,q\in\mathbb{Z}$. Let $m\in\mathbb{Z}$.
We say that $\rho$ is _of weight $m$_ if $\rho\circ
w:\mathrm{GL}_{1}\rightarrow\mathrm{GL}(V)$ is equal to $x\mapsto
x^{-m}\mathbf{id}_{V}$.

######  Remark 1.32.

If $\rho:\mathbb{S}\rightarrow\mathrm{GL}(V)$ is of weight $m$, we have
$V^{p,q}=0$ unless $p+q=m$, so the decomposition
$V_{\mathbb{C}}=\bigoplus_{p,q}V^{p,q}$ is a pure Hodge structure of weight
$m$ on $V$. In fact, representations of weight $m$ of $\mathbb{S}$ on $V$ are
in bijection with pure Hodge structures of weight $m$ on $V$.

####  1.3.2. Shimura data

######  Definition 1.33.

A _Shimura datum_ is a couple $(G,h)$, where $G$ is a connected reductive
algebraic group over $\mathbb{Q}$ and $h:\mathbb{S}\rightarrow G_{\mathbb{R}}$
is a morphism of real algebraic groups such that:

  * (a)

The image of $h\circ w:\mathrm{GL}_{1,\mathbb{R}}\rightarrow G_{\mathbb{R}}$
is central;

  * (b)

If $\mathfrak{g}=\operatorname{Lie}(G_{\mathbb{R}})$ and
$\mathfrak{g}_{\mathbb{C}}=\bigoplus_{p,q\in\mathbb{Z}}\mathfrak{g}^{p,q}$ is
the decomposition induced by the representation $\mathrm{Ad}\circ
h:\mathbb{S}\rightarrow\mathrm{GL}(\mathfrak{g})$, then we have
$\mathfrak{g}^{p,q}=0$ unless $(p,q)\in\\{(-1,1),(0,0),(1,-1)\\}$;

  * (c)

Conjugation by $h(i)$ induces a Cartan involution of
$G_{\mathrm{der}}(\mathbb{R})$ (see Theorem 1.3);

  * (d)

$G_{\mathrm{ad}}$ has no normal subgroup (defined over $\mathbb{Q}$) whose
group of $\mathbb{R}$-points is compact. 131313Note that
$G_{\mathrm{ad}}(\mathbb{R})$ could still have compact normal algebraic
subgroups, as long as they are not defined over $\mathbb{Q}$.

Note that condition (b) implies condition (a), because it implies that the
image of $h\circ w$ centralizes $\operatorname{Lie}(G_{\mathbb{R}})$.

Let $(G,h)$ be a Shimura datum. We denote by $K_{\infty}$ the centralizer of
$h$ in $G(\mathbb{R})$ and by $X$ the set of $G(\mathbb{R})$-conjugates of
$h$. Then $K_{\infty}$ contains the center of $G(\mathbb{R})$, and
$K_{\infty}\cap G_{\mathrm{der}}(\mathbb{R})^{0}$ is equal to the centralizer
of $h(i)$ in $G_{\mathrm{der}}(\mathbb{R})^{0}$, hence is a maximal compact
subgroup of $G_{\mathrm{der}}(\mathbb{R})^{0}$ by condition (c). We have
$X\simeq G(\mathbb{R})/K_{\infty}$, and Theorem 1.3 implies that there is a
$G(\mathbb{R})$-invariant complex structure on $X$ such that the connected
components of $X$ are Hermitian symmetric domains.

######  Remark 1.34.

There are many variants in the definition of a Shimura datum. The definition
we gave here is equivalent to Definition 5.5 of Milne’s notes [87], but it is
possible to impose additional conditions to make some statements simpler; see
for example the discussion of additional axioms on page 63 of [87]. On the
other hand, some authors use a slightly more general definition, where $X$ is
allowed to be finite over the set of morphisms $\mathbb{S}\rightarrow
G_{\mathbb{R}}$. This is for example the case of Pink in [96] (see Definition
2.1) and [97] (see (3.1)); in the first of these references, Pink also
generalizes the definition to include groups that are not necessarily
reductive.

######  Example 1.35.

Take $G=\mathrm{GSp}_{2d}$. Up to conjugation, there exists a unique morphism
$h:\mathbb{S}\rightarrow\mathrm{GSp}_{2d}$ satisfying conditions (a)-(c) of
Definition 1.33 and such that $(h\circ w)(x)=xI_{2d}$ for every
$x\in\mathbb{R}^{\times}$. An element of that class is given by

$h(a+ib)=\begin{pmatrix}aI_{d}&-bI_{d}\\\ bI_{d}&aI_{d}\end{pmatrix}.$

For this $h$, we have
$K_{\infty}=\mathrm{GSp}_{2d}(\mathbb{R})\cap\mathrm{GO}(2d)$, and we can
check that the map $\mathrm{GU}(d)\rightarrow K_{\infty}$ sending
$X+iY\in\mathrm{GU}(d)$ (with $X,Y\in M_{d}(\mathbb{R})$) to
$\begin{pmatrix}X&Y\\\ -Y&X\end{pmatrix}$ is an isomorphism of Lie groups. So
$K_{\infty}=\mathbb{R}_{>0}K^{\prime}_{\infty}$, where
$K^{\prime}_{\infty}=\mathrm{Sp}_{2d}(\mathbb{R})\cap\mathrm{O}(d)$ is the
maximal compact subgroup of $\mathrm{Sp}_{2d}(\mathbb{R})$ that was called
$K_{\infty}$ in Subsections 1.1 and 1.2. This implies that
$X\simeq\mathfrak{h}_{d}$.

The couple $(\mathrm{GSp}_{2d},h)$ is called a _Siegel Shimura datum_.

Let $K$ be an open compact subgroup of $G(\mathbb{A}_{f})$. We set

$M_{K}(G,h)(\mathbb{C})=G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K,$

where the group $K$ acts by right translations on the factor
$G(\mathbb{A}_{f})$, and the group $G(\mathbb{Q})$ acts by left translations
on both factors simultaneously. This is the _Shimura variety at level $K$_
associated to the Shimura datum $(G,h)$.

As in Subsection 1.1, if $(x_{i})_{i\in I}$ is a system of representatives of
the finite quotient $G(\mathbb{Q})\backslash G(\mathbb{A}_{f})/K$, and if
$\Gamma_{i}=G(\mathbb{Q})\cap x_{i}Kx_{i}^{-1}$ for every $i\in I$, then the
$\Gamma_{i}$ are arithmetic subgroups of $G(\mathbb{Q})$, and we have

$M_{K}(G,h)(\mathbb{C})=\coprod_{i\in I}\Gamma_{i}\backslash X.$

Hence it follows from Theorem 1.6 that $M_{K}(G,h)(\mathbb{C})$ is the set of
complex points of a quasi-projective algebraic variety over $\mathbb{C}$,
smooth if $K$ is small enough.

Again as in Subsection 1.1, we have Hecke correspondences between the
varieties $M_{K}(G,h)(\mathbb{C})$, which are finite maps, hence morphisms of
algebraic varieties. This defines an action of $G(\mathbb{A}_{f})$ on the
projective system $(M_{K}(G,h)(\mathbb{C}))_{K\subset G(\mathbb{A}_{f})}$, or
on its limit

$M(G,h)(\mathbb{C})=\varprojlim_{K}M_{K}(G,h)(\mathbb{C}).$

Under the additional assumption that the $\mathbb{R}$-split and
$\mathbb{Q}$-split maximal subtori of the center of $G$ are equal (which is
sometimes required in the definition of a Shimura datum), the limit is equal
to $G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})$ (see Proposition 4.19
of [87]). Moreover, for every open compact subgroup $K$ of
$G(\mathbb{A}_{f})$, we have $M_{K}(G,h)(\mathbb{C})=M(G,h)(\mathbb{C})/K$.

####  1.3.3. Morphisms of Shimura varieties

Let $(G_{1},h_{1})$, $(G_{2},h_{2})$ be Shimura data, and let
$u:G_{1}\rightarrow G_{2}$ be a morphism of algebraic groups such that $u\circ
h_{1}$ and $h_{2}$ are conjugated under $G_{2}(\mathbb{R})$; we say that $u$
is a _morphism of Shimura data_. Then $u$ induces a morphism of complex
manifolds $X_{1}\rightarrow X_{2}$. Hence, for all $K_{1}\subset
G_{1}(\mathbb{A}_{f})$, $K_{2}\subset G_{2}(\mathbb{A}_{f})$ open compact
subgroups such that $u(K_{1})\subset K_{2}$, we get a morphism of quasi-
projective varieties
$u(K_{1},K_{2}):M_{K_{1}}(G_{1},K_{1})(\mathbb{C})\rightarrow
M_{K_{2}}(G_{2},h_{2})(\mathbb{C})$. We can also think of the collection of
all $u(K_{1},K_{2})$ as a morphism of $\mathbb{C}$-schemes
$u:M(G_{1},h_{1})(\mathbb{C})\rightarrow M(G_{2},h_{2})(\mathbb{C})$.

######  Proposition 1.36 (Deligne, see Proposition 1.15 of [35]).

If $G_{1}$ is an algebraic subgroup of $G_{2}$ and $u$ is the inclusion, then,
for every open compact subgroup $K_{1}$ of $G_{1}(\mathbb{A}_{f})$, there
exists an open compact subgroup $K_{2}\supset K_{1}$ of
$G_{2}(\mathbb{A}_{f})$ such that
$u(K_{1},K_{2}):M_{K_{1}}(G_{1},K_{1})(\mathbb{C})\rightarrow
M_{K_{2}}(G_{2},h_{2})(\mathbb{C})$ is a closed immersion.

