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

The famous Mumford-Tate conjecture asserts that, for every prime number l, Hodge cycles are Q_l 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 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.