Chebotarev-Sato-Tate distribution for abelian surfaces potentially of $\rm{GL}_2$-type

TL;DR

This paper proposes a hybrid Chebotarev-Sato-Tate conjecture for abelian varieties and proves it in specific cases leveraging potential automorphy theorems, advancing understanding of their distribution properties.

Contribution

It introduces a new conjecture linking Chebotarev and Sato-Tate distributions for abelian varieties and provides proofs for particular cases using modern automorphy results.

Findings

Validation of the conjecture in specific cases

Application of potential automorphy theorems

Deeper insight into abelian surface distributions

Abstract

In this paper, we state a hybrid Chebotarev-Sato-Tate conjecture for abelian varieties and we prove it in several particular cases using current potential automorphy theorems.