####  1.3.4. Connected components

For $(G,h)$ equal to the Shimura datum of Example 1.35 and
$K=\operatorname{Ker}(\mathrm{GSp}_{2d}(\widehat{\mathbb{Z}})\rightarrow\mathrm{GSp}_{2d}(\mathbb{Z}/n\mathbb{Z}))$,
we have seen that the multiplier
$c:\mathrm{GSp}_{2d}\rightarrow\mathrm{GL}_{1}$ induces a bijection

$\pi_{0}(M_{K}(G,h)(\mathbb{C}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{Q}_{>0}\backslash\mathbb{A}_{f}^{\times}/c(K)=\mathbb{Q}^{\times}\backslash\mathbb{A}^{\times}/c(K_{\infty}K).$

In fact, it follows from real approximation and the Hasse principle that this
works for many Shimura varieties:

######  Theorem 1.37 (Deligne, see [35] 2.7).

Let $\nu:G\rightarrow T:=G/G_{\mathrm{der}}$ be the quotient morphism, and
suppose that $G_{\mathrm{der}}$ is simply connected. Then, for every open
compact subgroup $K$ of $G(\mathbb{A}_{f})$, the map $\nu$ induces a bijection

$\pi_{0}(M_{K}(G,h)(\mathbb{C}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T(\mathbb{Q})\backslash
T(\mathbb{A})/\nu(K_{\infty}\times K).$

###  1.4. Canonical models

In the situation of Example 1.35, we have seen that the algebraic varieties
$M_{K}(G,h)(\mathbb{C})$ and all the Hecke correspondences are defined over
$\mathbb{Q}$. We would like to generalize this kind of result to other Shimura
varieties.

####  1.4.1. Model of a Shimura variety

First we need to say what we mean by a model.

######  Definition 1.38.

Let $(G,h)$ be a Shimura datum, and $F$ be a subfield of $\mathbb{C}$. A
_model_ of the projective system $(M_{K}(G,h)(\mathbb{C}))_{K}$ over $F$ is
the data:

  * •

for every open compact subgroup $K$ of $G(\mathbb{A}_{f})$, of a quasi-
projective variety $M_{K}$ over $F$ and an isomorphism
$\iota_{K}:M_{K}\otimes_{F}\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M_{K}(G,h)(\mathbb{C})$;

  * •

for every $g\in G(\mathbb{A}_{f})$ and all open compact subgroups
$K,K^{\prime}$ of $G(\mathbb{A}_{f})$ such that $gK^{\prime}g^{-1}\subset K$,
of a morphism of $F$-varieties $T_{g,K,K^{\prime}}:M_{K^{\prime}}\rightarrow
M_{K}$,

such that:

  * (i)

For all $g,K,K^{\prime}$ as above, the morphism $\iota_{K}\circ
T_{g,K,K^{\prime},\mathbb{C}}\circ\iota_{K^{\prime}}^{-1}:M_{K^{\prime}}(G,h)(\mathbb{C})\rightarrow
M_{K}(G,h)(\mathbb{C})$ sends the class of $(x,h)$ in
$M_{K^{\prime}}(G,h)(\mathbb{C})$ to the class of $(x,hg)$ in
$M_{K}(G,h)(\mathbb{C})$;

  * (ii)

If $K$ is an open compact subgroup of $G(\mathbb{A}_{f})$ and $g\in K$, then
$T_{g,K,K}=\mathbf{id}_{M_{K}}$;

  * (iii)

If $K,K^{\prime},K^{\prime\prime}$ are open compact subgroups of
$G(\mathbb{A}_{f})$ and $g,h\in G(\mathbb{A}_{f})$ are such that
$gK^{\prime}g^{-1}\subset K$ and $hK^{\prime\prime}h^{-1}\subset K^{\prime}$,
then $T_{g,K,K^{\prime}}\circ
T_{h,K^{\prime},K^{\prime\prime}}=T_{gh,K,K^{\prime\prime}}$;

  * (iv)

If $K,K^{\prime}$ are open compact subgroups of $G(\mathbb{A}_{f})$ such that
$K^{\prime}$ is a normal subgroup of $K$, then the morphisms
$T_{g,K^{\prime},K^{\prime}}$ for $g\in K$ define an action of $K/K^{\prime}$
on $M_{K^{\prime}}$ (this follows from (ii) and (iii)), and
$T_{1,K^{\prime},K}:M_{K^{\prime}}\rightarrow M_{K}$ induces an isomorphism
$M_{K^{\prime}}/(K/K^{\prime})\rightarrow M_{K}$.

######  Remark 1.39.

We phrased Definition 1.38 in this way to stay close to the definition of a
canonical model given by Deligne in Définition 3.1 of [35]. In Deligne’s
words, a family of $E$-schemes $(M_{K})$ with a family of morphisms
$(T_{g,K,K^{\prime}})$ satisfying conditions (ii), (iii) and (iv) is a _scheme
over $E$ with a continuous action of $G(\mathbb{A}_{f})$_, and the family of
isomorphisms $(\iota_{K})$ satisfying condition (i) is _an isomorphism of
$(M_{K})\otimes_{F}\mathbb{C}$ with $M(G,h)(\mathbb{C})$ compatible with the
$G(\mathbb{A}_{f})$-actions_. However, we did not not phrase the conditions in
the most optimal way possible; for example, condition (i) implies conditions
(ii) and (iii).

If we have a model $(M_{K})_{K}$ of $(M_{K}(G,h)(\mathbb{C}))_{K}$ over $F$,
we write $M=\varprojlim_{K}M_{K}$ (where the transition morphisms are given by
the $T_{1,K^{\prime},K}$). This is an $F$-scheme with an action of
$G(\mathbb{A}_{f})$, and we have a $G(\mathbb{A}_{f})$-equivariant isomorphism
$M\otimes_{F}\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}M(G,h)(\mathbb{C})$.

In particular we get an action of $\operatorname{Gal}(\overline{F}/F)$ on
$\pi_{0}(M(G,h)(\mathbb{C}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{0}(M\otimes_{F}\overline{F})$,
which must commute with the action of $G(\mathbb{A}_{f})$. Under the
hypothesis of Theorem 1.37, we have
$\pi_{0}(M\otimes_{F}\overline{F})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{0}(T(\mathbb{A})/T(\mathbb{Q}))/\pi_{0}(K_{\infty})$
with $T=G/G_{\mathrm{der}}$, and $G(\mathbb{A}_{f})$ acts transitively on this
set of connected components (Proposition 2.2 of [35]). So every element of
$\operatorname{Gal}(\overline{F}/F)$ acts by translation by an element of
$\pi_{0}(T(\mathbb{A})/T(\mathbb{Q}))/\pi_{0}(K_{\infty})$, and the action of
$\operatorname{Gal}(\overline{F}/F)$ comes from a morphism of groups
$\operatorname{Gal}(\overline{F}/F)\rightarrow\pi_{0}(T(\mathbb{A})/T(\mathbb{Q}))/\pi_{0}(K_{\infty})$,
that necessarily factors through the maximal abelian quotient
$\operatorname{Gal}(\overline{F}/F)^{\mathrm{ab}}$.

Suppose that $F$ is a number field. Then global class field theory
141414Normalized so that local uniformizers correspond to geometric Frobenius
elements. gives an isomorphism

$\operatorname{Gal}(\overline{F}/F)^{\mathrm{ab}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\pi_{0}(F^{\times}\backslash\mathbb{A}_{F}^{\times})$

where $\mathbb{A}_{F}$ is the ring of adeles of $F$, so the action of
$\operatorname{Gal}(\overline{F}/F)$ on $\pi_{0}(M\otimes_{F}\overline{F})$
comes from a morphism of groups

$\lambda_{M}:\pi_{0}(F^{\times}\backslash\mathbb{A}_{F}^{\times})\rightarrow\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A}))/\pi_{0}(K_{\infty}),$

called the _reciprocity law_ of the model.

####  1.4.2. The case of tori

We consider the case where $G=T$ is a torus. Let $h:\mathbb{S}\rightarrow
T_{\mathbb{R}}$ be any morphism of real algebraic groups. Then $h$ trivially
satisfies the conditions of Definition 1.33, so we get a Shimura datum
$(T,h)$, and $X=T(\mathbb{R})/\operatorname{Cent}_{T(\mathbb{R})}(h)$ is a
singleton. For every open compact subgroup $K$ of $T(\mathbb{A}_{f})$,

$M_{K}(T,h)(\mathbb{C})=T(\mathbb{Q})\backslash T(\mathbb{A}_{f})/K$

is a finite group. Let $\overline{T(\mathbb{Q})}$ be the closure of
$T(\mathbb{Q})$ in $T(\mathbb{A}_{f})$; then

$M(T,h)(\mathbb{C})=\overline{T(\mathbb{Q})}\backslash T(\mathbb{A}_{f}),$

which is a profinite group isomorphic to $\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A}_{f}))$ by the obvious map. Giving a model of the Shimura variety
of $(T,h)$ over a subfield $F$ of $\mathbb{C}$ is the same as giving an action
of $\operatorname{Gal}(\overline{F}/F)$ over $\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A}_{f}))$ (commuting with the action of $T(\mathbb{A}_{f})$ by
translations), i.e. a morphism of groups
$\operatorname{Gal}(\overline{F}/F)\rightarrow\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A}_{f}))$. If $F$ is a number field, this is equivalent to giving a
morphism of groups

$\lambda_{M}:\pi_{0}(F^{\times}\backslash\mathbb{A}_{F}^{\times})\rightarrow\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A})).$

It is natural to construct such a morphism from a morphism of algebraic groups
$F^{\times}\rightarrow T$, where $F^{\times}$ is seen as an algebraic group
over $\mathbb{Q}$ (so that, for example, we have
$F^{\times}(\mathbb{A})=(\mathbb{A}\otimes_{\mathbb{Q}}F)^{\times}=\mathbb{A}_{F}^{\times}$).
We already have a morphism $h:\mathbb{S}\rightarrow T_{\mathbb{R}}$, which
gives a morphism of complex algebraic groups
$h_{\mathbb{C}}:\mathbb{S}_{\mathbb{C}}\rightarrow T_{\mathbb{C}}$. Remember
that
$\mathbb{S}_{\mathbb{C}}\simeq\mathrm{GL}_{1,\mathbb{C}}\times\mathrm{GL}_{1,\mathbb{C}}$,
and that we denoted by
$r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow\mathbb{S}_{\mathbb{C}}$ the embedding
of the first factor (see page 1.3.1). We get a morphism $h_{\mathbb{C}}\circ
r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow T_{\mathbb{C}}$. As $T$ is an
algebraic group over $\mathbb{Q}$, this morphism is defined over a finite
extension of $\mathbb{Q}$ in $\mathbb{C}$, and we call this extension $F$. We
get a morphism of $F$-algebraic groups $\mathrm{GL}_{1,F}\rightarrow T_{F}$,
hence a morphism of $\mathbb{Q}$-algebraic groups
$F^{\times}\rightarrow\operatorname{Res}_{F/\mathbb{Q}}T_{F}$, where
$\operatorname{Res}_{F/Q}T_{F}$ is the algebraic group that sends a
$\mathbb{Q}$-algebra $R$ to $T(R\otimes_{\mathbb{Q}}F)$. Composing this with
the norm $N_{F/\mathbb{Q}}:\operatorname{Res}_{F/\mathbb{Q}}T_{F}\rightarrow
T$, we finally get a morphism $r(h):F^{\times}\rightarrow T$, called the
_reciprocity morphism for $(T,h)$_. We take $\lambda_{M}$ to be induced by
$r(h)$. 151515This is the opposite of Deligne’s convention in [35] (3.9.1); I
have chosen to follow Pink’s convention in [96] Chapter 11.

