# The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture of Banaszak and Kedlaya

## TL;DR

This paper proves that the Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture for abelian varieties, bridging a gap between two important conjectures in algebraic geometry and number theory.

## Contribution

It establishes a logical implication from the Mumford-Tate conjecture to the algebraic Sato-Tate conjecture, expanding understanding of their relationship.

## Key findings

- Mumford-Tate conjecture implies algebraic Sato-Tate conjecture for abelian varieties
- The result links two major conjectures, reducing the cases needed to verify the Sato-Tate conjecture
- Enhances the theoretical framework connecting Hodge theory and Galois representations.

## Abstract

The algebraic Sato-Tate conjecture was initially introduced by Serre and then discussed by Banaszak and Kedlaya. This note shows 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.