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\title[On the generalised Andr\'e--Pink--Zannier conjecture]{On the generalised Andr\'e--Pink--Zannier conjecture.}

\author{\firstname{Rodolphe} \lastname{Richard}}
\address{UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK}
\email{r.richard@ucl.ac.uk}

\author{\firstname{Andrei} \lastname{Yafaev}\IsCorresp}
\address[1]{UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK}
\email{yafaev@ucl.ac.uk}

\thanks{This project was funded by Leverhulme Trust grant RPG-2019-180}
\CDRGrant[Leverhulme Trust]{RPG-2019-180}
\keywords{Shimura varieties, Hecke orbits, Zilber-Pink, Heights, Siegel sets, Mumford--Tate conjecture, Adelic linear groups}
\subjclass{03C64, 11G18, 11G50, 11F80, 14L30, 20G35, 15A16, 14G35}

\begin{abstract}
We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ``weakly adelic Mumford--Tate hypothesis'' and prove the generalised Andr\'e--Pink--Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila--Zannier strategy.
\end{abstract}

\begin{altabstract}
On introduit et \'etudie la notion d'orbite de Hecke g\'en\'eralis\'ee dans une vari\'et\'e de Shimura. On d\'efinit une notion de hauteur sur une telle orbite et \'etudie ses properi\'et\'es. On obtient une borne inf\'erieure pour la taille des orbites Galoisiennes de points dans une orbite de Hecke g\'en\'eralisee en termes de cette fonction hauteur en admettant \og la conjecture de Mumford--Tate faiblement ad\'elique\fg{} et on d\'emontre la conjecture de Andr\'e--Pink--Zannier g\'en\'eralis\'ee (un cas particulier de la conjecture de Zilber-Pink) en utilisant la strat\'egie de Pila--Zannier.
\end{altabstract}

\begin{document}
\maketitle

This note describes the results of~\cite{RY-APZ}. We refer to~\cite{Del79} for the language of Shimura varieties, and to~\cite[\S3]{UllmoAxLin} for the notion of weakly special subvarieties.

\section{Generalised Andr\'e--Pink--Zannier conjecture and Main results}Let~$(G,X)$ be a Shimura datum, let~$K\leq G(\AAA_f)$ be a compact open subgroup, and let~$S=Sh_K(G,X)=G(\QQ)\sous X\times G(\AAA_f)/K$ be the associated Shimura variety. Let~$x_0\in X$ (which we view as a morphism~$\SSS\to G_\RR$) and let~$M\leq G$ be its Mumford--Tate group.



In the following definition~$\Hom(M,G)(\QQ)$ denotes the set of algebraic group morphisms \emph{defined over~$\QQ$}.
\begin{defi}\label{defi:Hecke}
We define the \emph{Generalised Hecke orbit} $\cH(x_0)$ of $x_0$ \emph{in $X$} as
\[
\cH(x_0) := X\cap \{ \phi \circ x_0 : \phi \in \Hom(M,G)(\QQ)\}
\]
and the \emph{Generalised Hecke orbit} $\cH([x_0, g_0])$ of $[x_0, g_0]$ \emph{in $\Sh_K(G,X)$} as
\[
\cH([x_0,g_0]) := \{ [x,g] : x\in \cH(x_0), g\in G(\AAA_f) \}.
\]
\end{defi}

This notion of Hecke orbit is the most general; we refer to~\cite[\S2.5]{RY-APZ} for a comparison with some other notions of Hecke orbits. For instance, if~$A$ is an abelian variety with Mumford--Tate group~$M\leq G:=GSp(2\dim(A))$, every abelian variety~$B$ isogenous to~$A$ can be obtained from a morphism~$M\to G \in \cH(x_0)$. Definition~\ref{defi:Hecke} is also readily\footnote{Or combine~\cite[Prop.~3.6]{Pink} and~\cite[Prop.~2.15]{RY-APZ}.} at least as general than the notion of generalised Hecke orbit defined by~\cite[\S3]{Pink}.