So if $(G,h)$ is a Shimura datum with $G$ a torus, we get a canonically
defined model of the associated Shimura variety over the field of definition
of $h_{\mathbb{C}}\circ r$.

####  1.4.3. The reflex field

Let $(G,h)$ be a Shimura datum satisfying the hypothesis of Theorem 1.37, and
let $\nu:G\rightarrow T:=G/G_{\mathrm{der}}$ be the quotient morphism. For
every open compact subgroup $K$ of $G(\mathbb{A}_{f})$, we have a bijection,
induced by $\nu$:

$\pi_{0}(M_{K}(G,h)(\mathbb{C}))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T(\mathbb{Q})\backslash
T(\mathbb{A})/\nu(K_{\infty}\times K).$

Suppose that we have a model $(M_{K})_{K}$ of the Shimura variety of $(G,h)$
over a number field $F\subset\mathbb{C}$. We would expect the Shimura variety
of $(T,\nu\circ h)$ to also have a model over $F$, and the isomorphism above
to be $\operatorname{Gal}(\overline{F}/F)$-equivariant, where the action on
the right hand side is given by the morphism $r(\nu\circ
h):F^{\times}\rightarrow T$ constructed in 1.4.2.

Remember that
$r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow\mathbb{S}_{\mathbb{C}}\simeq\mathrm{GL}_{1,\mathbb{C}}\times\mathrm{GL}_{1,\mathbb{C}}$
is the embedding of the first factor (see page 1.3.1). By the previous
paragraph, we would expect $F$ to contain the field of definition of
$\nu_{\mathbb{C}}\circ h_{\mathbb{C}}\circ r$. In fact it would make sense to
take $F$ to be the field of definition of $h_{\mathbb{C}}\circ
r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow G_{\mathbb{C}}$, except that $h$ is
only significant up to conjugation. This motivates the following definition.

######  Definition 1.40.

Let $(G,h)$ be a Shimura datum. The _reflex field_ $F(G,h)$ of $(G,h)$ is the
field of definition of the conjugacy class of $h_{\mathbb{C}}\circ
r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow G_{\mathbb{C}}$.

Let $F=F(G,h)$. Then $F$ is a finite extension of $\mathbb{Q}$ in
$\mathbb{C}$, and, for every morphism $\rho$ of $G$ into a commutative
algebraic group, the morphism $\rho_{\mathbb{C}}\circ h_{\mathbb{C}}\circ r$
is defined over $F$. Note that $h_{\mathbb{C}}\circ r$ itself is not
necessarily defined over $F$.

######  Example 1.41.

Let $E=\mathbb{Q}[\sqrt{-d}]$ be an imaginary quadratic extension of
$\mathbb{Q}$, let $p\geq q\geq 1$ be integers, and set $n=p+q$. Let
$J=\begin{pmatrix}I_{p}&0\\\
0&-I_{q}\end{pmatrix}\in\mathrm{GL}_{n}(\mathbb{Z})$. For every commutative
ring $R$, we denote by $x\mapsto\overline{x}$ the involution of
$R\otimes_{\mathbb{Z}}\mathcal{O}_{E}$ induced by the nontrivial element of
$\operatorname{Gal}(E/\mathbb{Q})$, and, for every $Y\in
M_{n}(R\otimes_{\mathbb{Q}}E)$, we write $Y^{*}={}^{t}\overline{Y}$.

The general unitary group $\mathrm{GU}(p,q)$ is the $\mathbb{Z}$-group scheme
defined by

$\mathrm{GU}(p,q)(R)=\\{g\in\mathrm{GL}_{n}(R\otimes_{\mathbb{Z}}\mathcal{O}_{E})\mid\exists
c(g)\in R^{\times},g^{*}Jg=c(g)J\\}$

for every commutative ring $R$. Then $\mathrm{GU}(p,q)_{\mathbb{Q}}$ is a
connected reductive algebraic group, and we have a morphism of group schemes
$c:\mathrm{GU}(p,q)\rightarrow\mathrm{GL}_{1}$, whose kernel is the unitary
group $\mathrm{U}(p,q)$.

Let $h:\mathbb{S}\rightarrow\mathrm{GU}(p,q)_{\mathbb{R}}$ be the morphism
defined by

$h(z)=\begin{pmatrix}zI_{p}&0\\\
0&\overline{z}I_{q}\end{pmatrix}\in\mathrm{GU}(p,q)(\mathbb{R}).$

Then $(G,h)$ is a Shimura datum, and $K_{\infty}$ is the set of matrices
$\begin{pmatrix}g_{1}&0\\\ 0&g_{2}\end{pmatrix}$ such that
$g_{1}\in\mathrm{GL}_{p}(\mathbb{C})$, $g_{2}\in\mathrm{GL}_{q}(\mathbb{C})$
and there exists $c\in\mathbb{R}$ with $g_{1}^{*}g_{1}=cI_{p}$ and
$g_{2}^{*}g_{2}=cI_{q}$.

We use the isomorphism
$\mathbb{C}\otimes_{\mathbb{Q}}E\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathbb{C}\times\mathbb{C}$
sending $x\otimes 1+y\otimes\sqrt{-d}$ to $(x+\sqrt{-d}y,x-\sqrt{-d}y)$ to
identify $\mathrm{GU}(p,q)(\mathbb{C})$ to a subgroup of
$\mathrm{GL}_{n}(\mathbb{C})\times\mathrm{GL}_{n}(\mathbb{C})$; note that the
involution $g\mapsto\overline{g}$ of $\mathrm{GU}(p,q)(\mathbb{C})$
corresponds to switching the two factors. With this convention, we have
$h_{\mathbb{C}}\circ r(z)=\left(\begin{pmatrix}zI_{p}&0\\\
0&I_{q}\end{pmatrix},\begin{pmatrix}I_{p}&0\\\ 0&zI_{q}\end{pmatrix}\right)$
for every $z\in\mathbb{C}^{\times}$.

It is easy to see that $h_{\mathbb{C}}\circ r$ is defined over $E$ but not
over $\mathbb{Q}$. On the other hand, the reflex field of $(G,h)$ is $E$ if
$p>q$ and $\mathbb{Q}$ if $p=q$.

######  Example 1.42.

Let $(G,h)$ be the Shimura datum of Example 1.35 (so that
$G=\mathrm{GSp}_{2d}$). For every $z\in\mathbb{C}^{\times}$, we have

$h_{\mathbb{C}}\circ
r(z)=\begin{pmatrix}\frac{1}{2}(z+1)I_{d}&-\frac{1}{2i}(z-1)I_{d}\\\
\frac{1}{2i}(z-1)I_{d}&\frac{1}{2}(z+1)I_{d}\end{pmatrix}=P\begin{pmatrix}zI_{d}&0\\\
0&I_{d}\end{pmatrix}P^{-1},$

where

$P=\begin{pmatrix}\frac{1}{\sqrt{2}}I_{d}&\frac{i}{\sqrt{2}}I_{d}\\\
\frac{i}{\sqrt{2}}I_{d}&\frac{1}{\sqrt{2}}I_{d}\end{pmatrix}.$

So the reflex field of $(G,h)$ is $\mathbb{Q}$.

####  1.4.4. Canonical models

We are now ready to define canonical models.

######  Definition 1.43.

Let $(G,h)$ be a Shimura datum and let $F=F(G,h)$. A _canonical model_ of
$M(G,h)(\mathbb{C})$ is a model $(M_{K})_{K}$ over $F$ such that, for every
torus $u:H\subset G$ and every $h^{\prime}:\mathbb{S}\rightarrow
H_{\mathbb{R}}$ such $u\circ h^{\prime}$ and $h$ are
$G(\mathbb{R})$-conjugated (i.e. such that $u$ induces a morphism of Shimura
data from $(H,h^{\prime})$ to $(G,h)$), the morphism

$u:M(H,h^{\prime})(\mathbb{C})\rightarrow M(G,h)(\mathbb{C})$

is defined over the compositum $F\cdot F(H,h^{\prime})\subset\mathbb{C}$,
where we use as model of $M(H,h^{\prime})(\mathbb{C})$ over $F(H,h^{\prime})$
the one defined in 1.4.2.

######  Examples 1.44.

  * (1)

If $G$ is a torus, then the model of 1.4.2 is a canonical model of
$M(G,h)(\mathbb{C})$.

  * (2)

If $(G,h)$ is the Shimura datum of Example 1.35 (so that
$G=\mathrm{GSp}_{2d}$), then the schemes
$(\mathcal{M}_{d,K,\mathbb{Q}})_{K\subset\mathrm{GSp}_{2d}(\mathbb{A}_{f})}$
of 1.2.4 form a canonical model of $M(G,h)(\mathbb{C})$. This is not obvious
but follows from the main theorem of complex multiplication; see Section 4 of
[35].

