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\begin{document}


%\title{Introduction to Shimura varieties}


%\author{Victor Rotger}
%\footnote{?, Partially supported by Ministerio de Ciencia y
%Tecnolog\'{\i}a BFM2000-0627}


\vspace{15cm}

\begin{center}
{\bf SHIMURA VARIETIES AND THEIR CANONICAL MODELS}
\end{center}

\vspace{5cm}

\begin{center}
Victor Rotger\footnote{Partially supported by Ministerio
de Ciencia y Tecnolog\'{\i}a BFM2003-02914 and BFM2003-06768-C02-02.}
\end{center}

\vspace{5cm}

\begin{center}
{\bf  Centre de Recerca Matem\`{a}tica}
\end{center}
%$$ \bullet $$ \null\vfill
%\vskip 5cm
\begin{center}
20 / 23 - 9 - 2005
\end{center}

\newpage
%\maketitle

\section*{Introduction}

These are notes for a short seminar at the Centre de Recerca Matem\`{a}tica in Bellaterra, Barcelona, delivered from September 20th to 23rd 2005 as part of the activities organized during the {\em Special year on Shimura varieties and Arakelov Geometry}. The goal of these notes is offering a rough introduction to hermitian symmetric domains and Shimura varieties attached to a reductive algebraic group over $\Q $. Our main reference for Lie groups, Lie algebras and hermitian symmetric spaces has been the book of S. Helgason \cite{Hel}, while our main reference for algebraic groups and Shimura varieties was the notes by J. S. Milne \cite{Mil}. Comments and warnings of mistakes and misprints are welcome.   

\newpage

\section{Hermitian symmetric domains}

\subsection{Lie groups and Lie algebras}

\begin{definition}
A real Lie group is a $\mathcal C^{\infty}$-real manifold $G$ together with a group structure 
$$G\times G \stackrel{\cdot }{\longrightarrow }G\,,\quad G\stackrel{-1}{\longrightarrow } G$$
defined by $\mathcal C^{\infty}$-morphisms.
\end{definition}

By a Theorem of Lie, a real Lie group always admits a structure of real analytic manifold for which the group law is described by real 
analytic maps. 

Similarly, we define a complex Lie group to be a complex analytical manifold endowed with a group structure described by holomorphic maps.

A Lie subgroup $H\subset G$ of a real Lie group is a real analytic submanifold of $G$ for which the group law of $G$ inherits on $H$ the structure of a Lie group. 


\begin{definition}
Let $k$ be a field of characteristic $\ne 2$. A Lie algebra $\textsf{g}$ over $k$ is a $k$-vector space equipped with
a bracket operation $$[\, ,\, ] : \textsf{g}\times \textsf{g} \longrightarrow \g$$ such that $[X, Y] = -[Y, X]$ and $[X,[Y,Z]]+[Z,[X,Y]]+[Y,[Z,X]]=0$ 
for all $X, Y, Z\in \textsf{g}$.   
\end{definition} 

Throughout these notes, we always mean that $\g $ has finite dimension over $k$. A Lie subalgebra $\h$ of $\g $ is a vector subspace such that $[\h, \h ]\subseteq \h$.

If $G$ is a real (or complex) Lie group, the tangent space $\textsf{g}= \Lie (G) = T_e(G)$ at the identity element $e$ of $G$ is naturally a real (complex) Lie algebra:
$$\begin{matrix}
[\, ,\, ]: & \textsf{g}\times \textsf{g} & \longrightarrow & \g \\
         & (X, Y)& \mapsto & (\mathcal X \mathcal Y - \mathcal Y \mathcal X)_e,    
\end{matrix}$$
where for any two tangent vectors $X, Y \in \textsf{g}$, we let $\mathcal X, \mathcal Y$ denote the unique left $G$-invariant vector fields on $G$ such that 
$\mathcal X_e = X$ and $\mathcal Y_e = Y$.


For a Lie group $G$, let $G_0$ denote the connected component of $e\in G$. Note that $\sf{g} = \Lie (G) = \Lie  (G_0)$. 


\vspace{0.5cm}
{\bf Adjoint representations.} Inner conjugation $$G \lra \Aut (G), \qquad g \mapsto c_g(h)= g h g^{-1}$$ induces the adjoint representation of $G$:
$$\begin{matrix}
\mbox{Ad}: & G & \lra & \GL(\textsf{g}) \\
           & g & \mapsto & d_e(c_g) 
\end{matrix}$$
and the adjoint representation of $\textsf{g}$:
$$\begin{matrix}
\mbox{ad} = d_e(Ad): & \textsf{g} & \lra & \End(\textsf{g}).  
\end{matrix}$$

Let $Z(G) = \{ g\in G : g h = h g \mbox{ for all } h\in G\}$ and $z(\sf{g}) = \{ X \in \sf{g}: [X, Y] = 0 \mbox{ for all } Y \in \sf{g}\}$ 
denote the centers of $G$ and $\sf{g}$, respectively. Each are a Lie subgroup of $G$ and a Lie subalgebra of $\sf{g}$.

The group $Ad(G)= G/Z(G)\subseteq \GL(\textsf{g})$ is called the adjoint group of $G$ and $ad(\textsf{g}) = \sf{g}/z(\sf{g})$ the adjoint algebra of $\sf{g}$. 

\begin{proposition}
Let $G$ be a real Lie group and $\sf{g} = \Lie  (G)$ be its Lie algebra. There is a one-to-one correspondence
$$\{ H\subseteq G \mbox{ connected Lie subgroup }\} \leftrightarrow \{ \sf{h} \subseteq \sf{g} \mbox{ Lie subalgebra }\}.$$
\end{proposition}

Let $\sf{g}$ be a real Lie algebra. The adjoint algebra $ad(\sf{g})$ is naturally a subalgebra of 
$\End(\sf{g}) = \Lie  (\GL (\sf{g}))$ and thus there exists a connected Lie subgroup $\Int \subseteq \GL (\sf{g})$ such that 
$\Lie  (\Int ) = ad (\sf{g})$.

\begin{definition}
A real Lie algebra $\sf{g}$ is compact if $\Int $ is.

A Lie subalgebra $\sf{h}$ of $\sf{g}$ is compactly embedded in $\sf{g}$ if the Lie subgroup $H=\mathrm{Int}_{\sf{g}}(\sf{h})
\subset \GL (\sf{g})$ such that $\Lie  (H) = ad_{\sf{g}} (\sf{h})$ is compact.
\end{definition}

We have $\Int = Ad (G)_0 = (G/Z(G))_0$. Hence, if $G$ is compact then $\sf{g}$ is a compact Lie algebra. But there may exist non compact Lie groups $G$ such that $G/Z(G)$ is compact, and thus so $\sf{g}$ is. 


\vspace{0.5cm}
{\bf Simply connected Lie groups and the equivalence of categories.} Let $\sf{g}$ be a real Lie algebra.  By the above construction we know that if $z(\sf{g}) = \{ 0\}$ then $\sf{g} = ad (\sf{g})$ and we can realize $\sf{g} = \Lie  (\Int )$ as the Lie algebra of a Lie group. What about if $z(\sf{g})\ne \{ 0\}$?

Besides, there might be plenty of connected Lie groups $G$ such that $\Lie (G)=\sf{g}$. Indeed, if $\tilde G \,\lra \, G $ is a topological covering of Lie groups, then $G$ and $\tilde G$ are locally isomorphic and thus they share the same Lie algebra. Uniqueness is obtained when we consider the universal covering of $G$.

\begin{theorem}\label{cat}
There is an equivalence of categories between the category of \begin{center}
{\em Simply connected real Lie groups}
\end{center} and the category of \begin{center}
{\em Real Lie algebras}.
\end{center}
\end{theorem}


\vspace{0.5cm}
{\bf Semisimple Lie algebras.} Let $k$ be a field of characteristic $\ne 2$ and let $\sf{g}$ be a Lie algebra over $k$. 

An {\em ideal} of $\g$ is a vector subspace $\sf{a}\subseteq \g$ such that $[\g, \sf{a}]\subseteq \sf{a}$. The ideal is abelian if $[\sf{a}, \sf{a} ]=\{0\}$. Note that an ideal of $\g$ is also a subalgebra.

\begin{definition}
A Lie algebra $\g$ is simple if it is not abelian and it contains no ideals $\sf{a} \ne \{0\}, \g$. It is semisimple if it contains no abelian ideals $\sf{a} \ne \{ 0\}$.  
\end{definition} 

Define the Killing form on $\g$ to be
$$\begin{matrix}
B_{\g}: & \g \times \g & \lra & k \\
   & (X, Y)     & \mapsto & \Tr (ad(X)\cdot ad(Y)).
\end{matrix}$$

\begin{theorem}[Cartan]
Let $k$  be a subfield of $\C $.  
\begin{itemize}
\item[(i)] $\g$ is semisimple if and only if $B_{\g}$ is nondegenerate.
\item[(ii)] $\g$ is semisimple if and only if $\g \simeq \g_1\oplus ... \oplus \g_r$ is the direct sum of simple Lie algebras $\g_i$.
\end{itemize}
\end{theorem}

In fact, if $\sf{a}$ is an ideal of a semisimple Lie algebra $\g$, then $\g = \sf{a}\oplus \sf{a}^{\bot}$, where $\sf{a}^{\bot} = \{ X\in \g : B_{\g }(X, \sf{a}) = 0\}$. 


\begin{definition}
The Lie algebra $\g$ is reductive if for any ideal $\sf{a}\subseteq \g $ there exists an ideal $\sf{b}\subseteq \g $ such that $\g \simeq \sf{a} \oplus \sf{b}$.
\end{definition} 

\begin{theorem}\label{red}
The Lie algebra $\g$ is reductive if and only if $\g \simeq \sf{s}\oplus \sf{z}$, where $\sf{s}$ is a semisimple Lie algebra and $\sf{z}$ is an abelian Lie algebra. 

If $\g $ is reductive, in fact $s=[\g, \g]$ is semisimple and $\g \simeq s\oplus z(\g )$.
\end{theorem}


\begin{theorem}
Let $k$ be a field of characteristic $0$. Then $\g $ is reductive if and only if all representations $\varrho : \g\lra \End (V)$ on a finite dimensional vector space $V$ over $k$ are semisimple: if $W\subset V$ is $\varrho (\g)$-invariant, there exists a $\varrho (\g)$-invariant vector subspace $\tilde W\subset V$ such that $V = W\oplus \tilde W$.
\end{theorem}


We say that a connected real Lie group $G$ is :
\begin{itemize}
\item a {\bf torus} if it can be embedded as a subgroup of $D_n=\{ \mathrm{diag}(a_1, ..., a_n)\} \subset \GL_n(\C )$, $D_n\simeq \C^*\times \stackrel{n)}{...}\times \C^*$ for some $n$.
\item {\bf simple} if it is not abelian and contains no normal connected Lie subgroups $\ne \{ 1\}, G$.
\item {\bf semisimple} if it contains no normal connected abelian Lie subgroups $\ne \{ 1\}$.
\item {\bf reductive} if the only normal connected abelian Lie subgroups are tori.  
\end{itemize}


{\em Warning. } The definitions of simple and semisimple Lie group are those naturally equivalent for the same definitions on Lie algebras. Note however that the definition of reductive Lie group is more restrictive: the only commutative algebraic subgroups that we allow in a reductive Lie group are tori. For instance, $G=\R $ with the addition law is not a torus and thus also not reductive, although $\Lie (\R ) = \Lie (\R^*) = \R $.

\vspace{0.5cm}
{\em Examples:}
\begin{enumerate}
\item $\mathrm{SL}_n (\R )$, $\g = \mathrm{sl}_n(\R )=\{ X\in \M_n(\R ): \Tr (X) = 0\}$ is simple.


\item $\GL_n(\R )$, $\g = \M_n(\R ) = \langle 1_n\rangle \oplus \mathrm{sl}_n(\R )$ is reductive but not semisimple.


\item $\mathrm{SO}(n)=\{ A \in \mathrm{SL}_n(\R ): A^t A = 1\}$, $\\ \g = \{ X\in \M_n(\R ): X^t + X = 0\}$ is simple.

More generally, for any decomposition $n=p+q$, $p, q\ge 0$: $\\ \mathrm{SO}(p, q) = \{ A\in \SL_{n}(\R ): A^t \begin{pmatrix}
-1_p & 0 \\ 0 & 1_q
\end{pmatrix} A = \begin{pmatrix}
-1_p & 0 \\ 0 & 1_q
\end{pmatrix} \}$ is simple.

When $p, q>0$, $\mathrm{SO}(p, q)$ has two connected components.

\item $SU(p, q) = \{ A \in \SL_n(\C ): \bar A^t \begin{pmatrix}
-1_p & 0 \\ 0 & 1_q
\end{pmatrix} A = \begin{pmatrix}
-1_p & 0 \\ 0 & 1_q
\end{pmatrix}\}$ is simple.

It contains the Lie subgroup $S(U_p\times U_q) = \{ A = \begin{pmatrix}
A_p & 0 \\ 0 & A_q 
\end{pmatrix}: \bar A_p^t A_p = 1_p, \bar A_q^t A_q = 1_q, \det (A_p A_q) = 1 \}$.

\item $\mathrm{SO}^*(2n) = \{ A \in \mathrm{SU}(n, n): A^t \begin{pmatrix}
0 & 1_n \\ 1_n & 0
\end{pmatrix} A = \begin{pmatrix}
0 & 1_n \\ 1_n & 0
\end{pmatrix} \}$ is simple.

\item $\mathrm{Sp}_n(\R ) = \{ A\in \GL_{2 n}(\R ): A^t \begin{pmatrix}
0 & 1_n \\ -1_n & 0
\end{pmatrix} A = \begin{pmatrix}
0 & 1_n \\ -1_n & 0
\end{pmatrix}\}$ is simple.

\item $\mbox{Sp}(n) = \mbox{Sp}_n(\C )\cap \mbox{U}(2n)$ is simple.

\end{enumerate} 


\subsection{Hermitian symmetric manifolds}

A Riemannian real manifold is a $\mathcal C^{\infty}$-manifold $(M, \varrho )$ equipped with a metric
$$\varrho : T(M)\otimes T(M) \lra \, \R ,$$ that is, a positive definite $\mathcal C^{\infty}$-tensor field.

A hermitian complex manifold is a complex analytic manifold $(M, \varrho )$ equipped with a hermitian metric $$\varrho : T(M)\otimes T(M) \lra \, \C ,$$ that is, an analytic positive definite hermitian tensor field: $\C $-linear on the first variable, $g_p(u, v)= \overline {g_p(v, u)}$ for all $u, v\in T_p(M)$ and $g(u, u)>0$ for all $u\in T_p(M)\setminus \{ 0\}$.

Or equivalently, an $\R $-linear, symmetric, positive definite tensor field  $$\varrho _0: T(M)\otimes T(M) \lra \, \R $$ such that $\varrho _{0, p}(i u, i v)= \varrho _{0, p}(u, v)$ for all $u, v\in T_p(M)$. It induces $$\varrho (u, v):= \varrho _0(u, v) + i \varrho _0(u, i v).$$

\begin{definition}
A hermitian symmetric manifold is a connected hermitian manifold $(M, \varrho )$ such that:
\begin{itemize}
\item[(i)] $M$ is homogenous: the group $\Aut (M, \varrho )$ of holomorphic isometries acts transitively on $M$.
\item[(ii)] $M$ is symmetric: for all $p\in M$ there exists an involution $s_p\in \Aut(M, \varrho )$, $s_p^2=1$, such that $p$ is an isolated fixed point of $s_p$. 
\end{itemize}
\end{definition}


Similarly one defines Riemannian symmetric manifolds. An important remark is that for each point $p\in M$, the symmetry $s_p$ is always unique. Locally, $s_p$ is the geodesic involution $s_p(\gamma (t)) = \gamma(-t)$, where $\gamma $ denotes any geodesic on $M$ with $\gamma (0) = p$.

\vspace{1cm}
{\em Examples.} 
\begin{itemize}
\item[(a)] $M = \cH_1 = \{ z\in \C : \mbox{ Im }(z)>0\}$ with the hermitian metric $\varrho _0=\frac{dx dy}{y^2}$. The group $\mathrm{PSL}_2(\R )$ acts on $\cH_1$ by Moebius transformations: $$z\mapsto \begin{pmatrix}
a&b\\c&d
\end{pmatrix}\cdot z = \frac{a z + b}{c z + d}.$$ The group $\mathrm{PSL}_2(\R )$ acts transitively by holomorphic isometries on $\cH_1$ and a symmetry at $i\in \cH_1$ is given by the matrix $\begin{pmatrix}
0&-1\\1&0
\end{pmatrix}$.

\item[(b)] $M = \PP^1(\C )\subset \R^3$ with the hermitian metric induced by $\varrho _0 = dx dy dz$. The group $\mathrm{SO}_3(\R )$ acts by rotations transitively on $\PP^1(\C )$. These are holomorphic isometries and for each $p\in \PP^1(\C )$, there is an obvious rotation which leaves $p$ and $-p$ fixed.