The following conjecture extends questions of~\cite{Andre} of André,~\cite{Y} of the second author and~\cite{Pink} of Pink. Similar questions have been considered by Zannier~\cite{Zannier} in the context of abelian schemes. An argument of Orr (\cite{Orr}) shows that it is a special case of the Zilber-Pink conjecture for pure Shimura varieties.

\begin{conj}[Generalised André--Pink--Zannier, {\cite[Conj.~1.1]{RY-APZ}}] \label{APZ}
Let $S$ be a Shimura variety and $\Sigma$ a subset of a generalised Hecke orbit in $S$. Then the irreducible components of the Zariski closure of\/ $\Sigma$ are weakly special subvarieties.
\end{conj}

For a sufficiently large field~$E$ of finite type over~$\QQ$ we have the following (cf.~\cite{UY_General}):~$S$ and~$s_0=[x_0,1]$ are defined over~$E$ and there exists a Galois representation~$\rho_{x_0}:\Gal(\ol{E}/E)\to M(\AAA_f)\cap K$ such that
\[
\forall \sigma\in \Gal(\ol{E}/E),~g\in G(\AAA_f),~\sigma([x_0,g])=[x_0,\rho_{x_0}(\sigma)\cdot g].
\]

Our main result is the following.
\begin{theo}[{\cite[Thm.~1.2]{RY-APZ}}]\label{thm intro}
In the above situation, let~$U=\rho_{x_0}(\Gal(\ol{E}/E))$. We assume:
\begin{equation}\label{waMT}
\exists C>0, \forall p, [K\cap M(\QQ_p):U\cap M(\QQ_p)]\leq C.
\end{equation}
Then, for any~$\Sigma\subseteq \cH(x_0)$, every irreducible component of~$\ol{\Sigma}^{Zar}$ is weakly special.
\end{theo}

We call~\eqref{waMT} the weakly adelic Mumford--Tate hypothesis. It is related to a weak form of the conjecture~\cite[11.4?]{SerreConj} due to Serre. The assumption~\eqref{waMT} is known in many cases. This leads to the following unconditional results.

\subsection{}

Combining Theorem~\ref{thm intro} with~\cite[Thm.~A$\MK$(i)]{CM} we have the following, which strictly contains a 2005 result of Pink~\cite[\S7]{Pink} (and~\cite[Thm.~B]{CK}).

\begin{theo}[{\cite{RY-APZ}}]\label{th15}
The conjecture~\ref{APZ} is true if~$S$ is of abelian type, and~$\Sigma$ contains a point~$s$
which satisfies the Mumford--Tate conjecture (at some~$\ell$, in the sense of~\cite{UY_General}).
\end{theo}

The assumptions of Theorem~\ref{th15} are satisfied in the case where~$S=\mathcal{A}_g$ and~$\Sigma$ contains a point~$[A]$, where the abelian variety~$A$ satisfies the Mumford--Tate conjecture (at some prime~$\ell$). Examples of such abelian varieties include:~$\dim(A)\leq 3$; or when~$\dim(A)$ is odd and~$\End(A)\simeq \ZZ$. More examples are given in~\cite{Pink98}, and many examples are mentioned in~\cite[\S2.4]{DL}.

\subsection{}

The following is not restricted to Shimura varieties of abelian type.

\begin{theo}[{\cite[Thm.~1.2]{G}}]\label{Baldi MT}
We decompose the adjoint datum~$(M^{ad},X_{M^{ad}})$ of~$(M,X_M):={(M,M(\RR)\cdot x_0)}$ as a product
\[
(p_1,\dots,p_f):(M^{ad},X_{M^{ad}})\simeq (M_1,X_1)\times \ldots \times (M_f,X_f)
\]
with respect to the~$\QQ$-simple factors~$M_i$ of~$M^{ad}$.