At the time of Deligne’s paper [35], it was not known whether all Shimura
varieties have canonical models (spoiler: this is now known to be true, see
Theorem 2.30), but it was possible to prove their uniqueness. If
$u:(H,h^{\prime})\rightarrow(G,h)$ is a morphism of Shimura data as in
Definition 1.43 (so that $H$ is a subtorus of $G$), the points of the image of
$M(H,h^{\prime})(\mathbb{C})$ in $M(G,h)(\mathbb{C})$ are called _special
points_. The fact that canonical models are unique if they exist relies on the
following two points:

  * (i)

The set of all $G(\mathbb{A}_{f})$-translates of special points are dense in
$M(G,h)(\mathbb{C})$ (see Proposition 5.2 of [35]; in fact, we just need to
take $G(\mathbb{A}_{f})$-translates of one special point);

  * (ii)

For every finite extension $F^{\prime}\subset\mathbb{C}$ of $F(G,h)$, there
exists $u:(H,h^{\prime})\rightarrow(G,h)$ as above such that $F(H,h^{\prime})$
and $F^{\prime}$ are linearly disjoint over $F(G,h)$ (Théorème 5.1 of [35]).

From this, we can deduce:

######  Theorem 1.45 (Corollaire 5.4 of [35]).

Let $u:(G_{1},h_{1})\rightarrow(G_{2},h_{2})$ be a morphism of Shimura data,
and suppose that the Shimura varieties of $(G_{1},h_{1})$ and $(G_{2},h_{2})$
have canonical models. Then the morphism
$M(G_{1},h_{1})(\mathbb{C})\rightarrow M(G_{2},h_{2})(\mathbb{C})$ of Shimura
varieties corresponding to $u$ is defined over any common extension $F$ of
$F(G_{1},h_{1})$ and $F(G_{2},h_{2})$ in $\mathbb{C}$.

######  Corollary 1.46.

Let $(G,h)$ be a Shimura datum. Then a canonical model of $M(G,h)(\mathbb{C})$
is unique up to unique isomorphism if it exists.

######  Corollary 1.47.

Let $(G,h)$ be a Shimura datum satisfying the hypothesis of Theorem 1.37, let
$\nu:G\rightarrow T:=G/G_{\mathrm{der}}$ be the quotient morphism, and let
$F=F(G,h)$. Then the action of $\operatorname{Gal}(\overline{F}/F)$ on
$\pi_{0}(M(G,h)(\mathbb{C}))\simeq\pi_{0}(T(\mathbb{Q})\backslash
T(\mathbb{A}))/\pi_{0}(K_{\infty})$ is given by the reciprocity morphism for
$(T,\nu\circ h)$.

###### Proof.

Apply Theorem 1.45 to $\nu:(G,h)\rightarrow(T,\nu\circ h)$.

∎

Using the same techniques as for Theorem 1.45, we also get the following very
useful result.

######  Proposition 1.48 (Corollaire 5.7 of [35]).

Let $u:(G_{1},h_{1})\rightarrow(G_{2},h_{2})$ be a morphism of Shimura data
such that the underlying morphism of algebraic groups is a closed immersion
and that $F(G_{1},h_{1})\subset F(G_{2},h_{2})$. If
$M(G_{2},h_{2})(\mathbb{C})$ has a canonical model, then so does
$M(G_{1},h_{1})(\mathbb{C})$.

##  2\. Lecture 2: arithmetic Shimura varieties

In lecture 1, we have defined Shimura varieties over $\mathbb{C}$ and
introduced the notion of canonical model of a Shimura variety. We also
discussed in some detail the example of the Siegel modular varieties, which
have canonical models coming from their modular interpretation. In this
lecture, we first want to present different types of Shimura varieties, with
each type being contained in the next one:

  * •

The Siegel modular variety: it is a moduli space of principally polarized
abelian schemes with some level structure;

  * •

PEL type Shimura varieties: they have an interpretation as moduli spaces of
polarized abelian schemes with multiplication by the ring of integers
$\mathcal{O}$ of some number field and some level structure (here “P” means
“polarization”, “E” means “endomorphisms” in reference to the action of
$\mathcal{O}$ and “L” means “level structure”);

  * •

Hodge type Shimura varieties: they come from Shimura data $(G,h)$ that have an
injective morphism into a Siegel Shimura datum (Example 1.35);

  * •

Abelian type Shimura varieties: their Shimura datum is “isogenous” to a Hodge
type Shimura datum (in a way to be made precise later);

  * •

General Shimura varieties: all Shimura varieties.

The further we go down in the list, the less is known about the geometry of
the Shimura variety (and the higher the price for what we know), because many
of the techniques we have rely on the interpretation of the Shimura varieties
as moduli spaces of abelian schemes, and this is only really available for PEL
type Shimura varieties (there is a modular interpretation for Hodge type
Shimura varieties, but it is harder to use).

For example, PEL type Shimura varieties naturally come with an _integral
model_ defined over a localization of the ring of integers of their reflex
field, but it took a lot of effort to construct integral models for Hodge type
and abelian type Shimura varieties, and to formulate their properties (see
2.1.4 and Theorems 2.22 and 2.29); as far as we know, nothing is known for
general Shimura varieties.

We are not claiming that the classification above is the only measure of the
complexity of a Shimura variety, whatever that means. For example, as we will
discuss in lecture 3, if one wants to study the cohomology of Shimura
varieties and their zeta functions, then the simplest case is not the case of
Siegel modular varieties, but rather of compact PEL type Shimura varieties
whose group has no endoscopy and a simply connected derived subgroup; we will
introduce some examples of these, known as _Kottwitz’s simple Shimura
varieties_ , in 2.4.

###  2.1. PEL type Shimura varieties

These Shimura varieties were introduced by Kottwitz in [64], but we will
follow the presentation of Lan for the moduli problems (cf. [67] 1.4.1), which
is closer to our definition of the Siegel moduli problem. The equivalence
between the two definitions is proved in [67] 1.4.3.

####  2.1.1. PEL data

######  Definition 2.1 (see [68] 5.1 or Definition 1.2.13 of [67]).

An _(integral) PEL datum_ is a quintuple
$(\mathcal{O},*,\Lambda,\langle\cdot,\cdot\rangle,h)$, where:

  * (1)

$\mathcal{O}$ is an order in a finite-dimensional semisimple
$\mathbb{Q}$-algebra $B$ (that is, $\mathcal{O}$ is a subring of $B$ that is a
free $\mathbb{Z}$-module and spans the $\mathbb{Q}$-vector space $B$);

  * (2)

$*$ is a _positive involution_ of $\mathcal{O}$, i.e. an anti-automorphism of
rings of order $2$ such that, for every $x\in\mathcal{O}\setminus\\{0\\}$, we
have
$\operatorname{Tr}_{(B\otimes_{\mathbb{Q}}\mathbb{R})/\mathbb{R}}(xx^{*})>0$;

  * (3)

$\Lambda$ is an $\mathcal{O}$-module that is finitely generated and free as a
$\mathbb{Z}$-module;

  * (4)

$\langle\cdot,\cdot\rangle:\Lambda\times\Lambda\rightarrow\mathbb{Z}$ is a
$\mathbb{Z}$-bilinear alternating map such that, for all $x,y\in\Lambda$ and
$b\in\mathcal{O}$, we have

$\langle bx,y\rangle=\langle x,b^{*}y\rangle;$

  * (5)

$h:\mathbb{C}\rightarrow\mathrm{End}_{B\otimes_{\mathbb{Q}}\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})$
is an $\mathbb{R}$-algebra morphism such that:

    * (a)

For $z\in\mathbb{C}$ and $x,y\in\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$, we
have

$\langle h(z)(x),y\rangle=\langle x,h(\overline{z})(y)\rangle;$

    * (b)

The $\mathbb{R}$-bilinear pairing $\langle\cdot,h(i)(\cdot)\rangle$, which is
symmetric by (a), is also positive definite.

Let $(\mathcal{O},*,\Lambda,\langle\cdot,\cdot\rangle,h)$ be a PEL datum. We
define a group scheme $G$ over $\mathbb{Z}$ by

$G(R)=\\{g\in\mathrm{End}_{\mathcal{O}\otimes_{\mathbb{Z}}R}(\Lambda\otimes_{\mathbb{Z}}R)\mid\exists
c(g)\in R^{\times},\ \langle
g(\cdot),g(\cdot)\rangle=c\langle\cdot,\cdot\rangle\\}$

for every commutative ring $R$. We also get a morphism of group schemes
$c:G\rightarrow\mathrm{GL}_{1}$. The morphism of $\mathbb{R}$-algebras
$h:\mathbb{C}\rightarrow\mathrm{End}_{B\otimes_{\mathbb{Q}}\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})$
induces a morphism of $\mathbb{R}$-algebraic groups $h:\mathbb{S}\rightarrow
G_{\mathbb{R}}$.

######  Proposition 2.2.

The couple $(G_{\mathbb{Q}}^{0},h)$ satisfies conditions (a)-(c) in the
definition of a Shimura datum (Definition 1.33).

Let us explain why this proposition is true. Condition (a) of Definition 1.33
follows from the fact that
$h:\mathbb{C}^{\times}\rightarrow\mathrm{End}_{B\otimes_{\mathbb{Q}}\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})$
is a morphism of $\mathbb{R}$-algebras, and condition (c) of loc. cit. follows
from condition (5)(b) of Definition 2.1. We prove condition (b), on the
decomposition of $\operatorname{Lie}(G_{\mathbb{C}})$ into eigenspaces for the
action of $\mathrm{Ad}\circ h_{\mathbb{C}}$. Let
$V=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$, a finite-dimensional
$\mathbb{R}$-vector space. Then
$h:\mathbb{C}\rightarrow\mathrm{End}_{\mathbb{R}}(V)$ is a morphism of
$\mathbb{R}$-algebras, hence defines a structure of $\mathbb{C}$-vector space
on $V$. In particular, we have $V\otimes_{\mathbb{R}}\mathbb{C}=V_{1}\oplus
V_{2}$, where $\mathbb{C}$ acts on $V_{1}$ (resp. $V_{2}$) via multiplication
by $z\mapsto z$ (resp. $z\mapsto\overline{z}$). Condition (b) follows from
this, because $\mathrm{End}(V)\simeq V^{*}\otimes_{\mathbb{R}}V$ and
$\mathfrak{g}:=\operatorname{Lie}(G_{\mathbb{R}})\subset\mathrm{End}_{\mathbb{R}}(V)$.