\item[(c)] $M=\C /\Lambda $ where $\Lambda =\Z e_1 + \Z e_2 \subset \C $ is a lattice, together with the metric $\varrho _0 = dx dy$. The group $\C/\Lambda $ itself acts transitively by translations on $M$. For any point $p\in M$, the involution $s_p: q \mapsto 2 p - q$ is a symmetry which has exactly four fixed points, $p$ being one of them.

\end{itemize}

For a point $p$ in a $\mathcal C^{\infty }$-manifold $M$ and vector subspace $E\subseteq T_p(M)$, $\dim (E)=2$, recall Gauss' sectional curvature $K_p(E)$ of $p$ along $E$. It may be defined as
\begin{itemize}
\item $K_p(E)= - g_p(R_p(X, Y) X, Y)$ where $R$ is the $(3,1)$-curvature tensor field attached to the Riemannian connection $\nabla $ on $M$ and $\{ X, Y \}$ is an orthonormal basis of $E$.

\item Let $E\cap M$ be the local submanifold of $M$ around $p$ obtained by exponentiation of geodesics $\gamma $ such that $\gamma(0)=p$ and $\dot{\gamma }(0) \in E$. Let $C$ and $c$ be the maximum and minimum of the curvatures of the signed curves on $E\cap M$ obtained by cutting the surface $E\cap M$ with planes through a normal line at $p$ and define $K_p(E) = C\cdot c$. 

The curvature of a curve is $1/R$, where $R$ is the radius of the circle that best approximates the curve. We give the sign $+$ or $-$ to the curvature of the curve depending whether the curve bends towards the normal line or not.   
\end{itemize}


\begin{definition}
Let $(M, \varrho )$ be a hermitian symmetric manifold. If for all points $p\in M$ and all vector subspaces $E\subseteq T_p(M)$, $\dim (E)=2$, we have:
\begin{itemize}
\item[(a)] $K_p(E)<0$, we say that $(M, \varrho )$ is of noncompact type.
\item[(b)] $K_p(E)>0$, we say that $(M, \varrho )$ is of compact type.
\item[(c)] $K_p(E)=0$, we say that $(M, \varrho )$ is of euclidean type.
\end{itemize} 
\end{definition}

\begin{theorem}
Let $(M, \varrho )$ be a hermitian symmetric manifold. Then $$(M, \varrho )\simeq (M^-, \varrho^-)\times (M^+, \varrho^+)\times (M^0, \varrho^0)$$ is isometric to the product of a hermitian symmetric space of noncompact type, compact type and euclidean type.
\end{theorem}

The group of holomorphic isometries $G=\Aut (M, \varrho )$ of a hermitian symmetric manifold is equipped with the compact-open topology for which a basis of open subsets is given by
$$W(C, U) = \{ g\in G: g(C)\subset U\},$$ where $C\subset M$ is a compact and $U\subset M$ is open.

With this topology $G$ admits a unique structure of real analytic Lie group and for each point $p\in M$, the isotropy group 
$$K_p = \{ g \in G: g(p) = p\}$$ is compact.

Moreover, if $(M, \varrho )$ is of noncompact type or compact type, there exists an algebraic group $\mathcal G\subset \GL (\g)$ over $\R $ such that $$\mathcal G(\R )_0 = G_0 = \mathrm{Hol} (M)_0 = \Aut (M^{\R }, g^{\R })_0.$$

\vspace{0.5cm}
\begin{example}
For $M=\cH_1$, $g=\frac{dx dy}{y^2}$, we have $\Aut (M, \varrho ) = \mathrm{Hol }(M) = \mathrm{PSL}_2(\R )$, which is the set of real points of the algebraic group $\mathrm{PSL}_2$. However, $\Aut (M^{\R }, g^{\R }) = \mathrm{PSL}_2(\R ) \cup \mathrm{PSL}_2(\R )\cdot (z\mapsto \bar z ^{-1})$.    
\end{example} 

We focus now on hermitian symmetric manifolds of noncompact type. 

If $D\subset \C^n$ is a bounded open connected subset, there is canonical hermitian metric $g_{Bgm}$ on $D$, the so-called Bergman's metric, which has negative sectional curvatures. If $\mathrm{Hol}(D)$ acts transitively on $D$ and each point $p\in D$ admits a symmetry (in fact, if one then all), then it turns $(D, g_{Bgm})$ into a hermitian symmetric manifold of noncompact type. Conversely, any such space is isometric to $(D, g_{Bgm})$ for some bounded domain of $\C ^n$. 

Accordingly, hermitian symmetric manifolds of noncompact type are often called {\em hermitian symmetric domains}.


Let $(M, \varrho )$ be a hermitian symmetric domain, $G=\Aut (M, \varrho )$ be the Lie group of holomorphic isometries of $M$, $G_0$ be the connected component of $e\in G $ and $\g = \Lie (G)$. 

Fix a point $p\in M$ and let $K=K_p$ be the (compact) isotropy group of $M$ at $p$. 

The automorphism $$\begin{matrix}
\sigma : & G & \lra & G \\ & g & \mapsto & s_p g s_p
\end{matrix}$$ is an involution on $G$ and we let $\g = \g^+ \oplus \g ^-$ be the decomposition of $\g $ into $\pm 1$-eigenspaces with respect to $d_e(\sigma )$.

With the above notations we have the following fundamental result.

\begin{theorem}\label{main} Let $(M, \varrho )$ be a hermitian symmetric domain. Then
\begin{itemize}
\item[(i)] $M$ is simply connected and $G$ is semisimple and noncompact.

\item[(ii)] $Z(G)\subset K$ is a finite group, and there is a diffeomorphism
$$\begin{matrix}
\pi: & G_0/K & \stackrel{\simeq }{\lra } & M \\
    &   g    &  \mapsto & g(p)
\end{matrix}$$
\item[(iii)] $(K_{\sigma })_0 \subseteq K \subseteq K_{\sigma }:= \{ g\in G: \sigma (g) = g\}$ and $\Lie (K) = \g^+$.

\item[(iv)] $d_e \pi : \g^- \stackrel{\simeq }{\lra }\, T_p(M)$ and $\mbox{Id }\cdot \mbox{ exp }: K\times \g^- \stackrel{\simeq }{\lra }\, G_0$.

\item[(v)] The complex structure $J_p\in \End (T_p(M))$ belongs to $z(\Lie K)$. In particular, $K$ has non discrete center $Z(K)$. 

\end{itemize} 
\end{theorem}

In item (v) above, we regard $\Lie (K)$ as a subalgebra of $\End (T_p(M))$ through the representation $$\begin{matrix}
K & \lra & \GL (T_p(M)) \\ k & \mapsto & d_p(g K \mapsto k g K).
\end{matrix}$$

We say that hermitian symmetric domain $(M, \varrho )$ is {\em irreducible} if it is not isometric to the product of non zero hermitian symmetric domains. If $(M, \varrho )$ is a hermitian symmetric domain, then $$(M, \varrho )\,\simeq \,(M_1, \varrho_1)\times ... (M_r, \varrho_r),$$ where $(M_i, \varrho_i)$ are irreducible. 

If $(M, \varrho )$ is irreducible, then $\g = \Lie (G)$ and $\g \otimes \C $ are simple Lie algebras and $\Lie (K)$ is a maximal proper subalgebra of $\g $. 

\begin{remark}
Although a real Lie algebra $\g$ is semisimple if and only $\g\otimes \C $ is, there exist simple real Lie algebras $\g$ such that $\g \otimes \C $ are not. Indeed, let $\sf{c} /\C $ be a simple complex Lie algebra and let $\g = \sf{c}^{\R }$ be $\sf{c} $ regarded as a Lie algebra. Then $\g $ is simple but $\g \otimes \C $ is not.  
\end{remark}

\newpage 
\subsection{Cartan's classification}

\vspace{0.5cm}
{\bf Cartan involutions.}

\begin{definition}
Let $\g $ be a semisimple real Lie algebra. A Cartan involution on $\g $ is an automorphism $s\in \End (\g )$ such that $s\ne \mbox{ Id }$, $s^2 = \mbox{ Id }$ and the $+1$-eigenspace $\g^+ \subseteq \g $ is a compactly embedded subalgebra.
\end{definition}


\begin{definition}
We call $(\g , s)$ a symmetric Lie algebra, and we say that it is of compact or noncompact type depending whether $\g $ is compact or not. We say that $(\g, s)$ is an irreducible hermitian noncompact symmetric Lie algebra if 
\begin{itemize}
\item $\g $ is non compact, $\g $ and $\g \otimes \C $ are simple
\item $\g^+$ is a maximal proper subalgebra of $\g $ and $z(\g^+)\ne \{0\}$.
\end{itemize} 
\end{definition}


Next proposition can be found in \cite[p. 292, 303, 385]{Knapp}.

\begin{proposition} Let $\g $ be a semisimple real Lie algebra.
The Killing form $B_{\g}$ on $\g $ is $Ad (G)$-invariant:
$$B_{\g }( Ad(g) X, Ad(g) Y) = B_{\g }(X, Y), \quad \quad \mbox{ for all } g\in G, X, Y\in \g.$$
Moreover, it is negative definite on $\g^+$ and positive definite on $\g^-$.
\end{proposition}

\vspace{0.5cm}
{\bf Symmetric domains versus symmetric Lie algebras.}

If $(M, \varrho )$ is an irreducible hermitian symmetric domain, then Theorem \ref{main} implies that $(\g, d_e \sigma )$ is an irreducible noncompact hermitian symmetric Lie algebra, where $\g = \Lie (G)$, $G=\Aut (M, \varrho )$ and $\sigma : G\,\lra \,G, \,g \mapsto s_{p} g s_p$.

Conversely, let $(\g , s)$ be an irreducible noncompact hermitian symmetric Lie algebra. Let $\tilde G$ be the simply connected Lie group such that $\Lie (\tilde G)=\g $ (which can be obtained as the universal covering of $\Int $) and let $\sigma \in \Aut (\tilde G)$ be an involution of $\tilde G$ such that $d_e \sigma = s$. The existence of $\sigma $ is guaranteed by Theorem \ref{cat}. 

Let $\tilde K$ be the connected component of $e$ in $K_{\sigma } = \{ g\in \tilde G: \sigma (g) = g\}$. Since $\Lie (\tilde K) = \g^+$, it follows from the prescribed properties of $(\g , s)$ that $\tilde K$ is a maximal connected proper Lie subgroup of $\tilde G$. Moreover, $Ad_{\tilde G}(\tilde K)\subset \GL (\g)$ is compact.

The coset space $M = \tilde G/\tilde K$ is naturally a real analytic manifold. Let $\pi : \,\tilde G \,\lra \,M$ be the projection map and let $p_0=\pi (e)\in M$ be the base point. The Killing form $B_{\g}$ is positive definite on $\g^-\simeq T_{p_0}(M)$ and it is $\tilde G$-invariant: 
$$B_{\g }(X, Y) = B_{\g}(d_e Ad(g) X, d_e Ad(g) Y) \quad \mbox{ for any } g\in \tilde G.$$ By means of the transitive action of $\tilde G$ on $M$, it can be extended to a metric $\varrho $ on $M$ which  turns $(M, \varrho )$ into a Riemannian manifold on which $\tilde G$ acts by isometries.

The action of $\tilde G$ on $M$ is obviously transitive and for each $p\in M$ there is a symmetry $s_p\in \tilde G$ which leaves $p$ fixed. Indeed, the symmetry at $p_0$ is $s_{p_0} : g \tilde K \mapsto \sigma (g) \tilde K$ (and this is again an isometry because $B_{\g}$ is invariant under automorphisms of $\tilde G$).

Recall now that $Z(\tilde K)$ is a non discrete abelian (compact) Lie subgroup of $\tilde K$. Hence $Z(\tilde K)_0 = S^1\times ... \times S^1$, where $S^1 = \{ z\in \C : |z|=1\}$. In fact, we have the following lemma (see \cite[Thm 1.9]{Mil}).

\begin{lemma}
There exists a homomorphism 
$$u: S^1 \lra \, Z(\tilde K)$$ such that the isometry
$$\begin{matrix}
c_{u(-1)}: & \tilde G/\tilde K & \lra & \tilde G/\tilde K  & \\  & g \tilde K & \mapsto & u(-1) g \tilde K &( = u(-1) g u(-1)^{-1}\tilde K)
\end{matrix}$$ induced by conjugation by $u(-1)$ fixes $p_0$ and satisfies $$d_{p_0} c_{u(-1)} =  -\mbox{ Id }\in \End \,T_{p_0} (M).$$ 
\end{lemma} 

Since the symmetry $s_{p_0}$ satisfies the same properties, by uniqueness we have $c_{u(-1)} = s_{p_0}$ and thus also $\mbox{ad}(u(-1)) = s$. 

Also, $J_0 = d_e c_{u(i)} \in \End \,T_{p_0} (M)$ is an endomorphism such that $J_0^2 = -\mbox{ Id}$ and $d_e (c_k) J_0 = J_0 d_e(c_k)$ for all $k\in \tilde K$. 

It follows that there can be constructed a unique $\tilde G$-invariant almost complex structure $J$ on $M$ such that $J_{p_0} = J_0$ for which $\varrho $ is hermitian. In fact, $J$ is integrable and $(M, \varrho )$ is therefore an irreducible hermitian symmetric domain.  

\begin{remark}
And now the lemma can be reformulated to ensure the stronger statement that $d_{p_0} c_{u(z)} =  z\cdot \mbox{ Id }\in \End \,T_{p_0} (M)$ for all $z\in S^1$. It implies that in the representation $\mbox{Ad} (u)\otimes \C : S^1\lra \GL (\g \otimes \C )$, the only characters $\xi \in \mathrm{Hom} (S^1, \C^*)\simeq \Z $ that occur are $1$, $z$ and $z^{-1}$.
\end{remark}

We have thus established a one-to-one correspondence between 
\begin{center}
{\em Irreducible hermitian symmetric domains} 
\end{center} 
and
\begin{center}
{\em Irreducible noncompact hermitian symmetric Lie algebras}.
\end{center}

If we start with $(M, \varrho )$, Theorem \ref{main} produces $G=\Aut (M, \varrho )_0$, $K = K_p$ and $\sigma \in \Aut (G)$ so that $(\g = \Lie (G), s=d_e(\sigma ))$ is an irreducible noncompact hermitian symmetric Lie algebra. Starting now with the resulting pair $(\g, s)$, we have attached to it a simply connected Lie group $\tilde G$ and a compact subgroup $\tilde K$. The automorphism group $G$ of $M$ does not need to be simply connected, but we do have that $\tilde G/\tilde K$ is the universal covering of $G/K$ (see \cite[p. 178]{Hel}). Since $M$ is simply connected by Theorem \ref{main}, we deduce that $\tilde G/\tilde K\simeq G/K = M$.

\vspace{0.5cm}
{\bf Classification of simple Lie algebras.}

For a real symmetric Lie algebra $(\g_0, s)$, let $\g_0 = \g_0^+ \oplus \g_0^-$ be the decomposition into $\pm 1$-eigenspaces with respect to $s$. Define the dual symmetric Lie algebra $(\g_0^*, s^*)$ to be the subalgebra $\g_0^* = \g_0^+ \oplus i \g_0^-$ of the complex Lie algebra $\g := \g_0 \otimes \C $, and $s^*: X^+ + i X^- \mapsto X^+ - i X^-$.

\begin{theorem}
\begin{itemize}
\item[(i)] If $\g /\C $ is a semisimple Lie algebra, there exists a compact real Lie algebra $\g_0$ such that $\g = \g_0 \otimes \C $. Any two compact real forms of $\g $ are isomorphic and there is a one-to-one correspondence between $$\{ \mbox{ Semisimple Lie algebras } \g /\C \} /\simeq $$ and  $$\{ \mbox{ Compact Lie algebras } \g_0 /\R \} /\simeq $$ which preserves simplicity.

\item[(ii)] If $(\g_0, s)$ is a symmetric simple Lie algebra of compact type, then its dual symmetric Lie algebra $(\g_0^*, s^*)$ is of noncompact type, and conversely.   
\end{itemize}
\end{theorem}

Remark that in general it is not true that $\g_0$ is simple over $\R $ if and only if $\g = \g_0\otimes \C $ is simple over $\C $. This holds however true for compact real Lie algebras (see \cite[p. 308]{Hel}). 

