The Mumford—Tate conjecture implies the algebraic Sato—Tate conjecture

Abstract: The famous Mumford-Tate conjecture asserts that, for every prime number $\ell$, Hodge cycles are $\mathbb{Q}_{\ell}$-linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture, introduced by Serre and developed later by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato-Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.

Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture. This is a joint work with Johan Commelin.

number theory

Audience: researchers in the discipline

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POINT: New Developments in Number Theory

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