We keep the same notation. Then $A:=V/\Lambda$ is a complex torus. We would
like this torus to be an abelian variety, so we need a polarization on it,
that is, a positive definite Hermitian form $H$ on $V$ such that
$\operatorname{Im}(H)$ takes integer values on $\Lambda$. But we already have
an alternating form on $\Lambda$, so we know from 1.2.2 how to proceed: define
$H$ by

$H(v,w)=\langle h(i)(v),w\rangle+i\langle v,w\rangle.$

The fact that $\operatorname{Im}(H)$ takes integral values on $\Lambda$ is
clear, the pairing $H$ is Hermitian by condition (5)(a) and positive definite
by condition (5)(b). So the torus $A$ is an abelian variety, with dual abelian
variety $A^{\vee}=V/\Lambda^{\vee}$, where $\Lambda^{\vee}=\\{v\in
V\mid\forall w\in\Lambda,\langle v,w\rangle\in\mathbb{Z}\\}$ is the dual
lattice of $\Lambda$. The polarization $\lambda:A\rightarrow A^{\vee}$ defined
by $H$ is then just the map induced by $\Lambda\subset\Lambda^{\vee}$. We have
a morphism of algebraic groups $\mathbb{S}\stackrel{{\scriptstyle
h}}{{\rightarrow}}G_{\mathbb{R}}\subset\mathrm{GL}(V)$, and the decomposition
$V\otimes_{\mathbb{R}}\mathbb{C}=\bigoplus_{p,q\in\mathbb{Z}}V^{p,q}$ induced
by this morphism (see 1.3.1) is the same as the Hodge structure coming from
the isomorphism $V\simeq\mathrm{H}_{1}(A,\mathbb{R})$. Note also that the
action of $\mathcal{O}$ on $L$ defines a morphism of rings
$\iota:\mathcal{O}\rightarrow\mathrm{End}_{\mathbb{C}}(A)$ satisfying the
_Rosati condition_ , which means that, for every $b\in\mathcal{O}$, we have

$\lambda\circ i(b^{*})=i(b)^{\vee}\circ\lambda.$

######  Remark 2.3 ([64], sections 5 and 7).

Let $(\mathcal{O},*,\Lambda,\langle\cdot,\cdot\rangle,h)$ be a PEL datum, and
let $G$ be the associated group scheme. If the $\mathbb{Q}$-algebra
$B=\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ is simple, then all the simple
factors of $G_{\mathrm{der}}(\mathbb{C})$ are of the same type, which is $A$,
$C$ or $D$. If we are in type $A$ or $C$, then $G_{\mathbb{Q}}$ is connected
and reductive. In type $D$, the group $G_{\mathbb{Q}}$ is reductive and has
$2^{[F_{0}:\mathbb{Q}]}$ connected components, where $F_{0}$ is the field of
fixed points of $*$ in the center of
$\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$, so it is never connected.

We now give some examples covering all the types.

######  Example 2.4.

Take $\mathcal{O}=\mathbb{Z}$, $*=\mathbf{id}_{\mathcal{O}}$,
$\Lambda=\mathbb{Z}^{2d}$, $\langle\cdot,\cdot\rangle$ the perfect symplectic
pairing with matrix $\begin{pmatrix}0&I_{d}\\\ -I_{d}&0\end{pmatrix}$ in the
canonical basis of $\mathbb{Z}^{2d}$, and $h:\mathbb{C}\rightarrow
M_{2d}(\mathbb{R})$ defined by

$h(a+ib)=\begin{pmatrix}aI_{d}&-bI_{d}\\\ bI_{d}&aI_{d}\end{pmatrix}.$

Then we get a PEL datum, and the couple $(G_{\mathbb{Q}},h)$ is the Siegel
Shimura datum of Example 1.35.

######  Example 2.5.

Let $B=E\subset\mathbb{C}$ be an imaginary quadratic extension of
$\mathbb{Q}$, $\mathcal{O}$ be an order in $E$ (for example the ring of
integers $\mathcal{O}_{E}$), $*$ be the restriction to $\mathcal{O}$ of
complex conjugation, 161616If $(1,a)$ is a $\mathbb{Z}$-basis of
$\mathcal{O}_{E}$, then every order $\mathcal{O}$ of $E$ is contained in
$\mathcal{O}_{E}$ and of the form $\mathbb{Z}\oplus fa\mathbb{Z}$, where
$f=[\mathcal{O}_{E}:\mathcal{O}]$ is the _conductor_ of $\mathcal{O}$. In
particular, the order $\mathcal{O}$ is stable by the nontrivial element of
$\operatorname{Gal}(E/\mathbb{Q})$, which is the restriction of complex
conjugation to $E$. $\Lambda=\mathcal{O}^{p+q}$ with $p\geq q\geq 0$. Choose
$\varepsilon\in\mathcal{O}$ such that $-i\varepsilon\in\mathbb{R}_{>0}$, let
$H$ be the Hermitian pairing on $\Lambda$ with matrix
$\begin{pmatrix}I_{p}&0\\\ 0&-I_{q}\end{pmatrix}$ and
$\langle\cdot,\cdot\rangle$ be the alternating pairing
$\operatorname{Tr}_{\mathcal{O}/\mathbb{Z}}(\varepsilon H)$ on $\Lambda$.
Finally, define
$h:\mathbb{C}\rightarrow\mathrm{End}_{\mathbb{C}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})=M_{p+q}(\mathbb{C})$
by $h(z)=\begin{pmatrix}zI_{p}&0\\\ 0&\overline{z}I_{q}\end{pmatrix}$. Then we
get a PEL datum, and the couple $(G_{\mathbb{Q}},h)$ is the Shimura datum of
Example 1.41.

If $q=0$, then $(G,h)$ does not satisfy condition (d) of Definition 1.33.

######  Example 2.6.

At the other extreme of type A, we have the PEL data considered by Kottwitz in
[63] (see the beginning of Section 1 of that paper). Let $F_{0}$ be a totally
real number field, $F$ be a totally imaginary quadratic extension of $F_{0}$
and $B$ be a division algebra with center $F$; we suppose given a positive
involution $*$ on $B$ whose restriction to $F$ is the nontrivial element of
$\operatorname{Gal}(F/F_{0})$.

We take $\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}=B$ with its action by left
translations. The alternating form $\langle.,.\rangle$ on
$\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}$ is given by $\langle
x,y\rangle=\operatorname{Tr}_{B/\mathbb{Q}}(xby^{*})$, where $b$ is an element
of $B^{\times}$ such that $b^{*}=-b$. We have
$\mathrm{End}_{B}(\Lambda\otimes_{\mathbb{Z}}\mathbb{Q})=B^{\mathrm{op}}$, so
$G_{\mathbb{Q}}=\\{x\in(B^{\mathrm{op}})^{\times}|xx^{\ddagger}\in\mathbb{Q}^{\times}\\}$,
where $(.)^{\ddagger}$ is the involution of $B^{\mathrm{op}}$ define by
$x^{\ddagger}=bx^{*}b^{-1}$ (which is not positive in general).

Warning: What Kottwitz calls $D$ and $*$ in [63] is what we call
$B^{\mathrm{op}}$ and ${\ddagger}$ here, so in particular the involution $*$
of [63] is not assumed to be positive (and the PEL datum is not defined
there). We changed the notation to be consistent with the rest of the text and
with Kottwitz’s other paper [64].

We have $F_{0}\otimes_{\mathbb{Q}}\mathbb{R}\simeq\mathbb{R}^{r}$,
$F\otimes_{\mathbb{Q}}\mathbb{R}\simeq\mathbb{C}^{r}$ and
$B\otimes_{\mathbb{Q}}\mathbb{R}\simeq M_{n}(\mathbb{C})^{r}$, where
$n=\sqrt{\dim_{\mathbb{Q}}B}$; the involution $*$ is conjugate on each factor
to the positive involution $A\mapsto{}^{t}\overline{A}$. So, if
$G_{0}=\\{x\in(B^{\mathrm{op}})^{\times}|xx^{\ddagger}=1\\}$, then
$G_{0,\mathbb{R}}$ is isomorphic to a product
$\prod_{i=1}^{r}\mathrm{U}(p_{i},q_{i})$ of unitary groups, and
$G_{\mathbb{R}}$ is isomorphic to the corresponding general group (i.e. the
subgroup of $\prod_{i=1}\mathrm{GU}(p_{i},q_{i})$ where all the multipliers
are equal). We define $h:\mathbb{C}^{\times}\rightarrow G_{\mathbb{R}}$ by
sending $z\in\mathbb{C}^{\times}$ to element of $G(\mathbb{R})$ corresponding
to the family of matrices $\left(\begin{pmatrix}zI_{p_{i}}&0\\\
0&\overline{z}I_{q_{i}}\end{pmatrix}\right)_{1\leq i\leq r}$. This extend to a
morphism of $\mathbb{R}$-algebras $h:\mathbb{C}\rightarrow
B^{\mathrm{op}}\otimes_{\mathbb{Q}}\mathbb{R}$ sending $0$ to $0$; as
$\mathbb{C}$ is commutative, we can also see $h$ as a morphism of
$\mathbb{R}$-algebras $\mathbb{C}\rightarrow B\otimes_{\mathbb{Q}}\mathbb{R}$.

We have defined what is known as a rational PEL datum. To get an integral PEL
datum, we must suppose that there exists an order $\mathcal{O}$ in $B$ such
that $\mathcal{O}^{*}=\mathcal{O}$ and $\langle.,.\rangle$ takes integer
values on $\mathcal{O}$.