Combining the two statements of the above theorem we obtain that in order to classify symmetric Lie algebras of noncompact type and irreducible hermitian symmetric domains, we can proceed as follows:

\begin{enumerate}
\item Classify the isomorphism classes of simple Lie algebras $\g $ over $\C $.
\item Find a compact real form $\g_0$ for them and classify all possible Cartan involutions $s$ on them: $(\g_0^*, s^*)$ will recover all symmetric Lie algebras of noncompact type up to isomorphism.
\item Compute $z(\g_0^{* +})$. 
\end{enumerate}

\begin{theorem}[Cartan]
Let $(M, g)$ be an irreducible hermitian symmetric domain. Then $M$ is isometric to either
$$\begin{array}{|c|c|c|c|}
\mbox{Label}    & \mbox{Compact form} & \mbox{Domain} & \mbox{dim} \\
    \hline \\
A_{III} & \mbox{SU}(n)     & \mbox{SU}(p, q)/\mbox{S}(U_p\times U_q) & 2 p q \\
\hline \\
BD_{I}(q=2) & \mbox{SO} (n+2)  & \mbox{SO}_0(n,2)/\mbox{SO}(n)\times \mbox{SO}(2) & 2 n \\
\hline \\
C_{I} & \mbox{Sp}(n)    &  \mbox{Sp}_n(\R )/U(n)   & n (n+1) \\
\hline \\
D_{III}  & \mbox{SO(2n)} & \mbox{SO}^*(2 n)/U(n) & n(n-1) \\
\end{array}
$$
or to the exceptional cases $EIII$ of dimension $32$ or $EVII$ of dimension $54$.
\end{theorem}

\begin{itemize}
\item The compact real simple algebra of SO$(n)$ admits two Cartan involutions when $n$ is even. These give raise to two different noncompact forms. A similar phenomenon happens for the compact group $SU(n)$: for each decomposition $n=p+q$ with $p$, $q>0$ there is Cartan involution on $SU(n)$ which gives raise to each of the non-compact forms of the third column.

\item There are several coincidences among the above Lie groups. More precisely: $A_{III}(p=q=1)\simeq C_I(n=1)$, $BD_I(p=3, q=2)\simeq C_I(n=2)$, $A_{III}(p=q=2)\simeq BD_I(p=4, q=2)$, $A_{III}(p=3, q=1)\simeq D_{III}(n=3)$ and $BD_I(p=6, q=2)\simeq D_{III}(n=4)$.

\item The Siegel space is $\mathcal H_n = \{ Z\in \M_n(\C ): Z=Z^t, \mathrm{Im}(Z)>0\} = \mathrm{Sp}_n(\R )/\mathrm{U}(n)$. 
\end{itemize}


\newpage

\section{Locally symmetric varieties}

The aim of this section is considering quotients $D(\Gamma ):=\Gamma \backslash D$ of hermitian symmetric domains $D$ by discrete subgroups $\Gamma \subset \Aut (D)_0$ and studying under what conditions the coset space $D(\Gamma )$ is the set of complex points of an algebraic variety defined over a number field. 

The starting point is the following proposition.

\begin{proposition}\label{complexmanifold}
Let $D$ be a hermitian symmetric domain and let $G = \Aut (D)$ be the group of holomorphic isometries of $D$. Let $\Gamma \subset G_0$ be a discrete {\em torsion-free} subgroup. There is a unique complex analytic structure on $D(\Gamma )$ for which $\pi : D \twoheadrightarrow D(\Gamma )$ is a local isomorphism.

A map of complex analytic varieties $\DG \,\lra \,V$ is analytic if and only if the composition $D\twoheadrightarrow \DG \,\lra \,V$ is.
\end{proposition}  

{\em Proof.} With the quotient topology, $D(\Gamma )$ is a separated space\footnote{For any $p, q\in D$ not in the same orbit under $\Gamma $, we can find open subsets $p\in U_p$, $q\in U_q\subset D$ such that $\gamma U_p\cap U_q = \emptyset $ for all $\gamma \in \Gamma $.}.

For any $p\in D$, $\Gamma_p = \{ \gamma \in \Gamma: \gamma \cdot p = p\} \subset K_p$ is a discrete subgroup of a compact group, hence finite. If $\Gamma $ is torsion-free, $\Gamma_p=\{ 1\}$ for all $p\in D$. There exists $p\in U_p$ such that $\gamma U_p\cap U_p=\emptyset $ for all $\gamma \in \Gamma \setminus \{1\}$ so that $\pi_{|U_p}:\, U_p\,\lra \,\pi(U_p)$ is a homeomorphism, producing a complex analytic atlas on $D(\Gamma )$. $\Box $

Assume $\Gamma $ is torsion-free. Since $D$ is simply connected, it is the universal covering of the complex manifold $\DG $. For any $p\in D$ we have
$$\begin{matrix}
\Gamma & \stackrel{\simeq }\lra & \pi_1(\DG , \pi(p)) \\
g & \mapsto & [ \pi (c)]
\end{matrix}$$
where $c$ is any path on $D$ joining $p$ and $g\cdot p$.


\subsection{Algebraic groups}

Let $k$ denote a field of characteristic $0$ and $\bar k$ a fixed algebraic closure of $k$.

\begin{definition}
An algebraic group over a field $k$ is an algebraic variety $G$ over $k$ together with a group structure 
$$G\times G \stackrel{\cdot }{\longrightarrow }G\,,\quad G\stackrel{-1}{\longrightarrow } G$$
defined by algebraic morphisms defined over $k$.
\end{definition}

Examples of algebraic groups over a field $k$ are
\begin{itemize}
\item $\mathbb{G}_m = \mathrm{Spec} k[X, Y]/(X Y - 1)$ is the multiplicative group such that for any $k$-algebra $A$, $\mathbb{G}_m(A) = A^*$.
\item $\mathbb{G}_a = \mathrm{Spec} k[X]$ is the additive group such that for any $k$-algebra $A$, $\mathbb{G}_m(A) = A$.
\item $\M_n$, $\GL_n$, $\SL_n$, $\mathrm{SO}_n$, $\mathrm{Sp}_n$, ... 
\item Elliptic curves and abelian varieties over $k$.
\end{itemize}

The Lie algebra of $G$ is $\g = \Lie (G) = T_{e}(G)$, and it is a Lie algebra over $k$.

All examples of algebraic groups we will consider are affine. We will not consider abelian varieties, for instance. 
%By Yoneda's lemma, giving an affine algebraic group $G$ over $k$ is equivalent to giving a $k$-finitely generated algebra 
 
The notion of torus, semisimple group and reductive group are similarly defined as in the previous section for Lie groups. That is:

\begin{definition} A connected algebraic group $G$ over $k$ is 
\begin{itemize}
\item a {\bf torus} if $G\otimes \bar k\simeq \G_m\times \stackrel{n)}{...} \times \G_m$ for some $n$. The minimal field extension $K/k$ for which $G\otimes K \simeq \G_m\times ... \times \G_m$ is called the splitting field of $G$.
\item {\bf semisimple} if it contains no smooth connected normal commutative algebraic subgroups $\ne \{ 1\}$.
\item {\bf reductive} if it contains no smooth connected normal commutative algebraic subgroups other than tori.

\end{itemize}
\end{definition}

Note for instance that abelian varieties or $\G_a$ are not reductive algebraic groups.

A typical example of a non split torus is constructed as follows. Let $K/k$ be a finite Galois extension and $T = \mathrm{Res}_{K/k}(\mathbb{G}_m)$. This is an algebraic group over $k$ of dimension $n=[K:k]$ characterized by $T(A)= (A\otimes_k K)^*$ for any $k$-algebra $A$. In particular $T(k)=K^*$ and $T(K)=(K^*)^n$. It is split over $K$. 

For a reductive algebraic group $G$ over $k$, let $T$ stand for the largest commutative quotient of $G$. Since $T$ is connected, it is a torus. We define the derived group $G^{der}$ of $G$ to be the kernel of $\nu $. The Lie algebra of $G^{der}$ is $\Lie (G^{der}) = [\g, \g]$ and according to Theorem \ref{red} it is a semisimple group.

Let $Z$ denote the centre of $G$. Since $\g = [\g, \g ]\oplus z(\g )$, we have that $\Lie (Z) = \Lie (T)$. In fact $Z/Z'\simeq T$, where $Z' := Z\cap G^{der}$ is a finite group. We thus have exact sequences
$$\begin{matrix}
1 & \lra & G^{der} & \lra & G &\stackrel{\nu }\lra & T & \lra & 1 \\ 
1 & \lra & Z & \lra & G &\stackrel{ad}\lra & G^{ad} & \lra & 1 \\
1 & \lra & Z' & \lra & Z &\lra & T & \lra & 1. 
\end{matrix}$$

When $G=\GL_n$, these are
$$\begin{matrix}
1 & \lra & \SL_n & \lra & \GL_n &\stackrel{\det }\lra & \mathbb{G}_m & \lra & 1 \\ 
1 & \lra & \mathbb{G}_m & \lra & \GL_n &\stackrel{ad}\lra & \mathrm{PGL}_n & \lra & 1 \\
1 & \lra & \mu_n & \lra & \mathbb{G}_m &\stackrel{x\mapsto x^n}\lra & \mathbb{G}_m & \lra & 1. 
\end{matrix}$$

\subsection{Arithmetic and congruence groups}

Two subgroups $S_1, S_2\subset S$ of a group $S$ are {\em commensurable} if $S_1\cap S_2$ has finite index both in $S_1$ and $S_2$.

\begin{definition}
Let $G$ be an algebraic group over $\Q $. A subgroup $\Gamma \subset G(\Q )$ is {\em arithmetic} if it is commensurable with $G(\Q )\cap \GL_n(\Z )$ for some embedding $G\hookrightarrow \GL_n$.

A {\em congruence subgroup} of $G(\Q )$ is a subgroup $\Gamma \subset G(\Q )$ which for some embedding $G\hookrightarrow \GL_n$ contains 
$$
\Gamma (N) =G(\Q )\cap \{ g\in \GL_n(\Z ): g\equiv Id_n \mbox{ mod } N\}$$ as a subgroup of finite index. 
\end{definition}

For a reductive group there always exist an embedding $G\hra \GL_n$ and it can be shown that the above definitions do not depend on the chosen embedding $G\hookrightarrow \GL_n$.\footnote{Congruence groups are arithmetic and the {\em classical congruence problem} asks whether any arithmetic group $\Gamma \subset G(\Q )$ is congruence. By definition we know that $\Gamma $ is commensurable with $\Gamma (1)$ but we do not know whether $\Gamma (N)\subset \Gamma $ for some $N$. The answer is {\em yes} if $G$ is simply connected (that is, if any isogeny $G'\ra G$ from a connected algebraic group $G'$ is the identity) and $G\not \simeq \SL_2$. Otherwise, $\SL_2$ and non simply connected reductive groups have many non-congruence arithmetic subgroups. See \cite{PlRa}.}

Let $$\A_f = \hat \prod \Q_{\ell } = \{ (a_{\ell }): a_{\ell }\in \Q_{\ell }, a_{\ell }\in \Z_{\ell } \mbox{ for almost all }\ell \}$$ be the ring of finite adeles of $\Q $. It is a topological ring when we regard it as a subring of $\prod \Q_{\ell }$ with the product topology, in which a basis of (compact) open subsets of $0$ are $\{ K(N)= \prod K_{\ell }(N)\}_{N\ge 1}$, where $K_{\ell }(N) = \Z_{\ell }$ if $\ell \nmid N$; $K_{\ell }(N)= \ell ^{r_{\ell }}\Z_{\ell }$ if $r_{\ell } = \mbox{ord}_{\ell }(N)\ge 1$.

For an algebraic group $G$ over $\Q $, let $G(\A_f) = \hat \prod G(\Q_{\ell })$.\footnote{In order to talk about $G(\Z_{\ell })$, this definition implies the choice of a model of $G$ over $\Z $. However, any two such models will become isomorphic over $\Z [\frac{1}d]$ for some $d\ge 1$. Since there are finitely many primes $\ell \mid d$, there is no ambiguity in our definition.} If $G\hra \GL_n$ is an embedding, a basis of (compact) open neighbourhoods of $1$ is given by $K(N) = \prod K_{\ell }$ where $$K_{\ell } = \begin{cases}
G(\Z_{\ell }) & \mbox{ if } \ell \nmid N \\
\{ g\in G(\Z_{\ell }): g\equiv Id_n \mbox{ mod } \ell^{r_{\ell }}\} & \mbox{ if } r_{\ell } = \mbox{ord}_{\ell }(N)\ge 1.
\end{cases}$$

The topology does not depend of the choice of the embedding.\footnote{It does always exist for reductive groups.}
For instance $\G_a \simeq \begin{pmatrix}
1&*\\0&1
\end{pmatrix}\subset \GL_2$ and $\G_m=\GL_1$. 

\begin{proposition}\label{adeles}
Let $G$ be a reductive group over $\Q $. For any compact open subgroup $K\subset G(\A_f)$, $K\cap G(\Q )$ is a congruence subgroup of $G(\Q )$.\footnote{And every congruence subgroup arises in this way.}
\end{proposition}

{\em Proof. } Let $G\hookrightarrow \GL_n$ be an embedding. Then $K(N)$ is a compact open subgroup of $G(\A_f)$ and $$K(N)\cap G(\Q ) = \Gamma (N).$$ If $K$ is a compact open subgroup of $G(\A_f)$, there exists $N\ge 1$ such that $K\supseteq K(N)$ and $\Gamma := K\cap G(\Q )\supseteq \Gamma (N)$. Thus $\Gamma /\Gamma(N)\subset K/K(N)$ is a discrete subgroup of a compact group: the index of $\Gamma (N)$ in $\Gamma $ is finite and $\Gamma $ is congruence. $\Box $

\begin{definition}\label{arithmetic}
If $G$ is an arbitrary connected real Lie group, we still define {\em arithmetic subgroups} of $G$ as follows. A subgroup $\Gamma \subset G$ is {\em arithmetic} if there exists
\begin{itemize}
\item An algebraic group $\mathcal G$ over $\Q $.
\item An arithmetic subgroup $\tilde \Gamma \subset \mathcal G (\Q )$
\item A surjective morphism $\pi : \mathcal G (\R )_0 \lra \,G$ with compact kernel 
\end{itemize}
such that $\pi (\tilde \Gamma ) = \Gamma $.
\end{definition}

%\begin{proposition}\footnote{We may not state this on the blackboard, since it is somehow contained in Baily-Borel's theorem.}
%If $G$ is a connected semisimple real Lie group, every arithmetic subgroup $\Gamma \subset G$ is discrete, of finite covolume\footnote{We will not %define here what do we precisely mean by finite covolume. It is a necessary condition for $D(\Gamma )$ being an algebraic variety} and it contains a %torsion-free subgroup of finite index.
%\end{proposition}


\subsection{The theorem of Baily-Borel}


Let $D$ be a hermitian symmetric domain and $G = \Aut (D)$ be the (semisimple) real Lie group of holomorphic isometries of $D$. Let $\Gamma \subset G_0$ be an arithmetic torsion-free subgroup. This means that there exists an algebraic group $\mathcal G$ over $\Q $, an arithmetic subgroup $\tilde \Gamma \subset \mathcal G (\Q )$ and a surjective morphism $\pi : \mathcal G (\R )_0 \lra \,G$ with compact kernel 
such that $\pi (\tilde \Gamma ) = \Gamma $.

\begin{theorem}\label{BailyBorel} 

\begin{itemize}
\item[(i)][Baily-Borel] Then $D(\Gamma )$ has a canonical realization as a smooth Zariski-open subset of a projective algebraic variety $\DG^*$. If $\mathcal G(\Q )$ contains no unipotent elements\footnote{An unipotent element is an element $\gamma \in \mathcal G(\Q )$ such that $\varrho (\gamma )-1$ is nilpotent for all representations $\varrho : \mathcal G\,\hra \,\GL (V)$ of $\mathcal G$.}, then $\DG$ is compact. 

