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


Author: Gregorio Baldi
Journal: Proc. Amer. Math. Soc. 148 (2020), 95-102
MSC (2010): Primary 14G35, 14H30, 11F80
DOI: https://doi.org/10.1090/proc/14717
Published electronically: July 10, 2019
MathSciNet review: 4042833
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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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Additional Information

Gregorio Baldi
Affiliation: Department of Mathematics, London School of Geometry and Number Theory, University College London, Gower street, WC1E 6BT, London, United Kingdom
Email: gregorio.baldi.16@ucl.ac.uk

Received by editor(s): October 8, 2018
Received by editor(s) in revised form: January 6, 2019, and April 24, 2019
Published electronically: July 10, 2019
Additional Notes: This work was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], the EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London.
Communicated by: Rachel Pries
Article copyright: © Copyright 2019 American Mathematical Society