######  Example 2.7.

Let $B$ be a quaternion algebra over $\mathbb{Q}$ such that
$B\otimes_{\mathbb{Q}}\mathbb{R}\simeq\mathbb{H}$, and let $\mathcal{O}$ be an
order in $B$ that is stable by the involution of $\mathbb{H}$ defined by
$(x+iy+jz+kt)^{*}=x-iy-jz-kt$. Let $\Lambda=\mathcal{O}^{2n}$, let
$\langle\cdot,\cdot\rangle$ be
$\operatorname{Tr}_{\mathcal{O}/\mathbb{Z}}\circ H$, where $H$ is the skew-
Hermitian pairing on $\Lambda$ with matrix $\begin{pmatrix}0&I_{n}\\\
-I_{n}&0\end{pmatrix}$. Define
$h:\mathbb{C}\rightarrow\mathrm{End}_{B\otimes_{\mathbb{Q}}\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})=M_{n}(\mathbb{H})$
by $h(a+ib)=\begin{pmatrix}aI_{n}&-bI_{n}\\\ bI_{n}&aI_{n}\end{pmatrix}$.

The group $(G^{0})_{\mathbb{R},\mathrm{der}}$ is often denoted by
$\mathrm{SO}^{*}_{2n}$; it is a quasi-split outer form of the split orthogonal
group $\mathrm{SO}_{2n}$, hence is of type $D_{n}$. But note that the
algebraic group $G_{\mathbb{Q}}$ is not connected.

######  Definition 2.8.

We say that a Shimura datum is of _PEL type_ if it is of the form
$(G^{0}_{\mathbb{Q}},h)$, where $(G,h)$ comes from a PEL datum. The
corresponding Shimura varieties are called _PEL type Shimura varieties_.

####  2.1.2. PEL moduli problems

Just as in the case of the Siegel modular variety, PEL type Shimura varieties
are the solution of a moduli problem, 171717Well, almost. See Proposition 2.14
below for a more precise statement. known as a _PEL moduli problem_. We now
discuss these.

Let $(\mathcal{O},*,\Lambda,\langle\cdot,\cdot\rangle,h)$ be a PEL datum, and
let $(G,h)$ be defined as before. If
$V=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$, then we saw that the morphism
$h:\mathbb{C}\rightarrow\mathrm{End}_{\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{R}}(V)$
defines a decomposition $V\otimes_{\mathbb{R}}\mathbb{C}=V^{-1,0}\oplus
V^{0,-1}$. The _reflex field_ $F$ of the PEL datum is the field of definition
of the isomorphism class of $V^{-1,0}$ as an
$\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{C}$-module, that is, the subfield of
$\mathbb{C}$ generated by the elements $\operatorname{Tr}(b,V^{1,0})$, for
$b\in\mathcal{O}$. It is also equal to the field of the definition of the
conjugacy class of $h_{\mathbb{C}}\circ
r:\mathrm{GL}_{1,\mathbb{C}}\rightarrow G_{\mathbb{C}}$, that is, the reflex
field of $(G^{0}_{\mathbb{Q}},h)$ when this couple is a Shimura datum.

######  Definition 2.9.

We say that a prime number $p$ is _good_ for the PEL datum if:

  * •

$p$ is unramified in $\mathcal{O}$ (i.e. it does not divide the discriminant
of $\mathcal{O}/\mathbb{Z}$);

  * •

$p$ does not divide $[\Lambda^{\vee}:\Lambda]$, where $\Lambda^{\vee}$ is as
before the lattice $\\{v\in V\mid\forall w\in\Lambda,\langle
v,w\rangle\in\mathbb{Z}\\}$;

  * •

$p\not=2$ if the PEL datum has a factor of type $D$ (i.e. if
$\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{R}$ has a simple factor isomorphic to
an algebra $M_{n}(\mathbb{H})$ with its canonical positive involution).

If $p$ is not good we say that it is bad. Note that, if $p$ is good, then $F$
and $G_{\mathbb{Q}}$ are unramified at $p$; in fact, the group
$G(\mathbb{Z}_{p})$ is then a hyperspecial maximal compact subgroup of
$G(\mathbb{Q}_{p})$.

Let $T$ be a set of good primes (finite or infinite), and let
$\mathcal{O}_{F,T}$ be the localization $\mathcal{O}_{F}[\frac{1}{p},\
p\not\in T]$.

######  Definition 2.10.

Let $n$ be a positive integer that is prime to all the elements of $T$. Then
the PEL moduli problem at level $n$ defined by the fixed PEL datum is the
contravariant functor $\mathcal{M}_{n}$ from the category of
$\mathcal{O}_{F,T}$-schemes to the category of sets sending an
$\mathcal{O}_{F,T}$-scheme $S$ to the set of isomorphism classes of quadruples
$(A,\lambda,\iota,(\eta,\varphi))$, where:

  * •

$A$ is an abelian scheme over $S$;

  * •

$\lambda:A\rightarrow A^{\vee}$ is a polarization whose degree is prime to all
the elements of $T$;

  * •

$\iota:\mathcal{O}\rightarrow\mathrm{End}_{S}(A)$ is a morphism of rings
satisfying the Rosati condition: for every $b\in\mathcal{O}$, we have
$\lambda\circ\iota(b^{*})=\iota(b)^{\vee}\circ\lambda$;

  * •

$(\eta,\varphi)$ is a level $n$ structure on $A$, i.e.,
$\eta:A[n]\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\Lambda/n\Lambda)_{S}$
is an $\mathcal{O}$-equivariant isomorphism of group schemes and
$\varphi:\underline{\mathbb{Z}/n\mathbb{Z}}_{S}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mu_{n,S}$
is an isomorphism of group schemes such that
$\varphi\circ\langle\cdot,\cdot\rangle\circ\eta$ is the Weil pairing defined
by $\lambda$ on $A[n]_{S}$, and moreover $(\eta,\varphi)$ are liftable to
level $m$ structures for every prime-to-$T$ multiple $m$ of $n$, in the sense
of [67] Definition 1.3.6.2.

We furthermore require that this quadruple satisfy the following _determinant
condition_ (see Definition 1.3.4.1 of [67]): let
$\alpha_{1},\ldots,\alpha_{t}$ be a basis of the $\mathbb{Z}$-module
$\mathcal{O}$, and let $X_{1},\ldots,X_{t}$ be indeterminates. Then
$\det(\iota(\alpha_{1})X_{1}+\ldots+\iota(\alpha_{t})X_{t},\operatorname{Lie}(A))$
is a polynomial in $\mathcal{O}_{S}[X_{1},\ldots,X_{t}]$, and the condition
says that this polynomial is equal to the image by the map
$\mathcal{O}_{F,T}\rightarrow\mathcal{O}_{S}$ of the polynomial
$\det(\alpha_{1}X_{1}+\ldots+\alpha_{t}X_{t},V^{1,0})\in\mathcal{O}_{F,T}[X_{1},\ldots,X_{t}]$.
181818It is not totally obvious that the second polynomial, which is a priori
in $F[X_{1},\ldots,X_{t}]$, has its coefficients in $\mathcal{O}_{F,T}$. But
it is also not too hard to check. See for example pages 389-390 of [64].

######  Remark 2.11.

  * (1)

We did not specify the relative dimension of the abelian scheme $A$ in the
moduli problem, because it is necessarily equal to
$\dim_{\mathbb{C}}(V^{1,0})$ by the determinant condition.

  * (2)

For $n\in\\{1,2\\}$, the objects of the moduli problem $\mathcal{M}_{n}$ can
have nontrivial automorphisms, so it would make more sense to see
$\mathcal{M}_{n}$ as a functor with values in groupoids, i.e. as a stack. See
Remark 1.13.

  * (3)

As in Remark 1.17, it is also possible to define the moduli problem
$\mathcal{M}_{K}$ for more general levels $K$, i.e. for open compact subgroups
$K$ of $G(\mathbb{A}_{f})$, though there is a condition on $K$ corresponding
to the condition that $n$ be prime to $T$. Let $\mathbb{A}_{f}^{T}$ be the
ring of prime-to-$T$ adeles of $\mathbb{Q}$, i.e. the restricted product of
the $\mathbb{Q}_{p}$ for $p\not\in T$, and $\mathbb{A}_{T}$ be the restricted
product of the $\mathbb{Q}_{p}$ for $p\in T$. We have
$\mathbb{A}_{f}=\mathbb{A}_{T}\times\mathbb{A}_{f}^{T}$, and the condition on
$K$ is that $K=K_{T}K^{T}$, where $K^{T}\subset G(\mathbb{A}_{f}^{T})$ and
$K_{T}=\prod_{p\in T}K_{p}$ with $K_{p}=G(\mathbb{Z}_{p})$ for every $p\in T$.

The following theorem is a consequence of Mumford’s representability result
(Theorem 1.16), as explained in Section 5 of Kottwitz’s paper [64]. See also
Corollaries 1.4.1.12 and 7.2.3.10 of Lan’s book [67] for more details.

######  Theorem 2.12.

If $n\geq 3$, then the functor $\mathcal{M}_{n}$ is representable by a smooth
quasi-projective scheme over $\mathcal{O}_{F,T}$.

In fact, by Theorem 1.4.1.11 and Corollary 7.2.3.10 of [67], the functor
$\mathcal{M}_{K}$ is representable by a smooth quasi-projective scheme for $K$
small enough.

######  Remark 2.13.

We also have an action of the Hecke operators defined by elements $g\in
G(\mathbb{A}_{f}^{T})$ (i.e. the Hecke operators that are trivial at primes of
$T$) on the tower $(\mathcal{M}_{K})$; as for the Siegel moduli problem, the
element $g$ acts on the level structure. See Remark 1.4.3.11 of [67] (and the
comparison result of Proposition 1.4.3.4 of loc. cit.).

####  2.1.3. PEL moduli problems and canonical models

Consider the PEL datum of Example 2.4. The associated couple
$(G_{\mathbb{Q}},h)$ is the Siegel Shimura datum of Example 1.35, and we have
seen in Example 1.44(2) that the corresponding moduli problem, with
$T=\varnothing$, defines a canonical model of the Shimura variety of
$(G_{\mathbb{Q}},h)$.