\item[(ii)] [Borel] Let $V$ be a nonsingular quasi-projective variety over $\C $. Then every holomorphic map of complex analytic manifolds $$f: V(\C )\,\lra\,\DG(\C )$$ is regular algebraic.
\end{itemize}
\end{theorem}

\begin{itemize}

\item For $D=\cH_1$ the proof of (i) works as follows. 
\begin{enumerate}
\item Let $D^*=\cH_1 \cup \PP^1(\Q )$ with a suitable topology.
\item Let $\Gamma \subset \SL_2(\Q )$ be an arithmetic subgroup (commensurable with $\SL_2(\Z )$). It acts on $\cH_1^*$ by Moebius transformations on $\cH_1$ and by linear projective transformations on $\PP^1(\Q )$ so that $\DG^*=\Gamma \backslash \cH_1^*$ is a compact complex surface.
\item Modular forms (i.e. regular differentials) of large weight enough produce an embedding of $\DG^*$ into a projective space.
\item Chow's theorem asserts that $\DG^*$ is naturally a projective variety and $\DG = \DG^* \setminus (\Gamma \backslash\PP^1(\Q ))$, the complementary of a finite set of points.
\end{enumerate}     

\item For arbitrary $D$, the proof of (i) follows the same pattern.
\begin{enumerate}
\item $D^* = D \bigcup \, \cup B_i$, where $B_i$  are so-called rational boundary components, endowed with the Satake topology.
\item Automorphic forms of large weight embed $\DG^* = \Gamma \backslash D^*$ into a projective space, so that $\DG \subset \DG^*$ is a Zariski-open subvariety. 
\end{enumerate}

\begin{remark}\label{codim2}
\begin{enumerate}
\item If $\mathrm{PGL}_2$ is not a quotient of the algebraic group $\mathcal G$ over $\Q $, then the components of $\DG^*\setminus \DG $ have codimension $\ge 2$.\footnote{The example $\SL_2(\Z )\backslash \cH_1$ is explained as follows: the boundary is a nonempty set of finite points, thus of codimension $1$. We already expected this since $\mathrm{SL}_2(\Z )$ contains unipotent elements. But $\mathrm{PGL}_2$ is not a quotient of $\mathrm{SL}_2$ over $\Q $! That's true, but $\mathrm{PGL}_2(\R )_0 = \mathrm{SL}_2(\R )$ and we can also choose $\mathcal G = \mathrm{PGL}_2$ instead of $\SL_2$. With this choice we also have $\cH_1 = \mathrm{PGL}_2(\R )_0/K$ and now the remark makes sense. Since the natural inclusion $\cG = \mathrm{PSL}_2 \,\lra \,\mathrm{PGL}_2$ is an isomorphism on the connected components of $1$ of the real points, $\SL_2(\Z )$ is an arithmetic subgroup of $\mathrm{PGL}_2(\Q )$.} 

\item If $\Gamma $ is an arithmetic group with torsion, Proposition \ref{complexmanifold} does not apply and $\DG $ is even not a complex manifold. However, there exists a subgroup $\Gamma_0\subset \Gamma $ of finite index in $\Gamma $ which is torsion free. Thus $D(\Gamma_0)$ is by Baily-Borel a smooth algebraic variety and we can construct $\DG = D(\Gamma_0)/(\Gamma/\Gamma_0)$. By Hilbert's theorem on invariant algebras under finite groups, $\DG $ still has the structure of an algebraic variety, with {\em quotient} singularities at those points $p\in \DG $ for which the stabilizer $\Gamma_p$ is not trivial. Hence the singularities of $\DG^*$ are found at these points in $\DG $ and at the boundary $\DG^*\setminus \DG $. 
\end{enumerate}
\end{remark}

\item The proof of (ii) works as follows. Let $\mathcal D_1 = \{ z\in \C : |z|<1\} $ and $\mathcal D_1^* = \mathcal D_1\setminus 0$.
\begin{enumerate}
\item Big Picard Theorem: If a holomorphic function $f: \mathcal D_1^*\,\lra \, \C $ has an essential singularity at $0$, it takes all values of $\C $ except possibly one. (thus, if there are two values not in the image, $f$ has a pole at $0$).
\item Any holomorphic $f:\mathcal D_1^*\, \lra \,\PP^1(\C )\setminus \{p_1,p_2,p_3\}$ extends to an holomorphic function $f: \mathcal D_1\,\lra \,\PP^1(\C )$.
\item Borel's extension of big Picard's Theorem: every holomorphic map $\mathcal D_1^{* r}\times \mathcal D_1^s \,\hra \,\DG$ extends to a holomorphic map $\mathcal D_1^{r+s} \,\hra \,\DG^*$.
\item Hironaka: $V\subset V^*$ where $V^*$ is (possible singular) projective and $V^*\setminus V$ is a divisor with normal crossings: locally for the complex topology, $V\,\hra\,V^*$ is of the form $\mathcal D_1^{* r}\times \mathcal D_1^s \,\hra \,\mathcal D_1^{r+s}$.
\item By Borel's lemma, $f: V(\C )\,\hra \,\DG $ extends to a holomorphic map $V^*(\C )\,\hra \,\DG^*$.
\item Chow: any holomorphic map between projective varieties is algebraic. 
\end{enumerate}
\end{itemize}

\begin{corollary}
\begin{itemize}
\item[(i)] The structure of an algebraic variety on $\DG $ is unique.
\item[(ii)] For any other compactification $\DG \,\hra \,\DG^{\dag }$ with $\DG^{\dag }$ a projective variety and $\DG^{\dag}\setminus \DG $ a divisor with normal crossings, there is a unique regular map $\DG^{\dag }\,\lra \, \DG^*$ commuting with $\DG \,\hra \, \DG^*$.
\end{itemize}
\end{corollary}

{\em Proof: } As for (i), let $V$ be a complex algebraic variety such that $\DG \simeq V(\C )$ as complex manifolds. By Borel's theorem, this is an algebraic isomorphism. Statement (ii) follows similarly from the proof of Borel's theorem, considering $V=\DG \subset V^*=\DG^{\dag }$ and the map $f = Id : \DG \,\lra \,\DG $, which extends to $\DG^{\dag}\,\lra \, \DG^*$. $\Box $

For this reason, $\DG \,\hra\,\DG^*$ is often called the {\em minimal} compactification on $\DG $. Other names are the Satake-Baily-Borel or Baily-Borel compactification.

\vspace{0.5cm}

\newpage
{\em Examples: } 
\begin{enumerate}

\item $D = \cH_1$, $\Aut(\cH_1)=\mathrm{SL}_2(\R )$, $G=\SL_2$ is a semisimple group over $\Q $, $\Gamma = \mathrm{SL}_2(\Z )$, $\DG \,\stackrel{\simeq }\ra \, \A^1_{\C }$ is an algebraic variety over $\C $. It is not  projective: $\begin{pmatrix}1&1\\0&1\end{pmatrix}\in \Gamma $ is unipotent.  

\item $D = \cH_n$, $\Aut(\cH_n)=\mathrm{Sp}_n(\R )$, $G=\mathrm{Sp}_n$ is a semisimple group over $\Q $, $\Gamma = \mathrm{Sp}_n(\Z )$. Hence $\mathrm{Sp}_n(\Z )\backslash \cH_n$ is the set of complex points of a (non-projective) algebraic variety. We will see later that it is a moduli space defined over $\Q $. 

\item Let $F$ be a totally real field, $n=[F:\Q ]$. Let $B = F\oplus F i \oplus F j\oplus F k$, $i^2=a, j^2=b\in F^*$, $i j = -j i = k$ be a quaternion algebra over $F$. 

Let $v_1,..., v_n: F\,\hra \,\R $ be the real archimedean places of $B$ and write $B\otimes_{\Q }\R = B\otimes_F F\otimes_{\Q }\R= \bigoplus_v (B\otimes_F \R_{v}) = \M_2(\R )\stackrel{r}{...}\M_2(\R )\bigoplus \mathbb H\oplus \stackrel{s}{...} \oplus \mathbb H $, $n=r+s$.

Let $\n : B^*\,\lra \, F^*$ be the reduced norm, which coincides with $det$ on any matrix representation. Let $B^1=\{ b\in B^*: \n(b)=1\}$, which may be regarded as an algebraic group over $F$ because it is given by a polynomial equation in $F[x_1,x_2,x_3,x_4]$. Let $\cG_B = \mathrm{Res}_{F/\Q } B^1$, which as algebraic group over $\Q $ such that $\cG (K) = (B\otimes_{\Q }K)^1$ for any extension field $K/\Q $. In particular, $\cG_B(\R ) = \SL_2(\R )\times \stackrel{r}{...}\times \SL_2(\R )\times \mathbb H^1\times \stackrel{s}{...} \times \mathbb H^1 $. 

Let $\cO $ be a ring of integers of $B$: a subring of $B$ of rank $4$ over the ring of integers $\cO_F$ of $F$. Let $\cO^1\subset B^1$ the subgroup of elements of reduced norm $1$. This is a discrete subgroup of $\cG_B(\R )$ and according to Definition \ref{arithmetic}, it is an arithmetic subgroup of $G=\SL_2(\R )\times \stackrel{r}{...}\times \SL_2(\R )$ because the kernel of $\cG_B(\R )\twoheadrightarrow G$ is compact. Note that $\Aut (D)$, where we let $$D=\cH_1\times \stackrel{r}{...}\times \cH_1.$$

By the theorem of Baily-Borel $\DG $ this is a quasi-projective algebraic variety. 
\begin{itemize}
\item If $B\simeq \M_2(F)$, there are unipotent elements in $\Gamma $ and $\DG $ is not compact. 

If $F=\Q $, $\Gamma \subset \SL_2(\Z )$ and $X=\DG^*=\DG \cup \{\infty_1, ..., \infty_h\}$ is a {\em modular curve}. Prominent examples are the congruence subgroups $\Gamma(N)\subset \Gamma_1(N)\subset \Gamma_0(N)\subset \SL_2(\Z )$, giving raise to the modular curves $X(N)\twoheadrightarrow X_1(N)\twoheadrightarrow X_0(N)\twoheadrightarrow X(1)=\PP^1_{\C }$. 

If $F\ne \Q $, $\DG^*$ is called a {\em Hilbert(-Blumenthal) modular variety}, a singular projective variety of dimension $n=[F:\Q ]$. The difference with the case $F=\Q $ is that now $\mathrm{PGL}_2$ is not a quotient of $\cG_B$ over $\Q $ and we can apply Remark \ref{codim2}: the singular locus, which is contained in $\DG^*\setminus \DG $ has codimension $\ge 2$. 
 
\item If $B\not \simeq \M_2(F)$, there are no unipotent elements in $\Gamma $ and $\DG = \DG^*$ is already a {\em projective} variety of dimension $r$. It is smooth unless $\Gamma $ has torsion. Prominent examples are Shimura curves $X_D$ attached to a maximal order $\cO $ in the quaternion algebra $B$ over $\Q $ of discriminant $D=p_1\cdots p_{2 d}$, and their covers $X(D,N)\twoheadrightarrow X_1(D,N)\twoheadrightarrow X_0(D,N)\twoheadrightarrow X_D$, for any integer $N\ge 1$, $(D, N)=1$.
\end{itemize}  

\end{enumerate}


\newpage

\section{Shimura varieties}

\subsection{Connected Shimura varieties}

\begin{definition}\label{cShdatum}
A {\bf connected} Shimura datum is a pair $(G, D=\{ h\} )$ where
\begin{itemize}
\item $G$ is a semisimple algebraic group over $\Q $.
\item $h: S^1 \, \lra \, G_{\R }^{ad}$ is a homomorphism such that \begin{itemize}
\item[SV1: ] Only the characters $1$, $z$ and $z^{-1}=\bar z$ occur in the adjoint representation of $S^1$ on $V=\Lie(G^{ad})_{\C }$,\footnote{That is, $V=V^0\oplus V^+\oplus V^-$ in such a way that for any $z\in S^1$ we have $Ad \,h(z)\cdot v_0 = v_0$, $Ad \,h(z)\cdot v_+ = z\cdot v_+$, $Ad \,h(z)\cdot v_- = z^{-1}\cdot v_-$ for any $v_0\in V^0$, $v_+\in V^+$, $v_-\in V^-$.}
\item[SV2: ] $Ad(h(-1))$ is a Cartan involution on $\Lie G_{\R }^{ad}$,
\item[SV3: ] There exists no non-trivial normal subgroup $H$ of $G^{ad}$ over $\Q $ such that $H(\R )$ is compact. 
\end{itemize}
\item $D= \{ g\cdot h\cdot g^{-1}\} _{g\in G^{ad}(\R )_0}\subset \mathrm{Hom}(S^1, G_{\R }^{ad})$.

\end{itemize}

\end{definition}

Let $(G, D)$ be a connected Shimura datum. Write $$G^{ad}(\R ) \sim G_1\times ...\times G_r\times H_1\times ...\times H_s$$ as a product\footnote{This is an isomorphism up to a finite group, and the factors $G_i$, $H_j$ correspond to the decomposition of $\g $ into simple Lie algebras.} of simple {\em real} Lie groups, and label them so that $G_i$ are non-compact and $H_j$ are compact. 

By SV3 we have that $r\ge 1$: otherwise $G(\R )$ would be compact. 

If we write $h=(h_1, ..., h_{r+s})$, we know by SV2 that $s_i = Ad (h_i(-1)) \ne Id$ for the factors $G_i$.\footnote{This implies in particular that all three characters $1$, $z$, $z^{-1}$ do actually occur in the representation of $S^1$, because if only $1$ appeared, $G_i$ would be compact. By the way, the converse is also true (see Milne 1.17(a)): for $H_j$, $s_j=Id$.} 

For $i=1, ..., r$, let $K_i=K_{s_i}\subset G_{i, 0}$ and $\mathcal D_i = G_{i, 0}/K_i$ be the irreducible\footnote{$\mathcal D_i$ are irreducibles. See my comment previous to Remark 1.2.5 or Milne's Lemma 4.7.} hermitian symmetric domain attached to $(G_i, s_i)$. 

The $\mathcal D_i$ are indeed hermitian thanks to the existence of $h_i$ satisfying $SV1, 2$: $h_i(S^1)$ lies in the compact group $K_i$ because the elements obviously commute with $h(-1)$. 

Moreover, $h_i(S^1)$ is contained\footnote{If $k\in K_i$, we need to show that $h_k:=k h k^{-1}h^{-1} = 1\in \mathrm{Hom}(S^1,G_i^{ad})$, where we already know that $h_k(-1)=1$. But then $h_k$ factors through $S^1\stackrel{^2}\ra S^1$ and the only possible characters that may occur are $z^{2\Z }$. Since we only allow $1$, $z$ and $z^{-1}$, we obtain that $h_k=1$.} in the centre of $K_i$ and thus there is an integrable complex structure $J$ on $\mathcal D_i$.

The domain $\mathcal D_i$ is naturally identified with the $G_{i, 0}$-conjugacy class $D_i=\{ g h_i g^{-1}: \, g\in G_{i, 0}\} \subset \mathrm{Hom}(S^1, G_i^{ad})$ of $h_i$. The one-to-one correspondence is $$\begin{matrix}
\mathcal D_i = G_{i, 0}/K_i & \stackrel{\simeq }\lra & D_i \\
g  & \mapsto & g h_i g^{-1} 
\end{matrix}$$

This allows us to regard $D=\prod D_i$ as a product of irreducible hermitian domains and there is a natural surjective map $$G^{ad}(\R )_0 \, \lra \, \Aut (D)_0$$ whose kernel is $H_{1, 0}\times ... \times H_{s, 0}$, which is compact. 

Let $\Gamma \subset G^{ad}(\Q )_0$ be\footnote{By $G^{ad}(\Q )_0$ we mean $G^{ad}(\Q )\cap G^{ad}(\R )_0$.} an arithmetic subgroup of $G^{ad}$. It follows from Definition \ref{arithmetic} that the image $\bar \Gamma $ of $\Gamma $ in $\Aut (D)$ is again an arithmetic group. 

Moreover, since the kernel of $\Gamma \lra \bar \Gamma $ is finite (being discrete in a compact group), if $\Gamma $ is torsion-free we then have that $\Gamma \simeq \bar \Gamma $ and $\Gamma \backslash D \simeq \bar\Gamma \backslash D$ is a smooth algebraic variety. If $\Gamma'\subset \Gamma $, we obtain regular maps $D(\Gamma ') \lra \DG $ of algebraic varieties (by Borel's Theorem \ref{BailyBorel}).

Also, any $g\in G^{ad}(\Q )_0$ defines a holomorphic map $g: D\,\lra \,D$ and induces a regular morphism between algebraic varieties $$g: \DG \, \lra \, D(g\Gamma g^{-1}).$$

Let $\pi: \, G \,\lra \, G^{ad}$ denote the natural projection of algebraic groups.

\begin{definition}
Let $(G, D)$ be a connected Shimura data. The {\em connected Shimura variety} $Sh^0(G,D)$ is the inverse system of locally symmetric varieties $\DG $, where $\Gamma $ runs over the congruence subgroups of $G(\Q )_0$ such that\footnote{By $G(\Q )_0$ we mean those elements of $G(\Q )$ that map to $G^{ad}(\Q )_0$.} the image in $G^{ad}(\Q )_0$ is torsion-free.

For a given such congruence subgroup $\Gamma $, we also denote $Sh^0_{\Gamma }(G,D) = \DG $. For a compact open subgroup $K\subset G(\A_f )$, Proposition \ref{adeles} shows that $\Gamma = K \cap G(\Q )_0$ is congruence and we also set $Sh^0_K(G,D) = \DG $.
\end{definition}

Many examples are already obtained by those mentioned at the end of Chapter II. 

\begin{remark}

Let $\tilde \Gamma \subset G^{ad}(\Q )_0$ be an arithmetic subgroup. As it shown in Milne's easy Lemma 4.12, $\pi^{-1}(\tilde \Gamma )\subset G(\Q )_0$ is congruence if and only if $\tilde \Gamma $ contains the image $\pi(\Gamma )$ of a congruence subgroup $\Gamma \subset G(\Q )_0$. But usually the map $\pi: G(\Q )_0 \,\ra \, G^{ad}(\Q )_0$ is not surjective, and the family $$\{ \pi (\Gamma ): \Gamma \subset G(\Q )_0 \mbox{ congruence} \} \subset \{ \tilde \Gamma \subset G^{ad}(\Q )_0: \tilde \Gamma\supseteq \pi (\Gamma ), \Gamma \mbox{ congruence}\}$$ is smaller than the latter. A more general family of varieties is thus obtained when considering this second family of groups.
\end{remark}

A semisimple group $G$ over a field $k$ is said to be {\em simply connected} if any isogeny $G'\,\ra \, G$ with $G$ connected is an isomorphism. With this definition, $\SL_2$ is simply connected, whereas $\mathrm{PGL_2}$ is not simply connected because it admits the isogeny $\GL_2\,\ra \,\mathrm{PGL}_2$.

