\documentclass[a4,portrait]{seminar}
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\begin{document}
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\begin{slide*}
\begin{center} 
\Large \sc On the Andr\'e--Oort conjecture.
\end{center}
\bigskip
{\bf Conjecture.}
Let $(G,X)$ be a Shimura datum. Let $K$ be 
a compact open subgroup of~$G(\AAf)$ and let $S$ be a set of special points 
in $\Sh_K(G,X)(\CC)$. Then every irreducible component of the Zariski 
closure of $S$ in $\Sh_K(G,X)_\CC$ is a subvariety of Hodge type. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Shimura varieties.
\end{center}
\bigskip
$\SS:=\Res_{\CC/\RR}\GG_{{\rm m},\CC}$, algebraic group over $\RR$. 
Equivalence: representations of $\SS$ and $\RR$-Hodge structures. 

$G:=$ a reductive affine algebraic group over $\QQ$. 

$X:=$ a $G(\RR)$-conjugacy class in $\Hom(\SS,G_\RR)$, satisfying Deligne's 
three conditions. 

$\Sh_K(G,X)(\CC):=G(\QQ)\backslash(X\times G(\AAf)/K)$, for 
$K\subset G(\AAf)$ compact open subgroup. 

$\Sh_K(G,X)_\CC:=$ the corresponding complex algebraic variety. 

$\Sh(G,X)_\CC:=\lim_{K}\Sh_K(G,X)_\CC$, has $G(\AAf)$-action. 

\end{slide*}

\begin{slide*}
\begin{center} 
\sc Subvarieties of Hodge type.
\end{center}
\bigskip
A closed subvariety $Z\subset\Sh_K(G,X)_\CC$ is of Hodge type if it is 
an irreducible component of the image of some: 
$$
\begin{array}{cc}
\Sh(G',X')_\CC \overset{f}{\lto} \Sh(G,X)_\CC \overset{g}{\lto} & 
\Sh(G,X)_\CC \\
& \downarrow \\
& \Sh_K(G,X)_\CC
\end{array}
$$
with $f\colon G'\to G$ and $g$ in $G(\AAf)$. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Special points.
\end{center}
\bigskip
For $h$ in $X$, $\MT(h):=$ the smallest $H\subset G$ 
such that $h(\SS)\subset H_\RR$. 

$h$ in $X$ is special iff $\MT(h)$ is commutative. 

$x$ in $\Sh_K(G,X)_\CC$ is special iff $x$ is of Hodge type. 
\end{slide*}

\begin{slide*}
{\bf Conjecture.}
Let $(G,X)$ be a Shimura datum. Let $K$ be 
a compact open subgroup of~$G(\AAf)$ and let $S$ be a set of special points 
in $\Sh_K(G,X)(\CC)$. Then every irreducible component of the Zariski 
closure of $S$ in $\Sh_K(G,X)_\CC$ is a subvariety of Hodge type. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Results.
\end{center}
\bigskip
{\sc Moonen}: moduli space of abelian varieties; true for $S$ for which there 
exists a prime number $p$ at which all $s$ in $S$ have an ordinary reduction 
of which they are the canonical lift. 

{\sc Edixhoven}: moduli space of pairs of elliptic curves, i.e., products of 
two modular curves, assuming GRH. 

{\sc Andr\'e}: products of two modular curves. 

{\sc Yafaev}: products of two Shimura curves associated to quaternion 
algebras over $\QQ$ (GRH). 

{\sc Belhaj Dahman}: work in progress on jacobians of $y^n=x(x-1)(x-\lambda)$. 

{\sc Edixhoven}: Hilbert modular surfaces, products of modular curves (GRH). 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Two principles.
\end{center}
\bigskip
{\bf 1.} Level structures don't matter: 
for $K\subset K'$, $Z\subset\Sh_K(G,X)_\CC$ is of Hodge 
type if and only if its image in $\Sh_{K'}(G,X)_\CC$ is. 

