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

Abstract

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