The {\em Strong Approximation Theorem} asserts that if $G$ is a semisimple, simply connected algebraic group over $\Q $ of non compact type, then $G(\Q )$ is dense in $G(\A_f)$.\footnote{For instance, $G=\mathbb{G}_m$ and $\mathrm{PGL}_2$ are not simply connected, and it can be checked that $\Q^*\subset \A_f^* $ is not dense, nor it is $\mathrm{PGL}_2(\Q )$ in $\mathrm{PGL}_2(\A_f)$.}

\begin{proposition}\label{adelic}
Let $(G, D)$ be a connected Shimura datum with $G$ simply connected. Let $K$ be a compact open subgroup of $G(\A_f)$ and let $\Gamma =K\cap G(\Q )$. There is a homeomorphism $$\begin{matrix}
\Gamma \backslash D & \stackrel{\simeq }{\lra } & G(\Q )\backslash D\times G(\A_f)/K \\
x & \mapsto & (x,1)
\end{matrix}$$
\end{proposition}

In the proposition, $G( \Q)$ acts on $D\times G(\A_f)$ on the left, and $K$ acts on $G(\A_f)$ on the right (and trivially on $D$): for $(x,\{g_{\ell }\})\in D\times G(\A_f)$ the action described by $g\in G(\Q )$, $k\in K$ is $$g\cdot (x,\{g_{\ell }\}) \cdot k = (g x, g \{g_{\ell }\} k).$$

{\em Proof. } Since $G(\Q )$ is dense in $G(\A_f)$ and $K$ is open, $G(\A_f) = G(\Q )\cdot K$.\footnote{If $\{g_{\ell }\}\in G(\A_f)$, for any open set $\{g_{\ell }\}\in U$ there exists $g\in G(\Q )$ such that $g\in U$. For $U = \{g_{\ell }\}\cdot K$ this implies that $\{g_{\ell }\} = g\cdot k$ for some $k\in K$.} Thus every element in $G(\Q )\backslash D\times G(\A_f)/K$ is represented by $(x,1)$ for some $x\in D$ and this shows that our map is surjective. For $x, x'\in D$, we have $[x,1]=[x',1]$ if and only if there exists $g\in G(\Q )\cap K = \Gamma $ such that $g x = x'$: our map is a bijection of sets.

Since $G(\A_f)/K$ is discrete because $K$ is open, the map $D\,\lra \, D\times (G(\A_f)/K)$, $x\mapsto (x,[1])$ is a homeomorphism between $D$ and its image, which is open in $D\times G(\A_f)/K$. 
From this a routine exercise shows that the quotient map $$\Gamma \backslash D\,\lra \, G(\Q )\backslash (D\times (G(\A_f)/K))$$ is bi-continuous. $\Box $

\begin{remark}
As $K$ runs among compact open subgroups of $G(\A_f)$, the inverse limit of $Sh^0_K(G,D)$ is $G(\Q )\backslash D\times G(\A_f)$, something which contains $D$ and may be regarded as a kind of completion of it. We will not prove this.
\end{remark}

\subsection{Shimura varieties}

{\bf Alternative definition of connected Shimura datum.}

The exact sequence of real Lie groups $$1\,\ra \R^* \quad \stackrel{t\mapsto t^{-1}}{\lra }\, \C^* \quad \stackrel{z\mapsto z/\bar z}{\lra } S^1\,\ra \,1$$ arises from the exact sequence of algebraic groups $$1\,\ra \,\G_m \,\stackrel{w}{\lra }\,\mathbb{S} \lra \, U_1\, \ra \,1,$$
where $\mathbb{S} = \mbox{Res}_{\C/\R }\G_m$ and $U_1$ denotes simply the quotient of tori $\mathbb{S}/\G_m$. 

A connected Shimura datum can be alternatively defined to be a pair $(G, D=\{ h\})$, where $G$ and $D$ are as in Definition \ref{cShdatum} except that $$h: \mathbb{S}_{\R } \,\lra \, G_{\R }^{ad}$$ is a homomorphism of real algebraic groups satisfying the conditions

\begin{itemize}
\item[SV1: ] Only the characters $1$, $z/\bar z$ and $\bar z/z$ occur in the representation of $\mathbb{S}_{\R }$ on $\Lie(G^{ad})_{\C }$,
\item[SV2: ] $Ad(h(i))$ is a Cartan involution on $\Lie G_{\R }^{ad}$,
\item[SV3: ] There exists no nontrivial normal algebraic subgroup $H$ of $G^{ad}$ over $\Q $ such that $H(\R )$ is compact. 
\end{itemize}

Following this approach, we define (non-connected) Shimura varieties.  

\begin{definition}\label{Shdatum}
A Shimura datum is a pair $(G, D=\{ h\} )$ where
\begin{itemize}
\item $G$ is a {\bf reductive} algebraic group over $\Q $.
\item $h: \mathbb{S}_{\R } \,\lra \, G_{\R }$ is a homomorphism of algebraic groups over $\R $ such that \begin{itemize}
\item[SV1: ] Only the characters $1$, $z/\bar z$ and $\bar z/z$ occur in the representation of $\mathbb{S}_{\R }$ on $\Lie(G^{ad})_{\C }$,
\item[SV2: ] $Ad(h(i))$ is a Cartan involution on $\Lie G_{\R }^{ad}$,
\item[SV3: ] There exists no factor $H$ of $G^{ad}$ over $\Q $ such that $H(\R )$ is compact. 
\end{itemize}
\item $D= \{ g\cdot h\cdot g^{-1}\} _{g\in G_{\R }}\subset \mathrm{Hom}(S^1, G_{\R })$.

\end{itemize}
\end{definition}

Note that we extend the definition to arbitrary reductive algebraic groups over $\Q $. Another remarkable difference in our definition is that $D$ is the conjugation class of $h$ under the {\em possibly non-connected}\footnote{A theorem of Cartan asserts that if $G$ is reductive, $G(\R )$ has finitely many connected components for the real topology. In fact, a more powerful theorem of Whitney proves that the set of real points of an algebraic variety over $\R $ has finitely many connected components.} real Lie group $G_{\R }$. 

\begin{proposition}\label{cc}
Let $(G, D)$ be a Shimura datum. Let $D_0$ be a connected component of $D$ and $K\subset G(\A_f)$ be a compact open subgroup. Then
\begin{itemize}
\item The set $G(\Q )_0\backslash G(\A_f)/K$ is finite. Let $\mathcal C$ be a set of representatives of the double coset.

\item There is an homeomorphism $$\begin{matrix}
\bigcup_{c\in \mathcal C}\Gamma_c \backslash D_0 & \stackrel{\simeq }{\lra } & G(\Q )\backslash D\times G(\A_f)/K \\
x & \mapsto & (x,c_x),
\end{matrix}$$
where $\Gamma_c = c\cdot K \cdot c^{-1}\cap G(\Q )_0$ and $c_x$ denotes the connected component to which $x$ belongs.   
\end{itemize}
\end{proposition}

{\em Proof. }
In order to prove the first item, it suffices to show that $\\ G(\Q )\backslash G(\A_f)/K$ is finite, because $G(\Q )_0\backslash G(\Q )\,\hookrightarrow \,G^{ad}(\R )_0\backslash G^{ad}(\R )$ is already finite. If $G$ is semisimple and simply connected, we already showed that $G(\Q )\backslash G(\A_f)/K$ is finite: we actually saw that the cardinality is $1$. We will not explain here why the finiteness statement is also true in the general case (cf. \cite[p.\,48, 50]{Mil}). 

The second item follows similarly as in Proposition \ref{adelic}. $\Box $


\begin{example}\label{foot}
\begin{itemize}
\item $G=\GL_2, \\ $ $h: \mathbb{S}_{\R }\,\ra \, \GL_{2, \R }, \, a+b i\,\mapsto \, \begin{pmatrix}
a&b\\-b&a
\end{pmatrix}, \\ $ $D=\{ h_g:=g h g^{-1}\}_{g\in \GL_2(\R )} \stackrel{\cong }{\lra } \C \setminus \R \, = \, \cH_1^+ \cup \cH_1^-, \\ g\, \mapsto \, \mbox{ Fixed point $p$ of }h_g(\C^*)\mbox{ on }\C  \mbox{ such that }h(z) \mbox{ acts on the tangent} \\ \mbox{space of }p \mbox{ as }z/\bar z$ (and not as $\bar z/z$).\footnote{The pair $(\GL_2, \cH_1^{\pm })$ satisfies the conditions SV1, SV2, SV3. As for SV1, $h_{ad}:\C^*\,\lra \,\mathrm{PGL}_2(\R )$ factors through the circle unit $U_1$ by means of the map $\\ u: U_1\,\ra \, \mathrm{PGL}_2(\R )_0=\mathrm{PSL}_2(\R )$, $a+bi\mapsto \pm \sqrt{a+bi} = \pm (x+y i)\mapsto \begin{pmatrix}
x&y\\ -y&x
\end{pmatrix}$. $\\ $ The action of $h_{ad}(\C^*)\subset \mathrm{PSL}_2(\R )$ on the Lie algebra $\mathrm{sl}_2(\R ) = \mathrm{so}_2(\R )\oplus T_i(\cH_1)$ is trivial on the first factor, and acts on the second through the character $z/\bar z = \frac{a+bi}{a-bi}$, because $d(\tau \mapsto \frac{a\tau + b}{-b\tau +a})_{|i} = \frac{a(-b\tau +a)+b(a\tau +b)}{(-b\tau a)^2}_{|i} = \frac{a^2+b^2}{(a-bi)^2}=z/\bar z$. Hence the only characters that occur in $\Lie(G(\C )^{ad})$ are $1$, $z/\bar z$ and $\bar z/z$.}

\item $B$ quaternion algebra over a totally real number field $F$, $G = \mathrm{Res}_{F/\Q }(B^*)$, $\\ h(a+b i) = (\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}, \stackrel{r}{...},\begin{pmatrix}
a&b\\-b&a
\end{pmatrix},1, ..., 1), \quad D = \cH_1^{\pm}\times \stackrel{r}{...}\times \cH_1^{\pm }$.

\item $G = T$ a torus over $\Q $, $\quad h: \mathbb{S} \,\lra \, T$ any morphism. $\\ $ Since $T^{ad}=\{ 1\}$, conditions SV1, 2, 3 hold trivially true. $\\ D = \{ x \}$, a single point. $\\ $ For any compact open subgroup $K\subset G(\A_f)$, $$Sh_K(T, x) = T(\Q )\backslash \{ x \} \times T(\A_f)/K \simeq T(\Q )\backslash  T(\A_f)/K,$$ a finite set of points by Proposition \ref{cc} $\quad $-note that $T(\Q )_0 = T(\Q )$. 
\end{itemize}
\end{example}


\begin{definition}
Let $(G, D)$ be a Shimura data. The {\em Shimura variety} $Sh(G,D)$ is the inverse system $Sh_K(G,D) = G(\Q )\backslash D\times G(\A_f)/K$, where $K$ runs among compact open subgroups of $G(\A_f)$ such that the image of all congruence subgroups $\Gamma_c = c K c^{-1}\cap G(\Q )_0\,\subset \,G(\Q )$ in $G^{ad}(\Q )$ are torsion free.
\end{definition}

The Shimura variety $Sh(G, D)$ comes equipped with a right action of $G(\A_f)$. Indeed, if $g=\{ g_{\ell }\}\in G(\A_f)$, there is a natural morphism\footnote{It is an easy exercise that these maps are well defined.} $$\begin{matrix}
T_g: & Sh_K(D, G) & \lra & Sh_{g^{-1} K g}(D, G) \\
 & (x, a) & \mapsto & (x, a g)
\end{matrix}$$

\begin{remark}
Write $\mathcal C = \{ c_1, ..., c_h\}$. Any element of $Sh_K(D, G)$ can be written as $(x, c_i)$ for some $x\in D$, $1\le i\le h$. An element $g\in G(\A_f)$ induces a permutation $i\mapsto g(i)$ on $\{ 1, ..., h\}$ as follows: since $G(\A_f)=\bigcup G(\Q )_0 c_i K$, $c_i g = \gamma c_{g(i)} k$ for some $\gamma \in G(\Q )_0$ and $k\in K$. We then have that $T_g$ maps the connected component $\Gamma_{c_i}\backslash D$ to $\Gamma_{c_{g(i)}}\backslash D$. More precisely, $T_g(x, c_i) = (x, c_i g) = (x, \gamma c_{g(i)} k) = \gamma (\gamma^{-1} x, c_{g(i)}) k\equiv (\gamma^{-1} x, c_{g(i)})$.
\end{remark}

%\begin{remark}
%Note also that $T_g = Id$ when $g\in K$. Hence the group of automorphisms of $Sh_K(D, G)$ is $K\backslash \mbox{Norm}_{G(\A_f)}(K)$. Under our %torsion-free hypothesis, this is always a finite group (cf.\,\cite[Theorem 3.21]{Mil}). 
%\end{remark}

\vspace{0.5cm}
{\bf The weight homomorphism and additional axioms.}  

Recall that exact sequence of tori $$1\,\ra \,\G_m \,\stackrel{w}{\lra }\,\mathbb{S} \lra \, U_1\, \ra \,1.$$ Let $(G, D)$ be a Shimura datum. An element of $D$ is a homomorphism $h: \mathbb{S}_{\R } \,\lra \, G_{\R }$ that factors through $U_1$. When we restrict $h$ to $\G_{m, \R } = \R^*$, $h(r)$ must act trivially on $\Lie (G_{\R })\otimes \C $ for any $r\in \R^*$, because it lies in the kernel of the above exact sequence. Since the only elements of $G(\R )$ which act trivially by the adjoint representation are those lying in the center, we obtain that $h(r)\in Z(\R )$.

Hence $h_{| \G_m}$ does not depend of the choice of $h\in D$ and we can define the weight homomorphism of $(G, D)$ to be
$$\begin{matrix}
\omega : & \G_{m, \R } & \lra & G_{\R } \\
        & r  & \mapsto & 1/h(r)
\end{matrix}$$ 

Note that both $\G_{m, \R }$ and $G_{\R }$ are the sets of real points of algebraic groups over $\Q $. Since $\omega $ is a morphism of algebraic groups, it must actually be defined over some finite extension of $\Q $. Together with the axioms SV1, SV2 and SV3, sometimes the Shimura datum satisfies some or all of the following further axioms:

\begin{itemize}
\item[SV4: ] The weight homomorphism $\omega$ is {\em rational}, that is, it is defined over $\Q $,
\item[SV5: ] The group $Z(\Q )$ is discrete in $Z(\A_f)$,
\item[SV6: ] The torus $Z_0$ is split over a CM-field, that is, over a totally imaginary quadratic extension of a totally real number field. 
\end{itemize}

%Posar els exemples anteriors aqui i comentar SV4, 5 i 6? 

\subsection{The Siegel modular variety}

Let $(V, \Psi)$ be a symplectic space over $\Q $ of dimension $2 n$, for some $n\ge 1$. That is, a vector space over $\Q $ equipped with an alternating non-degenerate bilinear form  $\Psi : \, V\times V\, \stackrel{\Psi }{\lra } \Q $. 

Define $G=GSp(V, \Psi )\subset \GL (V)$ to be the algebraic group of transformations of $V$ preserving $\Psi $ up to scalar, so that
$$G(\Q )\, = \,\{ g\in \GL(V): \Psi (g u, g v) = \nu(g)\Psi(u, v)\} $$ for all $u, v\in V$ and some $\nu(g)\in \Q^*$.\footnote{One checks that automatically $\nu $ defines a homomorphism $\nu : G\,\ra \, \G_m$.}
Define $S=Sp(V, \Psi )$ by the exact sequence $$1\,\ra \, S\,\ra \,G\,\stackrel{\nu }{\ra }\, \G_m.$$ The center of $G$ is $\G_m$, $G^{ad} = S/\{ \pm 1\} $ and $G^{der} = S$.

A symplectic complex structure $J$ on $(V(\R ), \Psi )$ is an endomorphism $J\in S(\R )$ such that $J\ne Id $, $J^2=-Id$. For a given $J$, the bilinear form $$\begin{matrix}
\Psi_J: & V(\R )\times V(\R ) & \lra & \R \\ & (u, v) & \mapsto & \Psi (u, J v) 
\end{matrix}$$ is symmetric. We say that $J$ is positive (negative) if $\Psi_J$ is positive (negative) definite.

