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

Baldi, Gregorio

doi:10.1090/proc/14717

On the geometric Mumford-Tate conjecture for subvarieties of Shimura varieties
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by Gregorio Baldi PDF

Proc. Amer. Math. Soc. 148 (2020), 95-102 Request permission

Abstract:

We study the image of $\ell$-adic representations attached to subvarieties of Shimura varieties $\operatorname {Sh}_K(G,X)$ 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.

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