Assume that for some compact open subgroups~$K_i\leq M_i(\AAA_f)$
\[
\forall i\in\{1;\ldots;f\},\,[p_i\circ ad_M (x_0)]\in Sh_{K_i}(M_i,X_i)(\CC)\smallsetminus Sh_{K_i}(M_i,X_i)(\ol{\QQ}).
\]
Then~$U=\rho_{x_0}(\Gal(\ol{E}/E))$ satisfies~\eqref{waMT}.
\end{theo}

Theorems~\ref{th16} and~\ref{th17} follow from the combination of~Theorem~\ref{thm intro} with~Theorem~\ref{Baldi MT}.

\begin{theo}[\cite{RY-APZ}]\label{th16}
The conjecture~\ref{APZ} is true if\/ $\Sigma$ contains a $\ol{\QQ}$-Zariski generic point~$s$ of a special subvariety~$Z\subseteq S$, namely: for every proper subvariety~$V\subsetneq Z$ defined over~$\ol{\QQ}$, we have~$s\not\in V(\CC)$.
\end{theo}

\begin{theo}[\cite{RY-APZ}]\label{th17}
The conjecture~\ref{APZ} is true if\/ $M^{ad}$ is~$\QQ$-simple and~$\Sigma$ contains a point~$s$ in~$S(\CC)\smallsetminus S(\ol{\QQ})$.
\end{theo}

In the case~$M^{ad}=\{1\}$ we recover a result of~\cite{EdYa} and~\cite{KY}.

\begin{theo}[\cite{EdYa} and~\cite{KY}]\label{thmEdYa}
The conjecture~\ref{APZ} is true if\/ $\Sigma$ contains a special point.
\end{theo}

\subsection{Previous results towards Conjecture~\ref{APZ}}

For the history of the Conjecture and previous results, see the introduction of~\cite{RY-APZ}.

The results~\cite{Pink,RPHD1} towards~Conjecture~\ref{APZ}, based on equidistribution of Hecke points, are limited to the case where~$s$ is Hodge generic, and to an assumption similar to~\eqref{waMT} (\cite[Def.~6.3]{Pink}, \cite[\S6-7, pp.~57--59]{RPHD1}). The case of general~$S$, in the Hodge generic case~$Z=S$, can be treated using the results of~\cite{RPHD1} and an extension of~\cite[Prop 3.5]{R1}.

For general~$Z$ and~$S$, Conjecture~\ref{APZ} was obtained, under an assumption substantially weaker than~\eqref{waMT}, but for a much more restrictive notion of Hecke orbits (the~``$\mathcal{S}$-Hecke orbit'' for a finite set of primes~$\mathcal{S}$). See~\cite{APZS} for an approach of based on Ratner's theorems, and see~\cite{OrrPhD} for an approach based on the Pila--Zannier strategy.


\section{Some useful results}

The proof of Theorem~\ref{thm intro} in Section~\ref{Strategy outline} relies on the following results, which are of independent interest.

\subsection{Geometric Hecke orbits}

We define~$W=G\cdot \phi_0\subseteq \Hom(M,G)$ the conjugacy class of the injection~$\phi_0:M\hookrightarrow G$, as an algebraic variety. We define the geometric Hecke orbit as
\[
\cH^g(x_0) =X\cap \{ \phi \circ x_0 : \phi \in W(\QQ)\} \subset \cH(x_0).
\]
The following is an essential tool in reduction steps in the proof of Theorem~\ref{thm intro}.

\begin{theo}[{\cite[\S2.4]{RY-APZ}}]\label{theo general union finie geometrique}
The generalised Hecke orbit $\cH(x_0)$ is a union of finitely many geometric Hecke orbits.
\end{theo}

\subsection{Height functions and estimates}

Let~$\mathfrak{m}$ and~$\mathfrak{g}$ be Lie algebras over~$\QQ$, and choose $\ZZ$-structures on~$\mathfrak{m}$ and~$\mathfrak{g}$. We denote by~$\mathfrak{m}_{\widehat{\ZZ}}\leq \mathfrak{m}\tens\AAA_f$ and~$\mathfrak{g}_{\widehat{\ZZ}}\leq \mathfrak{g}\tens\AAA_f$ the corresponding~$\widehat{\ZZ}$-structures and define
\[
H_{\AAA_f}:\Hom(\mathfrak{m},\mathfrak{g})\tens\AAA_f\to \ZZ_{\geq1}\quad \text{by}\quad \Phi\mapsto\min\{n\in\ZZ_{\geq1} : n\cdot \Phi(\mathfrak{m}_{\widehat{\ZZ}})\subseteq \mathfrak{g}_{\widehat{\ZZ}}\}.
\]