Let $D = D^+\cup D^-$ be the set of positive or negative symplectic complex structures on $V(\R )$. The group $G(\R )$ acts on $D$ by $(g, J)\, \mapsto \, g J g^{-1}$. The stabilizer of $D^+$ in $G(\R )$ is its identity's connected component $G(\R )_0 = \{ g\in G(\R ): \nu(g)>0\}$. One easily shows also that $G(\R )$ acts transitively on $D$, and $S(\R )$ acts transitively on $D^+$.

Attached to any $J\in D$ there is the morphism\footnote{Indeed, $a+b J\in G(\R )$, with $\nu (a+bJ)=a^2+b^2=|z|^2$.}  $$\begin{matrix}
h_J: & \C^* & \lra & G(\R ) \\ & z = a + b i &\mapsto & a+b J.
\end{matrix}$$

Since $h_{g J g{-1}} = g h_J g^{-1}$ for any $g\in G(\R )$, there is a natural identification $$D\, \leftrightarrow \, \{ G(\R )-\mbox{Conjugation class of } h_J:\C^*\,\ra \,G(\R )\} .$$

\begin{lemma}
The Shimura datum $(G, D)$ satisfies the axioms SV1, SV2 and SV3.
\end{lemma}

{\em Proof. } {\bf SV1:} Only the characters $1$, $z/\bar z$ and $\bar z/z$ occur in the representation of $\mathbb{S}_{\R }$ on $\Lie(G^{ad})_{\C }: \\ $ Write $V(\C )=V^+\oplus V^-$ as a direct sum of $\pm i$-eigenspaces under the action of $J$. Then $h_J(z)$ acts on $V^+$ as multiplication by $z$, and it acts on $V^-$ as multiplication by $\bar z$. Thus\footnote{In order to check this, the exercise is a generalization from $\mathrm{PSL}_2$ (as in example \ref{foot}) to the algebraic group of symplectic similitudes of a vector space of arbitrary even dimension. Indeed, the matrix expression of $h_J(z)$ in $n\times n$-blocks is $\begin{pmatrix}
z&0\\ 0&\bar z 
\end{pmatrix}$, which is a complex form of $\begin{pmatrix}
a&b\\ -b&a
\end{pmatrix}$.}
$$\begin{matrix}
\End (V(\C )) & \simeq &  \End(V^+)\oplus & \mathrm{Hom} (V^+, V^-) \oplus &\mathrm{Hom}(V^-, V^+) \oplus &\End (V^-) \\ ad(h_J(z)) & \mapsto & (1,&z/\bar z,& \bar z/z,&1).
\end{matrix}\\ $$  
{\bf SV2:} $Ad(h(i))$ is a Cartan involution on $\Lie G_{\R }^{ad}$: It is an involution because $ad(h(i)): G^{ad}(\R )\,\ra \, G^{ad}(\R )$, $g\,\mapsto \,JgJ^{-1}$ and $J^2=-Id$. It is Cartan because the subgroup of fixed elements in $G(\R )$ is $\{ g\in G(\R ): J g = g J \} = \{ g\in G(\R ): \Psi_J(g u, g v) = \Psi (u, v)\}$ for all $u, v\in V(\R )$. Since $\Psi_J$ is positive definite, the subgroup is compact.
  
$\\ $ {\bf SV3:} There exists no factor $H$ of $G^{ad}$ over $\Q $ such that $H(\R )$ is compact: Indeed, $G^{ad}$ is simple over $\Q $ and $G^{ad}(\R )$ itself is not compact (a factor of it is Siegel's upper half space $\cH_n$). 
$\Box $

We define the {\em Siegel modular variety} attached to $(V, \Psi )$ to be the Shimura variety $Sh(G, D)$ associated to the Shimura datum $(G, D)$.

\subsubsection{Modular interpretation}

Let $(V, \Psi)$ be a symplectic space over $\Q $ of dimension $2 n$, for some $n\ge 1$ as before. Let $(G, D)$ be the Shimura datum attached to $(V, \Psi )$ and $K\subset G(\A_f)$ a compact open subgroup. Let $Sh_K(G, D)$ be the Shimura variety attached to $(G, D)$ and $K$.

Let $A/\C $ be a complex abelian variety of dimension $n$. There exists a lattice $\Lambda \subset T_0(A)$ of rank $2n$ over $\Z $ such that $A(\C )\simeq T_0(A)/\Lambda $. Under this isomorphism, there is a natural identification $H_1(A, \Z )=\Lambda $.

The isomorphism $V(\R )\simeq \Lambda \otimes \R \,\simeq \,T_0(A)$ induces a complex structure $J$ on the real vector space $V(\R )$.

\begin{definition}
A polarization on $A$ is a non-degenerate alternating form $s : \Lambda \times \Lambda \, \lra \, \Z $ such that $s(J u, J v) = \Psi (u, v)$ for all $u, v\in V(\R )$ and $s_J(u, v):=s(u, J v)$ is positive definite.
\end{definition}

Let $V(\A_f) = \Lambda \otimes_{\Z } \A_f \,\simeq \,\A_f^{2n}$. Let $V_f(A) = H_1(A, \A_f)\simeq \Lambda \otimes \A_f$ the Tate module of $A$.

Let $\mathrm{AV}^0$ be the category of abelian varieties up to isogeny and let $M_K(G, D) = \{ A, s, \eta \cdot K\}$ be the set of triples where 

\begin{itemize}
\item $A$ is a complex abelian variety of dimension $n$, $A(\C ) = V(\R )/\Lambda $,
\item $s$ is an alternating form on $H_1(A, \Z )$ such that $s$ or $-s$ is a polarization on $A$,
\item $\eta : V(\A_f)\, \stackrel{\simeq }\lra \, V_f(A)$ such that $\eta _*(\Psi ) = a\cdot s$ for some $a\in \A^*_f$.
\end{itemize} 

Two triples $(A, s, \eta \cdot K)$, $(A', s', \eta' \cdot K)$ are isomorphic if there is an isogeny $f: A\,\ra \, A'$ such that $f^*(s')=q\cdot s$, $q\in \Q^*$ and $f^*(\eta' K) = \eta K$.

\begin{theorem}
The Shimura variety $Sh_K(G, D)$ is the coarse moduli space over $\C $ that classifies triples in $M_K(G, D)$ up to isomorphism. In particular, there is a canonical bijection of sets $$M_K(G, D)/\simeq \quad \leftrightarrow \quad G(\Q )\backslash D\times G(\A_f)/K.$$  
\end{theorem}


\subsection{Shimura varieties of Hodge type}

\subsubsection{Hodge structures}

A {\em Hodge estructure} is a real vector space $V$ together with a decomposition of $V(\C )$ into complex vector subspaces $$V(\C )= \oplus V^{p, q}, \quad (p, q)\in \Z \times \Z : \quad \bar V^{p, q} = V^{q, p}.$$

The {\em type} of the Hodge structure is the set of pairs $(p, q)$ for which $V^{p, q}\ne \{ 0\}$. The {\em weight decomposition} of a Hodge structure is the decomposition $$V = \oplus_{n\in \Z } V_n$$ where $V_n$ is the real vector subspace of $V$ such that $V_n(\C )= \oplus_{p+q=n} V^{p, q}$.  If $V=V_n$, the Hodge structure is said to be of {\em weight $n$}.

A {\em rational} Hodge structure is a vector space $V$ over $\Q $ together with a Hodge structure for $V(\R )$ such that $V_n$ is defined over $\Q $ for any $n\in \Z $. 

\begin{example}\label{hodge}
\begin{itemize}
\item Let $V$ be a real vector space. To give a complex structure $J$ on $V$ is equivalent to give a Hodge structure $V(\C ) = V^{-1, 0}\oplus V^{0, -1}$ of type $(-1, 0)$, $(0, -1)$ on $V$.

\item  Let $\Q (r)$ denote the rational Hodge structure $V=\Q $, $V(\C )=\C^{-r, -r}$, the unique possible rational Hodge structure on $\Q $ of weight $-2r$.

\item Let $X$ be a non-singular projective variety. Let $V= H^n(X, \Q)$, a vector space over $\Q $. Then $V$ admits the following Hodge structure of weight $n$: $V(\C ) = \oplus_{p+q=n} H^q(X, \Omega^p)$.
\end{itemize}
\end{example}

A {\em morphism of Hodge structures} $t: V=\oplus_{(p, q)}V^{p, q}\,\lra \, W=\oplus_{(p,q)}W^{p, q}$ is a linear map $t: V\, \ra \, W$ such that $t(V^{p, q})\subseteq W^{p, q}$.


\subsubsection{Shimura varieties of Hodge type and its modular interpretation}
\begin{definition}
A Shimura datum $(G, D)$ is {\em of Hodge type} if there exists a symplectic vector space $(V, \Psi )$ over $\Q $ and a monomorphism $G\,\hookrightarrow \,GSp(V, \Psi)$ such that $D$ maps to $D_{(V, \Psi )}$. The Shimura variety $Sh(G, D)$ is then called {\em of Hodge type}.
\end{definition}

Recall the character $\nu : GSp(V, \Psi )\, \lra \, \G_m$. We shall still denote $\nu: G\, \hookrightarrow GSp(V, \Psi )\, \lra \, \G_m$. For any $r\in \Z $, denote (also) by $\Q (r) $ the vector space $\Q $ with the action $(g, q)\, \mapsto \, \nu (g)^r\cdot q$. 


\begin{proposition}
Let $(G, D)$ be a Shimura datum of Hodge type, $(G, D)\,\hookrightarrow \,GSp(V, \Psi )$ where $V$ is a vector space over $\Q $ of dimension $2 n$. Then there exist non-zero multilinear maps $$t_i: V\times \stackrel{2 r_i)}{...}\times V\, \lra \, \Q (r_i), \quad i=1, ..., k$$ such that for any field extension $k/\Q $, $$G(k)\, =\, \{ g\in \GL_k(V)\, : \,t_i(g v_1, ..., g v_{2 r_i})\,  = \, \nu(g)^{r_i}\cdot t_i(v_1, ..., v_{2 r_i}) \}$$ for all $v_j\in V(k)$, $i=1, ..., k$.
\end{proposition}

That is, we demand to $g$ to be equivariant with respect to the actions of $G$ on $V$ and $\Q (r_i)$, respectively. 

If $J\in D$ is a complex structure on $V(\R )$, it induces a Hodge structure of weight $-1$ on $V(\R )$. For any $r\ge 1$, there is a natural\footnote{$V(\R )^{\otimes r} = V(\R )^{-r, 0}\oplus V(\R )^{-r+1, -1}\oplus...\oplus V(\R )^{0, -r}$.} Hodge structure of weight $-r$ on $V(\R )^{\otimes r}$.
The map $t_i:\, V^{\otimes 2 r_i}\, \ra \, \Q (r_i)$ is a morphism of Hodge structures of weight $-2 r_i$.


Let $K\subset G(\A_f)$ be a compact open subset. Let $M_K(G, D)$ be the set of triples $(A, \{ s_i\}_{i=0, ..., k}\,, \eta \cdot K)$ where 
\begin{itemize}
\item $A$ is a complex abelian variety of dimension $n$, 
\item $s_0$ is an alternating form on $H_1(A, \Z )$ such that $s_0$ or $-s_0$ is a polarization on $A$,
\item $s_i\in H^{2 r_i}(A, \Q )\simeq \mathrm{Hom}(\wedge^{2 r_i} \Lambda , \Q )$ such that $V^{\otimes 2 r_i}\ra \wedge^{ 2 r_i}V\, \ra \, \Q (r_i)$ is a morphism of Hodge structures,\footnote{The map $s:\,V^{\otimes 2 r}\,\ra \,\Q(r)$ is a morphism of Hodge structures if and only if $s((V^{\otimes 2 r})^{p, q}) = 0$ for all pairs $(p, q)\ne (-r, -r)$, because the type of $\Q (r)$ is $\{ (-r, -r)\}$. In other words: if we write $H=H^{2 r}(A, \Q )\simeq \wedge^{ 2 r}V$, $H$ is endowed with a Hodge structure of weight $2 r$ (see Example \ref{hodge}) and we require that $s\in H^{2 r}(A, \Q )\cap H(\C )^{r, r}\subset H^{2r}(A, \C )$. The {\em Hodge conjecture} for abelian varieties asserts that all elements in this intersection are the cohomology classes of algebraic cycles on $A$ with coefficients in $\Q $. This is known for $r=1$.} for $i=1, ..., k$,
\item $\eta : V(\A_f)\, \stackrel{\simeq }\lra \, V_f(A)$ such that $\eta _*(\Psi ) = a\cdot s$ for some $a\in \A^*_f$, and $\eta _*(t_i)=s_i$ for $i=1, ..., k$,
\end{itemize}

satisfying the following condition: 

There exists an isomorphism $\alpha : H_1(A, \Q )\, \simeq V$ of vector spaces over $\Q $ such that $\alpha^*(\Psi ) = q\cdot s$ for some $q\in \Q^*$, $\alpha^*(t_i)=s_i$ for $i=1, ..., k$, and $\alpha_*(J)\in D_{(V, \Psi )}$.\footnote{$J$ is the complex structure on $H_1(A, \R )$ induced by the isomorphism with $T_0(A)$. We require that $\alpha_*(J)$ is a symplectic complex structure on $(V, \Psi )$ such that $\Psi_{\alpha_*(J)}$ is positive or negative definite.}

\begin{theorem}
The Shimura variety $Sh_K(G, D)$ is the coarse moduli space over $\C $ that classifies triples in $M_K(G, D)$ up to isomorphism. In particular, there is a canonical bijection of sets $$M_K(G, D)/\simeq \quad \leftrightarrow \quad G(\Q )\backslash D\times G(\A_f)/K.$$  
\end{theorem}

\subsubsection{Shimura varieties of PEL type}\label{PEL}

Let $(B, *)$ be a simple algebra over $\Q $ together with a positive involution $*: B\, \ra \, B$: an anti-involution such that $\Tr_{B\otimes_{\Q }\R /\R }(b^*\cdot b)>0$ for all $b\in B\setminus \{ 0 \}$. Let $F$ denote the center of $B$ and $F_0 = \{ b\in F: b^*=b\}$.

Assume that for every embedding $\varphi : F_0 \hookrightarrow \qbar $, $(B\otimes_{F_0}\qbar , *)$ is isomorphic to a product of algebras with involution either of the form $$(A)\quad \quad \M_n(\qbar )\times \M_n(\qbar ),\quad (b_1, b_2)^* = (b^t_2, b^t_1),$$ or of the form $$(C)\quad \quad \M_n(\qbar ),\quad b^* = b^t,$$but not a mixture of them. 

Let $(V, \Psi )$ be a $(B, *)$-module: a vector space over $\Q $ together with an action $B\subset \End(V)$ and a non-degenerate alternating bilinear form $$\Psi : V\times V\, \ra \, \Q ,\quad \quad \Psi (b u, v) = \Psi (u, b^* v), \quad u, v\in V, b\in B.$$ 

Let $G\subset \GL(V)$ be the algebraic group over $\Q $ such that for any field extension $k/\Q $: $$G(k)\,=\,\{g\in \Aut_B(V\otimes k):\, \Psi(g u, g v ) = \mu(g)\Psi (u, v)\}$$ for all $u, v\in V\otimes k$ and some $\mu(g)\in k^*$.



Let $G' = \{ g\in G:\, \mu(g)=1, \det(g)=1\} \subset G$. 

\begin{proposition}
The algebraic group $G$ is reductive and $G'$ is semisimple and simply connected.\footnote{There is a remaining possible type for $(B, *)$, which is usually called BD. In this case, $G$ is not connected, but its connected component is reductive. The group $G'$ is a special orthogonal group.}\label{BD}
\end{proposition}


In fact, we have $$G'_{\qbar} \simeq \SL_m^d \quad \quad \mbox{ if $B$ is of type A;}$$ $$G'_{\qbar} \simeq \mathrm{Sp}_m^d \quad \quad \mbox{ if $B$ is of type C;}$$ where $m=\dim_F(V)/\sqrt{[B:F]}$ and $d=[F_0:\Q ]$.

\begin{proposition}
There exists a (unique) $G(\R )$-conjugacy class $D$ of homomorphisms $h: \mathbb{S}\, \lra \, G_{\R }$ such that each $h$ induces a symplectic complex structure $J=h(i)$ on $V(\R )$ such that $\Psi_J$ is positive or negative definite. 

The Shimura datum $(G, D)$ satisfies the axioms SV1, SV2, SV3.
\end{proposition}

The corresponding Shimura varieties $Sh(G, D)$ are called of PEL-type (A or C).