We would like something like this to be true for general PEL data, but one
obstruction is that the moduli problem only depends on the completions of the
symplectic $\mathcal{O}$-module $(\Lambda,\langle.,,\rangle)$ at the places of
$\mathbb{Q}$; this is not obvious in Definition 2.10, but it becomes so if we
use the moduli problem of Definition 1.4.2.1 of [67] or of Section 5 of [64]
(the equivalence of the two moduli problems is proved in [67] 1.4.3). Suppose
that we have a finitely generated $\mathcal{O}$-module $\Lambda^{\prime}$ that
is free as a $\mathbb{Z}$-module, with a symplectic form
$\langle.,.\rangle^{\prime}$ satisfying condition (4) of Definition 2.1 and
such that, for every place $v$ of $\mathbb{Q}$,
$\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}_{v}$ and
$\Lambda^{\prime}\otimes_{\mathbb{Z}}\mathbb{Q}_{v}$ are isomorphic as
symplectic $B\otimes_{\mathbb{Q}}{\mathbb{Q}_{v}}$-modules. Then we can see
$h$ as a morphism
$\mathbb{C}\rightarrow\mathrm{End}_{B\otimes_{\mathbb{Q}}\mathbb{R}}(\Lambda^{\prime}\otimes_{\mathbb{Z}}\mathbb{R})$,
and it will satisfy condition (5) of Definition 2.1, so we get a PEL datum
$(\mathcal{O},*,\Lambda^{\prime},\langle.,.\rangle^{\prime},h)$ that defines
the same moduli problem as the original PEL datum. In particular, we will get
a Shimura datum $({G^{\prime}}^{0}_{\mathbb{Q}},h)$, and we should also see
the canonical model for the Shimura variety of
$({G^{\prime}}^{0}_{\mathbb{Q}},h)$ in the PEL moduli problem. A priori, there
is no reason for $G$ and $G^{\prime}$ to be isomorphic, and they won’t be in
general; we only know that $G_{\mathbb{Q}_{v}}\simeq
G^{\prime}_{\mathbb{Q}_{v}}$ for every place $v$ of $\mathbb{Q}$.

In fact, we have the following result.

######  Proposition 2.14.

Suppose that the semisimple $\mathbb{Q}$-algebra
$\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ has no simple factor of type $D$
(see Definition 2.9). Then:

  * (i)

(Remark 1.4.4.4 of [67].) Let $T\subset T^{\prime}$ be two sets of good prime
numbers, let ${K}^{T^{\prime}}$ be an open compact subgroup of
$G(\mathbb{A}_{f}^{T^{\prime}})$, and let $K=K^{T^{\prime}}\prod_{p\in
T^{\prime}}G(\mathbb{Z}_{p})$. Let $\mathcal{M}_{K}$ (resp.
$\mathcal{M}^{\prime}_{K}$) be the moduli problem over
$\operatorname{Spec}(\mathcal{O}_{F,T})$ (resp.
$\operatorname{Spec}(\mathcal{O}_{F,T^{\prime}})$) from Definition 2.10, where
we take the set of good primes to be $T$ (resp. $T^{\prime}$). Then the
forgetful functor
$\mathcal{M}_{K}\rightarrow\mathcal{M}^{\prime}_{K}\times_{\operatorname{Spec}(\mathcal{O}_{F,T^{\prime}})}\operatorname{Spec}(\mathcal{O}_{F,T})$
is an isomorphism.

  * (ii)

(Sections 7-8 of [64].) Let $\operatorname{Ker}^{1}(\mathbb{Q},G)$ be the
kernel of the diagonal map
$\mathrm{H}^{1}(\mathbb{Q},G)\rightarrow\prod_{v}\mathrm{H}^{1}(\mathbb{Q}_{v},G)$,
where we take the product over all places $v$ of $\mathbb{Q}$; this is a
finite set. Suppose that $(G_{\mathbb{Q}},h)$ is a Shimura datum, 191919As we
excluded case $D$, the group $G_{\mathbb{Q}}$ is connected. So, by Proposition
2.2, the only obstacle to $(G_{\mathbb{Q}},h)$ being a Shimura datum is
condition (d) of Definition 1.33. and use $T=\varnothing$ to define the moduli
problem of Definition 2.10, so that it is a moduli problem over
$\operatorname{Spec}(F)$. 202020By (i), this moduli problem is the generic
fiber of the moduli problems defined by nonempty sets $T$. Then the projective
system $(\mathcal{M}_{K})_{K\subset G(\mathbb{A}_{f})}$ is a disjoint union
indexed by $i\in\operatorname{Ker}^{1}(\mathbb{Q},G)$ of projective systems
$(\mathcal{M}_{K}^{(i)})_{K\subset G(\mathbb{A}_{f})}$, and each
$(\mathcal{M}_{K}^{(i)})_{K\subset G_{i}(\mathbb{A}_{f})}$ is a canonical
model of the Shimura variety of $(G_{\mathbb{Q}},h)$.

######  Remark 2.15.

  * (1)

For every $i\in\operatorname{Ker}^{1}(\mathbb{Q},G)$, let $G_{i}$ be the
corresponding inner form of $G_{\mathbb{Q}}$;212121See 2.4.1 for the
definition of inner forms. we have $G_{i,v}\simeq G_{v}$ for every place $v$
of $\mathbb{Q}$, and we denote by $h_{i}$ the morphism
$\mathbb{S}\stackrel{{\scriptstyle h}}{{\rightarrow}}G_{\mathbb{R}}\simeq
G_{i,\mathbb{R}}$. If $(G_{\mathbb{Q}},h)$ is a Shimura datum, then all
$(G_{i},h_{i})$ are, and the projective system
$(\mathcal{M}_{K}^{(i)})_{K\subset G(\mathbb{A}_{f})}$ in (ii) of the
proposition is actually a canonical model of the Shimura variety of
$(G_{i},h_{i})$ (note that $G_{i}(\mathbb{A}_{f})\simeq G(\mathbb{A}_{f})$).
But, as noted by Kottwitz at the end of Section 8 of [64], under our
hypothesis that $\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ has no factor of
type $D$, all the groups $G_{i}$ are isomorphic to $G_{\mathbb{Q}}$ and the
$h_{i}$ correspond to $h$. (In fact Kottwitz calculates
$\operatorname{Ker}^{1}(\mathbb{Q},G)$ in Section 7 of [64], and finds that in
our situation it is either trivial or isomorphic to
$\operatorname{Ker}^{1}(\mathbb{Q},Z(G))$ by the canonical map, where $Z(G)$
is the center of $G$.)

  * (2)

As mentioned in (1), there is some information about
$\operatorname{Ker}^{1}(\mathbb{Q},G)$ in Section 7 of [64], and ways to
calculate it. For example, if $\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ is
simple of type $C$, we have $\operatorname{Ker}^{1}(\mathbb{Q},G)=\\{1\\}$,
which explains why the moduli problem of Definition 1.15 gives a canonical
model of the Siegel Shimura variety and not of a finite disjoint union of
copies of it. If $\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ is simple of type
$A$, then $\operatorname{Ker}^{1}(\mathbb{Q},G)$ is automatically trivial in
“half” of the cases, and it is always isomorphic to
$\operatorname{Ker}^{1}(\mathbb{Q},Z(G))$; for example, it is trivial for the
Shimura datum of Example 1.41.

  * (3)

Kottwitz does not say much about the case where
$\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Q}$ has simple factors of type $D$.
The situation is complicated for many reasons: the group $G_{\mathbb{Q}}$ is
not connected, $\operatorname{Ker}(\mathbb{Q},G)$ is not trivial, and point
(i) of Proposition 2.14 is not true in general, so we must also be careful
about the choice of $T$. In any case, it is still true that the Shimura
varieties $M_{K}(G^{0}_{\mathbb{Q}},h)(\mathbb{C})$ are open and closed
subschemes of $\mathcal{M}_{K,\mathbb{C}}$ (see [66] 2.5), and we might even
get canonical models out of this, but we will not pursue this here because
there are other ways to construct canonical models for these Shimura data (see
2.2.1).

  * (4)

If we want $(G_{\mathbb{Q}},h)$ to be a PEL Shimura datum, then this puts
pretty strict conditions on the center of $G_{\mathbb{Q}}$. For example, if
$F$ is a nontrivial totally real extension of $\mathbb{Q}$, then the group
$\operatorname{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2}$ (defined by
$\operatorname{Res}_{F/\mathbb{Q}}\mathrm{GL}_{2}(R)=\mathrm{GL}_{2}(R\otimes_{\mathbb{Q}}F)$)
is part of a Shimura datum, but this Shimura datum cannot be PEL; more
generally, we have the issue with the group
$\operatorname{Res}_{F/\mathbb{Q}}\mathrm{GSp}_{2d}$. This is somewhat
annoying, as sometimes we really do want to consider the Shimura varieties for
these precise groups (see for example Nekovar and Scholl’s [93]). Fortunately,
these Shimura data are of abelian type (see 2.2.2), so their Shimura varieties
are still understood reasonably well.

####  2.1.4. Canonical integral models

PEL moduli problems don’t just give canonical models of Shimura varieties,
they also give models over various localizations of the ring of integers of
the reflex field, at least when $\operatorname{Ker}^{1}(\mathbb{Q},G)$ is
trivial. Here are some things that we can learn from this example:

  * (1)

The ring of integers over which we can expect to have a “good” integral model
depends on the level $K$. More precisely, to have a good integral model
defined over $\mathcal{O}_{F,(p)}$, we need $K$ to be of the form
$K^{p}K_{p}$, where $K^{p}\subset G(\mathbb{A}_{f}^{p})$ and $K_{p}$ is a
hyperspecial maximal compact subgroup of $G(\mathbb{Q}_{p})$. Likewise, the
Hecke correspondences that will extend to finite étale morphisms between
integral models over $\mathcal{O}_{F,(p)}$ are the ones that are trivial at
$p$, i.e. defined by elements of $G(\mathbb{A}_{f}^{p})$.