{\bf 2.} Irreducible components of intersections of 
subvarieties of Hodge type are again of Hodge type. 
\end{slide*}

\begin{slide*}
\begin{center} 
\Large \sc The case of $(\PP^1)^n$.
\end{center}
\bigskip
The subvarieties of Hodge type of $(\PP^1)^n$ are products of: 

{\bf 1.} a CM-point in $\PP^1$; 

{\bf 2.} closure of the image of $\HH^{\pm}\to(\PP^1)^m$ under 
$\tau\mapsto(j(g_1\tau),\ldots,j(g_m\tau))$, with the $g_i$ in $\GL_2(\QQ)$. 
\end{slide*}

\begin{slide*}
{\bf Theorem.} {\sl Assume GRH. Let $n\geq0$, let $d\geq0$ and 
$X\subset(\PP^1_\CC)^n$ a subvariety of dimension $d$, containing a dense 
subset of CM points. Then $X$ is of Hodge type.} 

The proof is by induction on $n$ and $d$. 
$$
\text{$X$ is an irreducible component of:} 
\bigcap_{\overset{I\subset\{1,\ldots,n\}}{|I|=d+1}}\pr_I^{-1}\pr_I X
$$
This reduces to the case: $d=n-1$ and all $\pr_I$ with $|I|=d$ are 
surjective. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc The case $n=2$.
\end{center}
\bigskip
$X_\QQ:=$ the image of $X$ in $(\PP^1_\QQ)^2$. 

$d_1$, $d_2$ $:=$ the degrees of the projections $X_\QQ\to\PP^1_\QQ$. 

Take $x=(x_1,x_2)$ in $X$ CM with $\max(|\discr(\End(x_i))|)$ big 
with respect to $d_1$ and $d_2$. 

Take (GRH) a prime number $p$ such that: 
\begin{enumerate}
\item $p\geq 5$, $p\geq d_i$;
\item $p$ split in $\End(x_i)$;
\item $p$ small w.r.t. $\max(|\discr(\End(x_i))|)$. 
\end{enumerate}

Then $\Gal(\Qbar/\QQ)\cdot x\subset X_\QQ\cap T_pX_\QQ$, 
and $|\Gal(\Qbar/\QQ)\cdot x| > X_\QQ\cdot T_pX_\QQ$, 
hence $X_\QQ\subset T_pX_\QQ$. 

Then one shows that $X$ is of Hodge type. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc The case $n=3$.
\end{center}
\bigskip
We have $d=2$, all $p_I$ with $|I|=2$ surjective. We should get a 
contradiction. 

As before: $X_\QQ$, $d_1$, $d_2$, $d_3$. 

Take $x$ in $X$ CM with $\Gal(\Qbar/\QQ)\cdot x$ big w.r.t. the~$d_i$. 

Take (GRH) $p_1$ prime such that: 
\begin{enumerate}
\item $p_1\geq 5$, $p_1\geq d_i$;
\item $\Gal(\Qbar/\QQ)\cdot x\subset X_\QQ\cap T_{p_1}X_\QQ$;
\item $p_1$ small w.r.t. $|\Gal(\Qbar/\QQ)\cdot x|$. 
\end{enumerate}
If $X_\QQ\subset T_{p_1}X_\QQ$, get a contradiction (use $T_{p_1}X_\QQ$ 
irreducible, and density of Hecke orbits). Hence suppose that 
$X_\QQ\not\subset T_{p_1}X_\QQ$. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc The case $n=3$, continued.
\end{center}
\bigskip
Take $X_1$ an irreducible component of $X_\QQ\cap T_{p_1}X_\QQ$ 
containing~$x$ ($X_1$ is a curve). 

For a suitable $p_2$ (GRH) one has $X_1\subset T_{p_2}X_1$, hence 
$X_1$ is of Hodge type. 

This gives a Zariski dense set of curves of Hodge type in $X$. 

One finishes by classifying such curves. Those which project surjectively 
under all $p_i$ are of the form: 
$$
C_{(n_1,n_2,n_3)}:=
\{(E_1,E_2,E_3)\;|\;\text{$\exists$ $E$, $f_i\colon E\to E_i$}\}
$$
with $\ker(f_i)\cong\ZZ/n_i\ZZ$ for a fixed triplet $(n_1,n_2,n_3)$. 

$\deg(\pr_i\colon C_{(n_1,n_2,n_3)}\to\pr_iC_{(n_1,n_2,n_3)})\geq\phi(n_i)$ 
gives the desired contradiction. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc The case $n>3$.
\end{center}
\bigskip
More successive intersections, the end is more difficult. 

As for $n=3$, get a dense set of curves of Hodge type. Then consider 
$X\cap\{x\}\times(\PP^1)^{n-1}$ with $x$ CM. 

\end{slide*}

\begin{slide*}
\begin{center} 
\Large \sc The case of Hilbert modular surfaces.
\end{center}
\bigskip
$K:=$ a real quadratic extension of $\QQ$. 