Let $b_1, ..., b_k$ be a set of generators of $B$ as a $\Q $-algebra and let $$\begin{matrix}
t_{b_i}: & V\times V & \lra & \Q \\
    & (u, v) & \mapsto & \Psi (u, b v).
\end{matrix}$$

Then $(G, D)$ is the Shimura datum of Hodge type associated to $(V, \Psi, \{ t_{b_i}\})$.

Let $K\subset G(\A_f)$ be a compact open subset. Let $M_K(G, D)$ be the set of quadruples $(A, \iota , s, \eta \cdot K)$ where 
\begin{itemize}
\item $A$ is a complex abelian variety, 
\item $\iota : B\hookrightarrow \End (A)\otimes \Q $,
\item $s$ is an alternating form on $H_1(A, \Z )$ such that $s$ or $-s$ is a polarization on $A$,
\item $\eta : V(\A_f)\, \stackrel{\simeq }\lra \, V_f(A)$ such that $\eta _*(\Psi ) = a\cdot s$ for some $a\in \A^*_f$.
\end{itemize}

satisfying the following condition: 

There exists a $B$-linear isomorphism $\alpha : H_1(A, \Q )\, \simeq V$ of vector spaces over $\Q $ such that $\alpha^*(\Psi ) = q\cdot s$ for some $q\in \Q^*$.

\begin{theorem}
The Shimura variety $Sh_K(G, D)$ is the coarse moduli space over $\C $ that classifies quadruples in $M_K(G, D)$ up to isomorphism. 
\end{theorem}


\newpage

\section{Canonical models of Shimura varieties}


\subsection{The reciprocity map for abelian extensions of number fields}

Let $E$ be a number field and let $E^{ab}$ be the maximal abelian extension of $E$ inside a fixed algebraic closure of $E$. Global class field theory provides a continuous surjective homomorphism $$rec_E:\, \A_E^* \, \twoheadrightarrow \Gal (E^{ab}/E)$$ which is called the {\em Artin reciprocity map} and it is such that for any finite abelian extension $E'/E$ of $E$, we have a commutative diagram:

$$
\begin{matrix}
E^*\backslash \A_E^* & \stackrel{rec_E}{\twoheadrightarrow } & \Gal (E^{ab}/E) \\
\downarrow & \circlearrowleft & \downarrow \\
E^*\backslash \A_E^*/N(\A_{E'}^*) & \stackrel{\stackrel{rec_{E'/E}}{\simeq }}{\longrightarrow } & \Gal (E'/E)
\end{matrix} 
$$

The identity component of $\A_E^*$ (and thus in particular the identity component of $\prod_{v\mid \infty }E_v^*$) lies in the kernel of the reciprocity map $rec_E$ \footnote{because $\Gal(E^{ab}/E)$ is totally disconnected and $1\in \Gal(E^{ab}/E)$ is its own connected component.}.

Hence, if $E$ is totally imaginary, $rec_E$ factors through $\A_{E, f}^*$.  

When $E=\Q $, $rec_{\Q }$ factors through $\{ \pm \} \times \A_{\Q , f}$ and $\Q^{ab} = \bigcup_{N\ge 1} \Q (\zeta_N)$, where $\zeta_N$ is a primitive $N^{th}$-root of the unity. 

There is a commutative diagram

$$
\begin{matrix}
\Q ^*\backslash (\{ \pm 1\}\times \A_{\Q ,f}^*) & \stackrel{rec_{\Q }}{\twoheadrightarrow } & \Gal (\Q^{ab}/\Q ) \\
\downarrow & \circlearrowleft & \downarrow \\
\Q^*\backslash \A_{\Q }^*/N(\A_{\Q(\zeta_N)}^*) = (\Z/ N\Z )^*& \stackrel{\stackrel{a\mapsto \{ \zeta_N\mapsto \zeta_N^{a^{-1}} \} }{\simeq }          }{\longrightarrow }& \Gal (\Q(\zeta_N)/\Q )
\end{matrix}
$$

\subsection{Abelian varieties with complex multiplication}

A CM-field is a totally imaginary quadratic extension $E$ of a totally real number field $F$. Let $[E:\Q ] = 2 g$ for some $g\ge 1$. Each of the $g$ embeddings $F\hookrightarrow \R $ extends to two conjugate embeddings $\varphi , \bar \varphi : E \hookrightarrow \C $. A CM-type $\Phi $ for $E$ is a subset $\Phi =\{ \varphi_1, ..., \varphi_g\}\subset \mathrm{Hom} (E, \C )$ such that $\mathrm{Hom} (E, \C ) = \Phi \sqcup \bar \Phi $, that is, a full set of representatives of embeddings $E\hookrightarrow \C $ up to complex conjugation.

An abelian variety of CM-type $(E, \Phi )$ is an abelian variety $A/\C $ of dimension $g$ such that there exists a monomorphism $$i: E \hookrightarrow \End^0(A)$$ such that for any $a\in E$, we have $Tr(a_{|T_0(A)}) = \sum_{\varphi \in \Phi }\varphi (a)$.\footnote{For any $i: E \hookrightarrow \End^0(A)$, there always exists a CM-type $\Phi $ for $E$ such that $A$ is of CM-type $(E, \Phi )$. Indeed, write $A=\C /\Lambda $. Then $\Lambda \otimes \Q $ is an $E$-vector space of dimension $1$. We have that $\Lambda \otimes \C $ is a 1-dimensional module over $E\otimes \C \simeq \oplus_{\varphi\in \mathrm{Hom}(E, \C )} \C_{\varphi} $, where $E$ acts on $\C_{\varphi }$ through $\varphi $. Since $\Lambda \otimes \R = T_0(A)$ and it is well known that $\Lambda \otimes \C = T_0(A)\oplus \overline {T_0(A)}$, we obtain that any $a\in E$ acts on $T_0(A)$ as diag$(\varphi_1(a),...,\varphi_g(a))$ for some CM-type $\Phi = \{ \varphi_i,...,\varphi_g\} $.} 

\begin{definition}
Let $(E, \Phi )$ be a CM-type. The reflex field of $(E, \Phi )$ is the number field $\tilde E$ characterized for any of the following equivalent conditions:
\begin{itemize}
\item $\tilde E$ is the fixed field of $\{ \sigma \in \Gal(\qbar /\Q ): \Phi^{\sigma } = \Phi \} $.
\item $\tilde E = \Q (\sum_{\varphi \in \Phi }\varphi (a)\,:\,a\in E)$.
\item It is the smallest subfield of $\qbar $ for which there exists an $\tilde E$-vector space $V$ together with a monomorphism $E\hookrightarrow \End_{\tilde E}(V)$ for which $Tr(a) = \sum_{\varphi \in \Phi }\varphi (a)$ for all $a\in E$. 
\end{itemize}
\end{definition}

Observe that when $E/\Q $ is a Galois extension, then $\tilde E\subseteq E$ by the second definition.

As it follows from the third definition of $\tilde E$, if $(A, i)/k$ is an abelian variety of CM-type $(E, \Phi )$ defined over a number field $k$, then $V = T_0(A)$ is a $k$-vector space for which $Tr(a) = \sum_{\varphi \in \Phi }\varphi (a)$ for all $a\in E$. Hence $\tilde E\subset k$ for any possible field of definition of $A$. 

Note that in the definition above, $V$ can also be regarded as an $E$-vector space on which $\tilde E$ acts. Under this point of view, for any $a\in \tilde E$ we shall denote $\Tr_E(a)$ and $\det_E(a)$ for the trace and determinant of the $E$-linear endomorphism $a: V\, \ra \, V$, $v\mapsto a\cdot v$.



\begin{theorem}[Shimura-Taniyama]\label{ShiTan}
Let $(A, i)$ be an abelian variety of CM-type $(E, \Phi )$ and let $\sigma \in \Aut (\C / \tilde E)$. For any idele $s\in \A^*_{\tilde E, f}$ such that $rec_{\tilde E}(s) = \sigma _{|\tilde E^{ab}}$, there exists a unique $E$-linear isogeny $$\alpha : A \, \lra \, A^{\sigma }$$ such that $$x^{\sigma } = \alpha \,(\mathrm{det}_{\A_{E, f}}(s^{-1})\cdot x), \quad \mbox{ for all } x\in V_f(A).$$
\end{theorem}

In particular, Theorem \ref{ShiTan} asserts that the isogeny class of $(A, i)$ is defined over $\tilde E$.



\subsection{The reflex field of a Shimura datum}

Let $(G, D)$ be a Shimura datum. For any subfield $k\subset \C $, define $$\mathcal C (k) = G(k)\backslash \mathrm{Hom}_k(\G_m, G_k),$$ that is, the set of co-characters of $G$ defined over $k$ up to inner conjugation by elements in $G(k)$.

The group $\Aut (\C /k)$ naturally acts on $\mathcal C(k)$: $c^{\sigma }(a):= c(a)^{\sigma }$ for any $c\in \mathcal C(k), a\in \G_m(\C^*)$ and $\sigma \in \Aut(\C /k)$. 

%\begin{example}
%\begin{itemize}
%\item weight?? SV4... pero encara no se si parlare sobre el weight i tal
%\item 
As an example, let $x\in D$ and $h: \mathbb{S}\,\lra \,G_{\R }$ be the associated morphism. It induces $$\begin{matrix}
\mu_x: & \C^* & \ra & \G_{m, \C }=\C^*\times \C^* & \lra  & G_{\C } \\
  &   z & \mapsto & (z, 1)   & \mapsto & h_{\C }(z,1)
\end{matrix}$$
%\end{itemize}
%\end{example} 

Note for any other point in $D$, the associated morphism is conjugated to $h$ and therefore $\mu _x\in \mathcal C(\C )$ is independent of the choice of $x$ and we shall denote the co-character as $c_D\in \mathcal C(\C )$. It can be seen that actually $c_D$ is defined over a number field\footnote{See \cite[Lemma 12.1]{Mil}.}.

\begin{definition}
The reflex field $E(G, D)$ of the Shimura datum $(G, D)$ is the field of definition of $c_D$ as an element of $\mathcal C(\qbar )$, that is, the field fixed by $\{ \sigma \in \Gal(\qbar /\Q ): c_D^{\sigma }\equiv c_D\in \mathcal C(\qbar )\}$.
\end{definition}

\begin{example}
\begin{itemize}
\item $(T, h)$ where $T$ is a torus and $h: \mathbb S \,\lra \,T_{\R }$ is any morphism. Then $E(T, h)$ is simply the field of definition of $\mu_h$, because it is the single element in its conjugacy class under $T(\C )$.

\item Let $(E, \Phi )$, $\Phi =\{ \varphi_1,...,\varphi_g\}$, be a CM-type. $\\ T = \mathrm{Res}_{E/\Q }\G_m$, $\, T(\Q )=E^*$, $\, T(\R )=(E\otimes_{\Q } \R )^*=\C^*_{\varphi_1}\times ...\C^*_{\varphi_g}$.\footnote{For any choice of $g$-inequivalent embeddings $\{ \varphi_1,...,\varphi_g\}$ we have an isomorphism $E\otimes_{\Q }\R \,\simeq \,\C^g$, $\, a\otimes r\,\mapsto (\varphi_1(a)r,...,\varphi_g(a)r)$.} $\\ \mbox{Define } h_{\Phi }: \mathbb{S}(\R ) = \C^*\,\ra \, T(\R )$, $z\, \mapsto \, (z, \stackrel{g}{...}, z)$. Then 
$$h_{\Phi }\otimes \C : \mathbb{S}(\C ) = \C^*\times \C^*\,\ra \, T(\C ), z\, \mapsto \, (z, \stackrel{g}{...}, z, \bar z, \stackrel{g}{...}, \bar z)$$ and 
$$\begin{matrix}
\mu_h: & \C^* & \ra & \mathbb{S}(\C ) = \C^*\times \C^* & \ra & T(\C )= \C^*_{\varphi_1}\times ...\C^*_{\varphi_g}\times \C^*_{\bar \varphi_1}\times ...\C^*_{\bar \varphi_g}\\   & z&\mapsto &(z, 1)& \mapsto & (z,\stackrel{g}{...},z,1,\stackrel{g}{...},1).
\end{matrix}$$  

Hence $E(T,h_{\Phi})$ is the field fixed by $\{ \sigma \in \Aut (\qbar /\Q)\,:\,\Phi^{\sigma } = \Phi \}$, that is, the reflex field $\tilde E$ of $(E, \Phi )$ that we already defined. $\\ $ At the level of $\tilde E$-rational points, we have 
$\mu_h: \, \tilde E^* \, \ra \, T(\tilde E)= (E\otimes_{\Q }\tilde E)^*$.  
 
\item {\em The Siegel modular variety}. Let $(V, \Psi )$ be a symplectic vector space over $\Q $ of dimension $2 n$, $G=GSp(V, \Psi )$, $D = D^+\cup D^-$ be the set of positive or negative symplectic complex structures $J$ on $V(\R )$. 

Let $V= W\oplus \tilde W$ be a decomposition into totally isotropic vector spaces over $\Q $: $\Psi (W, W) =\{ 0\}$, $\Psi (\tilde W, \tilde W) = \{ 0\}$. Let $W=<e_1, ..., e_n>$, $\tilde W=<\tilde e_1, ..., \tilde e_n>$ be symplectic basis and define $J\in \End(V)$, $J(e_i)=\tilde e_i$, $J(\tilde e_i)=-e_i$.

We had $h_J(a+bi)=a+bJ$. Hence $(h_J\otimes \C )(z_1, z_2) = (a_1+b_1 J, a_2-b_2 J)$ and $$\begin{matrix}
c_{h_J}: & \C^* & \lra & G_{\C }\subset \End(V(\C )) = \End (V^+\oplus V^-) \\ & z & \mapsto & (h_J\otimes \C )(z, 1) = \begin{pmatrix}
z & 0 \\ 0 & 1
\end{pmatrix},
\end{matrix}$$ 
where $V^+$, $V^-$ are the $\pm i$-eigenspaces under $J$.

Since the decomposition $V= W\oplus \tilde W$ is defined over $\Q $, the conjugation class of $c_{h_J}$ remains invariant under the action of $\Aut(\C /\Q )$. Thus $E(G, D) = \Q $.


 
\item Let $(G, D)$ be the Shimura datum attached to a quaternion algebra $B$ over a totally real number field $F$. $\\ h: \mathbb{S}\,\lra \,G_{\R }, \, h(a+b i) = (\begin{pmatrix}
a&b\\-b&a
\end{pmatrix}, \stackrel{r}{...},\begin{pmatrix}
a&b\\-b&a
\end{pmatrix},1, \stackrel{s}{...}, 1)$. $\\ \mbox{Up to inner conjugation: }h_{\C }(z_1, z_2) = (\begin{pmatrix}
z_1&0\\0&z_2
\end{pmatrix}, \stackrel{r}{...},\begin{pmatrix}
z_1&0\\0&z_2
\end{pmatrix},1, \stackrel{s}{...}, 1)$. $\\ \begin{matrix}
\mu_h: & \C^* &\lra &\GL_2(\C )\times \stackrel{g=r+s}{...}\times \GL_2(\C ) \\
&z&\mapsto &(z,1)\mapsto (\begin{pmatrix}
z&0\\0&1
\end{pmatrix}, \stackrel{r}{...},\begin{pmatrix}
z&0\\0&1
\end{pmatrix},1, \stackrel{s}{...}, 1).
\end{matrix}$ $\\ E(G, D) = \qbar^{\{ \sigma \in \Gal(\qbar /\Q ) : \{ \infty_1,...,\infty_r\}^{\sigma } = \{ \infty_1,...,\infty_r\}\}} \, \subseteq \, F $
\begin{itemize}
\item If $B=\M_2(F)$, $r=g, s=0$, $\dim_{\C }(D)=g$, $E(G, D) = \Q $.
\item If $B\otimes_{\Q }\R \simeq \M_2(\R )\times \stackrel{g}{...}\times \M_2(\R )$, $\dim_{\C }(D)=g$, $E(G, D) = \Q $.
\item If $B\otimes_{\Q }\R \simeq \M_2(\R )\times \mathbb{H}\stackrel{g-1}{...}\times \mathbb{H}$, $\dim_{\C }(D)=1$, $E(G, D) = F $.
\end{itemize}
\item Let $i: (G, D)\hookrightarrow (G', D')$ be an inclusion of Shimura data (i.e. $G\hookrightarrow G'$ is a monomorphism and $i$ maps $D$ to $D'$). Then there is a natural morphism of Shimura varieties $Sh(G, D)\hookrightarrow Sh(G',D')$ which is a closed immersion (Deligne) and it is an easy exercise to show that $E(G, D)\supseteq E(G', D')$.
\end{itemize}
\end{example}

\subsection{Canonical models of Shimura varieties}

Let $(G, D)$ be a Shimura datum.

\begin{definition} A point $x\in D$ is a {\em special point} if there exists a torus $T\subset G$ such that $h_x(\C^*)\subset T(\R )$.
\end{definition}

The pair $(T, x)$ is also called a {\em special pair} and $(T, x)\subset (G, D)$ is an inclusion of Shimura data. Note that since $h_x(\C^*)\subset T(\R )$, we have $t h_x t^{-1}= h_x $ for any $t\in T(\R )$ and thus $T(\R )$ fixes the point $x\in D$. Almost conversely, if $T\subset G$ is a {\em maximal} torus and $x\in D$ is a point fixed by the elements of $T(\R )$, then $h(\C^*)\subset \{ g\in G(\R ): g t = t g \mbox{ for all }t\in T(\R )\} = T(\R )$, because $T$ is its own centralizer in $G$; and hence $(T, x)$ is a special pair.

\begin{example}
\begin{itemize}
\item $G=\GL_2$, $D=\cH_1^{\pm }$. The tori in $\GL_2$ are $\G_m \stackrel{a\mapsto a\cdot Id_2}{\hookrightarrow }\mathrm{GL}_2 $ and $T=\mathrm{Res}_{E/\Q } \G_m \stackrel{i}{\hookrightarrow }\mathrm{GL}_2$, for any embedding $E\stackrel{i}{\hookrightarrow }\GL_2(\R )$ of a quadratic field $E/\Q $. Among these, only imaginary quadratic fields provide special pairs (because only in this case we have $T(\R ) = (E\otimes \R )^* = \C^*$), and for each $E\stackrel{i}{\hookrightarrow }\GL_2(\R )$, $T(\R )$ has exactly two fixed points $x, \bar x$ on $\cH_1^{\pm }$.