\subsubsection{}
We choose $\ZZ$-structures in such a way that~$\mathfrak{g}_{\widehat{\ZZ}}$ is invariant under the adjoint action of~$K$ and~$\mathfrak{m}_{\widehat{\ZZ}}=\mathfrak{m}\tens \AAA_f\cap \mathfrak{g}_{\widehat{\ZZ}}$.

\begin{prop}[{\cite[\S4.3]{RY-APZ}}]\label{Gal invariance}
There is a well defined function~$H_{[x_0,1]}:\cH([x_0,1])\to \ZZ_{\geq1}$ given by
\[
[\phi\circ x_0,g]\mapsto H_{\AAA_f}(d(g^{-1}\cdot \phi\cdot g)).
\]
This function is constant on the orbits of~$\Gal(\ol{E}/E)$ in~$\cH([x_0,1])$.
\end{prop}

For a set~$A$ and two functions~$f,g:A\to \RR_{\geq0}$, we write~$f\dom g$, resp.~$f\approx g$ when
\[
\exists b,c,d\in\RR_{>0}, \forall a\in A, f(a)\leq d+ b\cdot g(a)^c,
\quad\text{resp. } f\dom g\text{ and }g\dom f.
\]
The following estimate requires the assumption~\eqref{waMT}.

\begin{theo}[{\cite[Thm.~6.4 and~Prop.~3.6]{RY-APZ}}]
As~$s=[\phi\circ x_0,g]$ ranges through~$\cH([x_0,1])$, we have
\begin{equation}\label{Galois estimate}
\abs{\Gal(\ol{E}/E)\cdot s}\approx H_{\AAA_f}(d(g^{-1}\cdot \phi\cdot g)).
\end{equation}
\end{theo}

\subsubsection{}
Only the lower bound~$H_{\AAA_f}(d(g^{-1}\cdot \phi\cdot g))\dom \abs{\Gal(\ol{E}/E)\cdot s}$ is needed for the proof of Theorem~\ref{thm intro}. This lower bound is derived from the following.
\begin{theo}[{\cite[Thm.~B.1]{RY-APZ}}]\label{general global bounds}
Let~$M\leq GL(N)$ be a linear algebraic subgroup defined over~$\QQ$, denote by~$\phi_0: M\to GL(N)$ the identity morphism and~$W$ the~$GL(N)$-conjugacy class of~$\phi_0$. We choose a basis of~$\mathfrak{m}$ such that~$\mathfrak{m}_{\widehat{\ZZ}}=\mathfrak{m}\tens \AAA_f\cap \mathfrak{gl}(N,\widehat{\ZZ})$, and define~$ M(\widehat{\ZZ})=M(\AAA_f)\cap GL(N,\widehat{\ZZ})$.

There exists~$c=c(\phi_0)\in\RR_{>0}$ such that, as~$\phi$ ranges through~$W(\AAA_f)$, we have
\begin{align}\label{explicit global adelic bound}
[\phi(M(\widehat{\ZZ})):\phi(M(\widehat{\ZZ}))\cap GL(N,\widehat{\ZZ})]
\geq \frac{1}{c^{\omega(H_{\AAA_f}(d\phi))}}\cdot H_{\AAA_f}(d\phi).
\end{align}
(Where~$\omega(n)$ is the number of prime factors of~$n$.)
\end{theo}

A main tool in proving~\eqref{explicit global adelic bound} is~Theorem~\ref{lemme_exp}. We establish~Theorem~\ref{lemme_exp} using estimates~\cite[Prop.~A.1]{RY-APZ} on $p$-adic norms of exponentials of a matrix~$X\in M_d(\QQ_p)$.
\begin{theo}[Lemma of the exponentials, {\cite[Thm.~A.3]{RY-APZ}}] \label{lemme_exp}
Let $X \in M_d(\QQ_p)$ be such that $\exp(X)$ converges and denote by~$\exp(X)^\ZZ$ the subgroup generated by~$\exp(X)$ in~$GL_d(\QQ_p)$. We define~$H_p(X)=\max\{1;\norm{X}\}$. Then we have
\[
[\exp(X)^\ZZ:\exp(X)^\ZZ\cap GL_d(\ZZ_p)]\geq H_p(X)/d.
\]
If~$p>d$, we have $[\exp(X)^\ZZ:\exp(X)^\ZZ\cap GL_d(\ZZ_p)]\geq H_p(X).$
\end{theo}

\subsection{Comparison with the global height}

Let~$\mathfrak{S}\subseteq G(\RR)^+$ be a finite union of Siegel sets, and denote by~$\mathfrak{S}\cdot \phi_0\subseteq W(\RR)$ its image.