  * (2)

We need a notion of what a “good” integral model is. If the Shimura varieties
for $(G,h)$ are compact, then we can just ask for the integral model to be
projective smooth over the localization of $\mathcal{O}_{F}$ that we are
using. But this does not suffice in the noncompact case.

######  Remark 2.16.

In these notes, we will only talk about integral models at places of good
reduction, so we want these models to be smooth over the base ring and this is
why we impose the conditions of (1). There is of course a (much more difficult
and very interesting) theory for places of bad reduction.

To solve problem (2), Milne suggested only looking at models with a certain
extension property. Let $(G,h)$ be a Shimura datum, let $F=F(G,h)$, let $p$ be
a prime number at which $G$ is unramified, and let $K_{p}\subset
G(\mathbb{Q}_{p})$ be a hyperspecial maximal compact subgroup. 222222The
condition on $p$ means that $G$ extends to a reductive group scheme
$\mathcal{G}$ over $\mathbb{Z}_{p}$, and then we can take
$K_{p}=\mathcal{G}(\mathbb{Z}_{p})$. For every level $K$, write
$M_{K}=M_{K}(G,h)$. We want to define the notion of a canonical integral model
$\mathcal{M}_{K_{p}}$ over $\mathcal{O}_{F,(p)}$ of the projective system
$(M_{K_{p}K^{p}})_{K^{p}\subset G(\mathbb{A}_{f}^{p})}$, or of its limit
$M_{K_{p}}$. The idea, first suggested by Milne in [85] (see also Moonen’s
paper [90]), is to require that, for every $S$ in a class of “admissible test
schemes” over $\mathcal{O}_{F,(p)}$, any morphism
$S\otimes_{\mathcal{O}_{F,(p)}}F\rightarrow M_{K_{p}}$ should extend to a
morphism $S\rightarrow\mathcal{M}_{K_{p}}$. The problem is to decide what
class of admissible test schemes one should use. We will follow Kisin’s
presentation in [54].

######  Definition 2.17 (See [54] 2.3.7.).

A _canonical integral model_ of the projective system
$(M_{K_{p}K^{p}})_{K^{p}\subset G(\mathbb{A}_{f}^{p})}$ (or of its limit
$M_{K_{p}}$) over $\mathcal{O}_{F,(p)}$ is a projective system
$(\mathcal{M}_{K_{p}K^{p}})_{K^{p}\subset G(\mathbb{A}_{f}^{p})}$ of smooth
$\mathcal{O}_{F,(p)}$-schemes with finite étale transition maps, given with
finite étale morphisms
$T_{g,K^{p},{K^{\prime}}^{p}}:\mathcal{M}_{K_{p}{K^{\prime}}^{p}}\rightarrow\mathcal{M}_{K_{p}K^{p}}$
for all $g\in G(\mathbb{A}_{f}^{p})$ and $K^{p},{K^{\prime}}^{p}$ open compact
subgroups of $G(\mathbb{A}_{f}^{p})$ such that ${K^{\prime}}^{p}\subset
K^{p}\cap gK^{p}g^{-1}$, and with an isomorphism of projective systems
$\iota:(\mathcal{M}_{K_{p}K^{p}})\otimes_{\mathcal{O}_{F,(p)}}F\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(M_{K_{p}K^{p}})$,
such that:

  * (a)

The morphisms $T_{g,K^{p},{K^{\prime}}^{p}}$ satisfy the analogues of
conditions (ii), (iii) and (iv) of Definition 1.38, and they correspond to the
morphisms $T_{g,K_{p}K^{p},K_{p}{K^{\prime}}^{p}}$ between canonical models by
the isomorphism $\iota$ (in other words, $\iota$ is
$G(\mathbb{A}_{f}^{p})$-equivariant);

  * (b)

The scheme $\mathcal{M}_{K_{p}}:=\varprojlim_{K^{p}}\mathcal{M}_{K_{p}K^{p}}$
satisfies the following extension property: if $S$ is a regular formally
smooth $\mathcal{O}_{F,(p)}$-scheme, then any morphism
$S\otimes_{\mathcal{O}_{F,(p)}}F\rightarrow M_{K_{p}}$ extends to a morphism
$S\rightarrow\mathcal{M}_{K_{p}}$.

As in the definition of canonical models, the properties listed in this
definition are redundant, see Remark 1.39.

In particular, by applying the extension property with
$S=\mathcal{M}_{K_{p}}$, we see that integral canonical models are unique up
to unique isomorphism. Now the problem is existence. For PEL type Shimura
varieties of type $A$ or $C$ satisfying the condition that
$\operatorname{Ker}^{1}(\mathbb{Q},G)$ is trivial, the PEL moduli problem will
give a canonical integral model, though that is not trivial: the main
ingredient is the Néron-Ogg-Shafarevich criterion on the good reduction of
abelian varieties. For more details, see Theorem 2.10 of [85], and also
Corollary 3.8 and Example 3.12(ii) of [90].

But what about other Shimura varieties ?

###  2.2. Hodge type and abelian type Shimura varieties

####  2.2.1. Hodge type Shimura varieties

For every $d\geq 1$, we denote by $(\mathrm{GSp}_{2d},h_{d})$ the Siegel
Shimura datum of Example 1.35.

The following condition was introduced in Deligne’s paper [37] (Section 2.3)
and named in Milne’s paper [84] (at the end of Section 3).

######  Definition 2.18.

A Shimura datum $(G,h)$ is _of Hodge type_ if there exists an integer $d\geq
1$ and a morphism of Shimura data
$u:(G,h)\rightarrow(\mathrm{GSp}_{2d},h_{d})$ such that the underlying
morphism of algebraic groups $G\rightarrow\mathrm{GSp}_{2d}$ is injective. In
this case, we also say that the corresponding Shimura varieties are of Hodge
type.

######  Example 2.19.

  * (1)

Every Shimura datum of PEL type is of Hodge type, pretty much by definition
(or by Proposition 2.3.2 of [37]): if
$(\mathcal{O},*,\Lambda,\langle\cdot,\cdot\rangle,h)$ is a PEL datum and $G$
is the corresponding group scheme, then $G$ embeds into the group scheme $H$
defined by

$H(R)=\\{g\in\mathrm{End}_{R}(\Lambda\otimes_{\mathbb{Z}}R)\mid\exists c(g)\in
R^{\times},\ \langle
g(\cdot),g(\cdot)\rangle=c(g)\langle\cdot,\cdot\rangle\\}$

for every commutative ring $R$. As the alternating pairing
$\langle\cdot,\cdot\rangle$ is nondegenerate by condition (5)(b) of Definition
2.1, and as all nondegenerate alternating pairings on
$\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}$ are equivalent, we have
$H_{\mathbb{Q}}\simeq\mathrm{GSp}_{2d,\mathbb{Q}}$ for
$2d=\dim_{\mathbb{Q}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{Q})$. Let
$h^{\prime}$ be the composition of $h:\mathbb{S}\rightarrow G_{\mathbb{R}}$
and of the embedding $G_{\mathbb{R}}\rightarrow
H_{\mathbb{R}}=\mathrm{GSp}_{2d,\mathbb{R}}$. We have seen in the discussion
after Proposition 2.2 that
$h:\mathbb{C}\rightarrow\mathrm{End}_{\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})$
induces a Hodge structure of type $\\{(-1,0),(0,-1)\\}$ on
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}$, 232323Which means that, in the
decomposition
$\Lambda\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{p,q\in\mathbb{Z}}V^{p,q}$
induced by $h$, we have $V^{p,q}=0$ unless $(p,q)\in\\{(-1,0),(0,-1)\\}$. so
$h^{\prime}$ satisfies condition (b) of Definition 1.33. It also satisfies
condition (c) of loc. cit. because the $\mathbb{R}$-bilinear pairing
$\langle\cdot,h(i)(\cdot)\rangle$ on $\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$
is symmetric definite positive. Finally, for every $a\in\mathbb{R}^{\times}$,
the element $h(a)$ of
$\mathrm{End}_{\mathbb{R}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{R})$ is
$a\cdot\mathbf{id}$ (because $h$ is a morphism of $\mathbb{R}$-algebras), so
$h^{\prime}(a)=aI_{2d}\in\mathrm{GSp}_{2d}(\mathbb{R})$. This implies that
$h^{\prime}$ and $h_{d}$ are conjugated by $\mathrm{GSp}_{2d}(\mathbb{R})$.

  * (2)

The list of groups $G$ that have Shimura data of Hodge type is given (at least
in theory) in Section 2.3 of [37]. For example, the group $G$ can be of type
$B$, while that is not possible for PEL type Shimura data.

The following result is due to Deligne; it follows from Corollaire 5.7 of
[35], which was already cited as Proposition 1.48.

######  Proposition 2.20.

Every Shimura variety of Hodge type admits a canonical model.

######  Remark 2.21.

There is a general philosophy that Shimura varieties should be moduli spaces
of motives (the conditions that we put on a Shimura datum $(G,h)$ are
basically there to force the $G(\mathbb{R})$-conjugacy class of $h$ to be a
parameter space for Hodge structures); see Section 3 of [84] for more precise
hopes.

For $(G,h)$ of Hodge type, we are a bit closer to that hope: Milne has proved
that $M_{K}(G,h)(\mathbb{C})$ is a moduli space of abelian varieties with
Hodge cycles of a certain type and level structure (see Theorem 3.11 of [84]).
As Hodge cycles on complex abelian varieties are absolute by a theorem of
Deligne (see Theorem 2.11 of Chapter I of [38]), we can also see the action of
$\mathrm{Aut}(\mathbb{C}/F)$ (where $F=F(G,h)$) on this modular
interpretation. So far, this has not allowed people to give a moduli
interpretation of the integral models of the Shimura variety of $(G,h)$, but
it does help with the construction of integral models, that we now discuss.

Let $(G,h)$ be a Shimura datum of Hodge type, and let
$u:G\rightarrow\mathrm{GSp}_{2d}$ be an injective morphism inducing a morphism