$G:=\Res_{O_K/\ZZ}(\GL_2(O_K))$. 

$G(\RR)=\GL_2(\RR\otimes O_K)=\GL_2(\RR)^2$. 

$S_\CC:=G(\QQ)\backslash ((\HH^{\pm})^2\times G(\AAf)/G(\Zhat))$ 

$S_\QQ:=$ the canonical model over $\QQ$; it is the moduli space for 
abelian surfaces with $O_K$-action. 

{\bf Theorem.} {\sl Assume GRH. Let $C\subset S_\CC$ be an irreducible curve 
containing infinitely many CM points. Then $C$ is of Hodge type. }
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Generic Mumford-Tate groups.
\end{center}
\bigskip
The generic Mumford-Tate group of $S_\CC$ is the group $G'$ given by: 
$$
\begin{array}{ccc}
G' & \into & G_\QQ \\
\downarrow & \square & \downarrow\scriptstyle{\det} \\
\Gm_\QQ & \into & \Res_{K/\QQ}\Gm_K
\end{array}
$$

$M:=$ the generic Mumford-Tate group of $C$. 

If $M\neq G'$, then $C$ is of Hodge type, because of dimensions. 

Hence suppose that $M=G'$. Then $C$ is not of Hodge type, and we have to get 
a contradiction. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Algebraic monodromy.
\end{center}
\bigskip
$H:=$ the connected component of the Zariski closure of the image of 
$\pi_1(C')$ ($C'$ a suitable finite cover of $C$), under the monodromy map. 

{\bf Theorem (Andr\'e).} $H=M^\der$ ($=G^\der=\Res_{K/\QQ}\SL_{2,K}$). 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Finite monodromy.
\end{center}
\bigskip
{\bf Theorem (Nori).} The closure of the image of $\pi_1(C')$ in 
$H(\Zhat)$ is open. 
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Irreducibility.
\end{center}
\bigskip
For $p$ prime, let $T_p$ be the Hecke correspondence on $S_\QQ$, 
induced by 
$(\begin{smallmatrix}p&0\\0&1\end{smallmatrix})$ in $G(\QQ_p)$. 

{\bf Corollary.} {\sl For $p$ big enough, $T_pC$ is irreducible.}
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Galois action.
\end{center}
\bigskip
Let $A/\CC$ be an abelian surface with $O_K$-action, and with CM. 

Then $R:=\End_{O_K}(A)$ is an order in a CM field $L$ of degree 2 over $K$. 

The set of $A'/\CC$ with $\End_{O_K}(A')=R$ and with the same CM type 
as $A$ is a $\Pic(R)$-torsor. 

$E:=$ the reflex field of $A$ with its $O_K$ action. 

$\Gal(\Qbar/E)$ acts via the reflex type norm: 
$E^*\backslash\AA_{\rf,E}^*\lto L^*\backslash\AA_{\rf,L}^*$
\end{slide*}

\begin{slide*}
\begin{center} 
\sc Analytic number theory.
\end{center}
\bigskip
{\bf Theorem (Siegel).} {\sl $|\Pic(R)| = |\discr(R)|^{1/2+o(1)}$ as 
$|\discr(R)|\to\infty$. }

{\bf Corollary.} 
{\sl There exists $\eps>0$ and $c>0$ such that for $x$ on $S$ a CM point 
we have 
$$
|\Gal(\Qbar/\QQ)\cdot x|\geq c|\discr(R)|^\eps. 
$$ 
}

{\bf Theorem (L.O.M.).} 
{\sl Assume GRH. For $x$ with $|\discr(R)|\gg0$ there are many $p$ split in 
$R$ with $p$ at most polynomial in $\log|\discr(R)|$.}
\end{slide*}

\begin{slide*}
\begin{center} 
\sc End of the proof.
\end{center}
\bigskip
For $x$ with $|\discr(R)|$ big enough, there is a $p$ such that: 
$$
T_px\ni \sigma x, \quad \text{$T_pC$ is irreducible}
$$
for some $\sigma$ in $\Gal(\Qbar/\QQ)$. 

Hence $T_pC_\QQ=C_\QQ$, which contradicts the density of 
the $T_p$-orbit of $x$ in its connected component of~$S(\CC)$. 

\end{slide*}



\begin{slide*}

\end{slide*}

\end{document}