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INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  07 June 2018

Anna Cadoret  and
Ben Moonen Open the ORCID record for Ben Moonen [Opens in a new window]
Anna Cadoret
Affiliation:
IMJ-PRG – Sorbonne Université, Paris, France ([email protected])
Ben Moonen
Affiliation:
Radboud University, IMAPP, Nijmegen, The Netherlands ([email protected])
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Abstract

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

Keywords

Galois representationsMumford-Tate conjectureShimura varietiesAbelian varietiesK3 surfaces

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G35: Modular and Shimura varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 3 , May 2020 , pp. 869 - 890
Copyright
© Cambridge University Press 2018

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