\item $G=\mathrm{Res}_{F/\Q }(B^*)$, $B$ a quaternion algebra over $F$.  Embeddings $E\hookrightarrow B$ of quadratic extensions $E/F$ with at least one non-real archimedean place provide special pairs $(T, x)$.

\end{itemize}

\end{example} 


Let $(T, x)\subset (G, D)$ be a torus in $(G, D)$. Let $E(x) = E(T, x)$ be the field of definition of $\mu_x$, which is a finite extension of the reflex field $E=E(G, D)$.\footnote{Watch out that we still have not defined the notion of field of definition of a Shimura variety, though clearly $E$ will become a field over which $Sh(G, D)$ admits a canonical model and $E(x)$ will turn out to be the field generated by the coordinates of the point $x\in Sh(G, D)$.} Define the homomorphism 
$$\begin{matrix}
r_x: & \A^*_{E(x)} & \lra    &   T(\A_{\Q }) & \stackrel{\pi }{\twoheadrightarrow } & T(\A_{\Q , f})  \\
     &     s       & \mapsto & \prod_{\sigma : E(x)\hookrightarrow \qbar }\mu_x(s)^{\sigma } & \mapsto & r_x(s).
\end{matrix}$$ 


\begin{definition}\label{canonical} Let $(G, D)$ be a Shimura datum and let $K\subset G(\A_{\Q , f})$ be a compact open subgroup. A {\em canonical model} of $Sh_K(G, D)$ is an algebraic variety $M_K=M_K(G, D)$ defined over $E(G, D)$ with $$M_K(G, D)(\C )\simeq Sh_K(G, D)$$ such that for any special pair $(T, x) \subset (G, D)$ and any $a\in G(\A_f)$:
\begin{itemize}
\item $(x, a)\in M_K(E(x))^{ab}$,
\item $(x, a)^{\mathrm{rec}(s)} = (x, r_x(s^{-1})\cdot a)$ for any $s\in \A^*_{E(x)}$.
\end{itemize}


\end{definition}

\begin{example}
\begin{itemize}
%\item 
\item $(T, x)\,$, $K\subset T(\A_f)\,$, $E=E(x)\,$, $\\ Sh_K(T, x)=T(\Q )\backslash T(\A_f)/K. \\ $
A model over $E$ of a finite set of points $\,=\,$ Action of $\Gal(\bar E/E)$ on it. $\\ $ Define it through
$$\Gal(\bar E/E)\twoheadrightarrow \Gal(E^{ab}/E)\twoheadrightarrow \Gal(H_K/E)\stackrel{\stackrel{\mathrm{rec}_{H_K/E}}{\simeq }}{\leftarrow }E^*\backslash \A^*_{E}/K$$ by $a^s := r_x(s^{-1})\cdot a, \,\quad a\in T(\A_f), \, s\in \A^*_E$.


\item $(E, \Phi )$ CM-type, $\,T\,=\,\mathrm{Res}_{E/\Q }\G_m \,$, $E(h_{\Phi }) \,=\, \tilde E$, $\\ Sh_K(T, h_{\Phi })=E^*\backslash \A^*_{E, f}/K.\\ $ The action of $\tilde E^*\backslash \A^*_{\tilde E, f}/K\,\stackrel{\mathrm{rec}}{\simeq }\,\Gal(H_K/\tilde E)$ on $Sh_K(T, h_{\Phi })$ is provided by $$a^s := r_{h_{\Phi }}(s^{-1})\cdot a = \prod_{\sigma : \tilde E\hookrightarrow \qbar } \mu_{\Phi }(s^{-1})^{\sigma }\cdot a = \mathrm{det}_{\A_{E, f}}(s^{-1})\cdot a $$ for $a\in \A^*_{E, f}, s\in \A^*_{\tilde E, f}$, as in the theorem of Shimura-Taniyama.\footnote{The first equality is the definition of $r_{h_{\Phi }}$. The second equality follows by tracing the definitions of $\mu_{\Phi }$, $\tilde E$ and $\mathrm{det}_{E}$, but it is not immediate. Check the example $E\subset \C $ imaginary quadratic, $\Phi =\{ \varphi \}$ given by the inclusion: Then $\tilde E = E$ and 
$$\begin{matrix}
\mu_{\Phi }: & \tilde E^* & \lra & T(\tilde E)=(E\otimes_{\Q }\tilde E)^* = \tilde E^*\times \tilde E^*\\ & s & \mapsto & (s,1). 
\end{matrix}$$ The Galois action of $\Gal(\tilde E/\Q )$ on $T(\tilde E)$ is given by $\overline{(z_1, z_2)} = (\bar{z_2}, \bar{z_1})$, so that $T(\Q )=E^*\,\stackrel{\iota }{\hookrightarrow }\,T(\tilde E)$ as $\{ (s, \bar s): \,s\in \tilde E\}$. Thus $\prod_{\sigma:\,\tilde E\,\hookrightarrow \qbar }\mu_{\Phi}(s)^{\sigma } = (s,1)\cdot (1, \bar s) = (s, \bar s)=\iota (s)$. Since we can choose $V=\tilde E = E$, we also have $\mathrm{det}_E(V\stackrel{\cdot s}{\ra }V) = s$, as we wished to show. For general $E$, the details are those of the proof of Theorem \ref{ShiTan} of Shimura-Taniyama.}


\end{itemize}
\end{example}

{\bf Modular interpretation of the example:} For $T\,=\,\mathrm{Res}_{E/\Q }\G_m \,$ as above, and any compact open subgroup $K\subset T(\A_f)$, $Sh_K(T, h_{\Phi })$ is a finite set of points which can be regarded as a Shimura variety of PEL-type associated to the commutative algebra $E$. 

It is the moduli space of triples $(A, \iota, s, \eta K)$ as in Section \ref{PEL}, where we recall that $(A, \iota )$ is an abelian variety of CM-type $(E, \Phi )$ up to isogeny.

The automorphism group $\Aut (\C /\tilde E)$ acts on the finite set $Sh_K(T, h_{\Phi })(\C ) = \{ [A, \iota, s, \eta K] \}$: 

$(A, \iota, s, \eta K)^{\sigma }=(A^{\sigma }, \iota^{\sigma }: E\,\hookrightarrow\,\End(A^{\sigma }), s^{\sigma }, \eta^{\sigma }: V(\A_f)\ra V_f(A)\ra V_f(A^{\sigma }))$.

Since $\sigma $ fixes $\tilde E$ one checks that $(A^{\sigma, \iota^{\sigma }})$ is again an abelian variety of CM-type $(E, \Phi )$.

A reformulation of the main Theorem of Complex Multiplication \ref{ShiTan} shows the following proposition:

\begin{proposition}\label{rule}
By means of the identification of sets $$Sh_K(T, h_{\Phi }) = T(\Q )\backslash \{ x\}\times T(\A_f)/K \stackrel{\sim }{\lra }\{ [A, \iota, s, \eta K] \},$$ the Galois action induced on $Sh_K(T, h_{\Phi })$ is: $$(x, a)^{\sigma } = (x, r_{h_{\Phi }}(s^{-1})\cdot a),$$ where $\sigma _{|\tilde E^{ab}} = \mathrm{rec}(s)$.
\end{proposition}

We are now able to state the main theorem of this section. 





\begin{theorem}
Let $(G, D)$ be a Shimura datum. For any compact open subgroup $K\subseteq G(\A_f)$, there exists a unique canonical model $M_K(G, D)$ of $Sh_K(G, D)$ over $E(G, D)$, up to (unique) isomorphism over $E(G, D)$. 
\end{theorem}

$\\ $
{\bf Outline of the proof for Shimura varieties of Hodge type.}
$\\ $

We first discuss the case of {\bf Siegel modular varieties} $\mathcal S_K(G, D)$ attached to a symplectic vector space $(V, \Psi )$ of dimension $2 n$:

We already saw that the reflex field of $(G, D)$ is $E(G, D)=\Q $. Thus we wish to prove that $\mathcal S_K(G, D)$ admits a canonical model $\mathcal M_K / \Q $ over $\Q $. 

Thanks to the moduli interpretation, we are able to describe an action of $\Aut (\C /\Q )$ on the set of complex points $\mathcal S_K(\C )$.

Indeed, there is a one-to-one correspondence $$\mathcal S_K(\C )\quad \leftrightarrow \quad \{ [A, s, \eta K] \}$$ and $\Aut (\C /\Q )$ acts on it: 

$$[A, s, \eta K] \, \mapsto \, [A^{\sigma }, s^{\sigma }, \eta^{\sigma } K], \mbox{ for any }\sigma \in \Aut (\C /\Q ).$$

Recall that we already saw that PEL-Shimura varieties are of Hodge type and hence Shimura subvarieties of a Siegel modular variety. 

In our case, we have that for any CM-type $(E, \phi )$ with $[E:\Q ]=2n$, the Shimura variety of PEL-type attached to $(\mathrm{Res}_{E/\Q }\G_m, h_{\Phi })$ is a subvariety of $\mathcal S_K$. These are finite sets of points and they are special. Proposition \ref{rule} above shows that the action of $\Aut (\C /\Q )$ on these special points on $\mathcal S_K$ behaves according to the rule required in the definition of a canonical model for $\mathcal S_K$.

Almost conversely, any special point $[A, s, \eta K]\in \mathcal S_K(\C )$ corresponds to an abelian variety $A/\C $ such that $\End (A) = E_1\times ...\times E_m$, where $E_j$ are CM-fields and $\sum [E_j:\Q ] = 2 \dim (A) = 2 n$. Proposition \ref{rule} readily extends to this more general case and hence we obtain that the action of $\Aut (\C /\Q )$ on all special points of $\mathcal S_K(\C )$ is as it should according to Definition \ref{canonical}

$\\ $
{\bf Descent criteria.} It remains left proving that the action of $\Aut (\C /\Q )$ on $\mathcal S_K(G, D)$ is a true Galois action on a certain algebraic variety $\mathcal M_K / \Q $ such that $\mathcal M_K\otimes \C \simeq \mathcal S_K(G, D)$. 


Let $k$ be a field of characteristic $0$ and let $\mathcal K / k$ be an algebraically closed field containing $k$. Let $X/\mathcal K$ be an algebraic variety. 

For any $\sigma \in \Aut (\mathcal K/k)$ there is a well-defined algebraic variety $X^{\sigma }$ and a map of sets $$\begin{matrix}
X(\mathcal K) & \lra & X^{\sigma }(\mathcal K) \\ x & \mapsto & x^{\sigma }. 
\end{matrix}$$

Locally at affine open subsets, $X^{\sigma }$ is obtained from $X$ by conjugating by $\sigma $ the defining polynomials of $X$. If $x\in X(\mathcal K)$, the coordinates of the point $x^{\sigma }\in X^{\sigma }$ are the conjugate coordinates of $x$ by $\sigma $.

Assume that $X$ is equipped together with an action of $\Aut (\mathcal K/k)$ on the set $X(\mathcal K)$. Let us denote the action by $(\sigma , x )\, \mapsto \, \sigma (x)\in X(\mathcal K)$. 

For instance, if $X_0$ is a variety over $k$, then $\sigma (x):=x^{\sigma }$ defines an action of $\Aut (\mathcal K/k)$ on $X=X_0\otimes \mathcal K$.

\begin{theorem}\label{Weil} Let $X$ be a quasi-projective variety over $\mathcal K$ together with an action of $\Aut (\mathcal K/k)$ on $X(\mathcal K)$ such that
\begin{itemize}


\item (Regularity) The morphism of sets $$\begin{matrix}
f_{\sigma }: & X &\, \lra \, & X^{\sigma } \\ & x & \mapsto & \sigma ^{-1}(x^{\sigma })
\end{matrix}$$ is a regular algebraic isomorphism.


\item (Continuity) There exist points $x_1$, ..., $x_n\in X(\mathcal K)$ and a finitely generated extension $L/k$ in $\mathcal K$ such that \begin{itemize}
\item $\sigma (x_i)=x_i$ for all $\sigma \in \Aut (\mathcal K/L)$,
\item The only automorphism $\alpha \in \Aut(X)$ fixing all $x_i$ simultaneously is $\alpha = Id$.
\end{itemize}  

\end{itemize}

Then there exists a model $X_0$ over $k$ of $X$.
\end{theorem}

Theorem \ref{Weil} applies to the Siegel modular variety $\mathcal S_K=\mathcal S_K(G, D)/\C $ in order to show that it admits a model over $\Q $:

\begin{itemize}
\item $\mathcal S_K$ is a quasi-projective variety over $\C $ by Baily-Borel's Theorem \ref{BailyBorel}.

\item {\bf Regularity: } Checking this condition exploits the moduli interpretation of $\mathcal S_K$ and uses the theory of local systems on topological manifolds and families of abelian varieties. We refer the interested reader to \cite[p.\,100-103]{Mil}.

\item {\bf Continuity:} Let $x\in D$ be a special point. The real approximation Theorem (cf.\,\cite[Theorem 7.7]{PlRa}) asserts that for any connected algebraic group over $\Q $, $G(\Q )$ is dense in $G(\R )$. Hence, since $D$ is a quotient of $G(\R )$, the set of points $\{ [x, a] : a\in G(\A_f)\}\subset G(\Q )\backslash D\times G(\A_f)/K$ is dense in $\mathcal S_K$ for the analytic and thus also the Zariski topology. 

Hence only the trivial automorphism $Id\in \Aut (\mathcal S_K)$ fixes all points $\{ [x, a] : a\in G(\A_f)\}$ simultaneously. It can be shown that $\Aut (\mathcal S_K)$ is a finite group -recall we only consider $K\subset G(\A_f)$ such that the resulting congruence groups are torsion free!. Therefore there exists a finite set $\{ [x, a_1], ..., [x, a_n]\}$ such that only $\alpha = Id$ fixes all them.

Finally, the main Theorem of Complex Multiplication implies that $[x, a_1]$, ..., $[x, a_n]$ are fixed by $\Aut (\C /E(x)')$, where $E(x)'$ denotes a finite abelian extension of $E(x)$.

\end{itemize}

Hence $\mathcal S_K$ admits a model $\mathcal M_K$ over $\Q $, which is canonical in the sense of Definition \ref{canonical}.

The proof of the existence of a canonical model for Shimura varieties of Hodge type follows the same pattern, since they are subvarieties of Siegel modular varieties and they also have a moduli interpretation in terms of abelian varieties. 

The main technical difficulty is the definition of the Galois action on the Hodge tensors $s_i$ on the abelian varieties $A$ arising in the moduli interpretation. If the Hodge conjecture were true, any Hodge tensor $s$ on $A$ would be the cohomology class $c(Z)$ of an algebraic cycle $Z$ on $A$ and one could define $s^{\sigma } = c(Z^{\sigma })$. But the conjecture is not known to hold and Deligne succeeded to give a definition of $s^{\sigma }$ not in terms of algebraic cycles - check \cite[Theorem 14.13]{Mil}.  

\begin{thebibliography}{99}



\bibitem{Hel}
S.\ Helgason, Differential Geometry and Symmetric spaces, Academic Press, 1962.

\bibitem{Knapp}
A.\ W.\ Knapp, Lie groups. Beyond an Introduction, Birkhauser {\bf 140}, 1996.

\bibitem{Mil} 
J.\ S.\ Milne, Introduction to Shimura varieties, available at www.jmilne.org/math

\bibitem{PlRa} 
V.\ Platonov, A.\ Rapinchuk, Algebraic groups and number theory, {\em Pure and Applied Mathematics} {\bf 139}, Academic Press, 1994.


\end{thebibliography}
\end{document}
 
