On the geometric Mumford-Tate conjecture for subvarieties of Shimura varieties

TL;DR

This paper proves a geometric version of the integral $ ext{ell}$-adic Mumford-Tate conjecture for certain subvarieties of Shimura varieties, showing their associated $ ext{ell}$-adic Galois representations have large images.

Contribution

It establishes that for large enough primes, the $ ext{ell}$-adic Galois image of these subvarieties contains the integral points from the simply connected cover of the derived subgroup of the Shimura datum.

Findings

The image of $ ext{ell}$-adic representations is large for sufficiently large $ ext{ell}$.

The result applies to subvarieties not contained in smaller Shimura subvarieties.

Provides a geometric analogue of the Mumford-Tate conjecture.

Abstract

We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $Sh_K(G,X)$ that are not contained in a smaller Shimura subvariety and have no isotrivial components. We show that, for $\ell$ large enough (depending on the Shimura datum $(G,X)$ and the subvariety), such image contains the $\mathbb{Z}_\ell$-points coming from the simply connected cover of the derived subgroup of $G$. This can be regarded as a geometric version of the integral $\ell$-adic Mumford-Tate conjecture.