There exists a closed affine embedding~$W\to \AAA^n$ defined over~$\QQ$, say~$\iota:\phi\mapsto d\phi:W\to\Hom(\mathfrak{m},\mathfrak{g})$. For~$(q_1,\ldots,q_n)\in\QQ^n$, we denote by~$H_\QQ(q_1,\ldots,q_n)\in\ZZ_{\geq1}$ the usual height of~$(q_1,\ldots,q_n)$ and we denote by~$H_f(q_1,\ldots,q_n)\in\ZZ_{\geq1}$ the lowest common multiple of the denominators of the~$q_i$. We recover~$H_{\AAA_f}(d\phi)= H_f(\iota(\phi))$ on~$W(\QQ)$.

\begin{theo}[{\cite[Thm.~5.16]{RY-APZ}}]\label{thm type Orr}
As~$\phi$ ranges through~$\mathfrak{S}\cdot \phi_0\cap W(\QQ)$, we have
\begin{equation}\label{Type Orr}
H_f(\iota(\phi))\approx H_\QQ(\iota(\phi)).
\end{equation}
\end{theo}

Theorem~\ref{thm type Orr} was~\cite[Thm.~1.1]{OrrS} obtained by M. Orr, \emph{in the case~$W=G$}. This is insufficient for us. We replaced the ``block version of Pila--Wilkie'' (used by Orr) by Theorem~\ref{PilaWilkie}.

The above~\eqref{Type Orr} is used to relate the height used in~\eqref{Galois estimate} with the height used in Theorem~\ref{PilaWilkie}. We deduce Theorem~\ref{PilaWilkie} from~\cite[Thm.~1.7]{Pila}. The referee informed us that~Theorem~\ref{PilaWilkie} can also be deduced from~\cite[Cor.~7.2]{PH}. Theorem~\ref{PilaWilkie} and its proof in~\cite{RY-APZ} are both much simpler.

\begin{theo}[{\cite[Thm.~7.1]{RY-APZ}}]\label{PilaWilkie}
Let~$W\subseteq \AAA^N$ be a closed subvariety defined over~$\QQ$, let~$X$ be a semi-algebraic set, and let~$p:W(\RR)\to X$ be a semialgebraic map.

Let~$Z\subseteq X$ be a definable subset, and denote~$Z^{\alg}$ be the union of the semialgebraic subsets of~$X$ which are contained in~$Z$ and of non-zero dimension.

Then, for every~$\epsilon\in\RR_{>0}$, there exists~$C(\epsilon,Z)\in\RR_{>0} $, such that
\[
\forall T\gg0, \quad \abs{(Z\smallsetminus Z^{\alg})\cap p(\{w\in W(\QQ) : H_\QQ(w)\leq T\})}\leq C(\epsilon,Z)\cdot T^{\epsilon}.
\]
\end{theo}


\section{Outline of the proof of Theorem~\ref{thm intro}}\label{Strategy outline}

We reduce the Conjecture~\ref{APZ} to the case where~$V:=\ol{\Sigma}=\ol{\{s_0;s_1;\ldots\}}$ is irreducible, $G$ is adjoint and~$V$ is Hodge generic in~$S$. We rely on functoriality properties of geometric and generalised Hecke orbits.\footnote{This avoids one difficulty in the approach~\cite{Orr} of Orr.} Theorem~\ref{theo general union finie geometrique} allows us to use geometric and generalised Hecke orbits interchangeably. We also prove, cf.~\cite[\S6.3]{RY-APZ}, functoriality properties of the assumption~\eqref{waMT}.


