Tate module tensor decompositions and the Sato-Tate conjecture for certain abelian varieties potentially of $\mathrm{GL}_2$-type

TL;DR

This paper introduces a tensor decomposition of the Tate module for certain abelian varieties and uses it to analyze their Sato--Tate groups, proving the conjecture in specific cases for varieties of potentially e4-type over totally real fields.

Contribution

It provides a new tensor decomposition framework for Tate modules and applies it to establish the Sato--Tate conjecture for certain abelian varieties of e4-type.

Findings

Describes the Sato--Tate group for these abelian varieties.

Proves the Sato--Tate conjecture in specific cases.

Introduces a novel tensor decomposition method.

Abstract

We introduce a tensor decomposition of the $\ell$-adic Tate module of an abelian variety $A_0$ defined over a number field which is geometrically isotypic. If $A_0$ is potentially of $\GL_2$-type and defined over a totally real number field, we use this decomposition to describe its Sato--Tate group and to prove the Sato--Tate conjecture in certain cases.