The final objective of the proof is to apply the geometric part of the André--Oort conjecture~\cite{UllmoAxLin} (or~\cite{RU}), and use induction on the number of simple factors of~$M^{ad}$. For every $n$ large enough, we construct a weakly special subvariety~$Z_n\subseteq V$ of non-zero dimension such that~$s_n\in Z_n$. Then~\cite{UllmoAxLin,RU} describes~$\ol{\bigcup Z_n}$, and we deduce Conjecture~\ref{APZ}.

In order to construct the non-zero dimensional~$Z_n$, we use the Pila--Zannier strategy. The generalised Hecke orbit is naturally related to~$W(\QQ)$ where~$W=G\cdot \phi_0\simeq G/Z_G(M)$ (cf.~\cite[Lem.~2.5]{RY-APZ}.)

The goal is to apply the variant Theorem~\ref{PilaWilkie} of the Pila--Wilkie theorem, after constructing many rational points of small height in some set definable in an o-minimal structure. This definable set is
\[
\wt{V}=\left(\stackrel{-1}{\pi}(V)\cap \mathfrak{S}\right)/Z_{G(\RR)}(M)\subseteq W(\RR)
\]
where~$\pi:G(\RR)\to X\to S$ is the uniformisation map, and~$\mathfrak{S}\subseteq G(\RR)$ is a finite union of Siegel sets such that~$S=\pi(\mathfrak{S})$.


Let~$E$ be field of definition of~$V$. Then~$V$ contains the Galois orbits~$\Gal(\ol{E}/E)\cdot s_n$. Each point~$s'\in \Gal(\ol{E}/E)\cdot s_n$ lifts to a rational point~$\wt{s'}\in\wt{V}\cap W(\QQ)$.

By~Proposition~\ref{Gal invariance}, the value of~$H_{[x_0,1]}$ is constant as~$s'$ ranges through~$\Gal(\ol{E}/E)\cdot s_n$. It follows from~\eqref{Galois estimate} that there are~$\#\Gal(\ol{E}/E)\cdot s_n \approx H_{[x_0,1]}({s_n})$ such points.\footnote{This is where the assumption~\eqref{waMT} is needed.} By~\eqref{Type Orr}, we have~$H_{[x_0,1]}({s_n}) \approx H_\QQ(\widetilde{s_n})$. We introduce
\[
Q_n:=\{\phi\in \mathfrak{S}\cdot \phi_0\cap W(\QQ):[\phi\circ x_0:1]\in \Gal(\ol{E}/E)\cdot s_n\}\subseteq \widetilde{V}.
\]
For~$\phi\in Q_n$, we have~$H_{\AAA_f}(d\phi)=H_{[x_0,1]}([\phi\circ x_0,1])=H_{[x_0,1]}({s_n})\approx H_\QQ(\wt{s_n})$.

Denote by~$p$ the map~$G(\RR)\cdot \phi_0\to X$ with~$G(\RR)\cdot \phi_0\subseteq W(\RR)$. We have surjections~$Q_n\to p(Q_n)\to \Gal(\ol{E}/E)\cdot s_n$. Thus~$\#Q_n\geq \#\Gal(\ol{E}/E)\cdot s_n \approx H_\QQ(\widetilde{s_n})$.

Thus~$\widetilde{V}$ contains~$\#Q_n\approx H_\QQ(\wt{s_n})$ points of height~$\approx H_\QQ(\wt{s_n})$.

By~Theorem~\ref{PilaWilkie}, for sufficiently large~$n$, there exist~$\phi_n$ in~$Q_n$ such that~$p(\phi_n)\in Z^{\alg}$, with~$Z=p(\widetilde{V})$. By Ax--Lindemann--Weierstrass theorem~\cite{KUY}, it follows that~$s'_n=[\phi_n,1]\in Z_n\subseteq V$, for a non-zero dimensional weakly special subvariety~$Z_n$. Using Galois action, we may assume~$s'_n=s_n$.

This concludes the proof of Theorem~\ref{thm intro}.

%\section{Acknowledgements} This project was funded by Leverhulme Trust grant RPG-2019-180.

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