Sato-Tate Distributions on Abelian Surfaces

TL;DR

This paper advances the understanding of Sato-Tate distributions for abelian surfaces by proving new cases using automorphy theorems and analyzing trace asymptotics in unproven cases.

Contribution

It introduces new automorphy results for abelian surfaces and provides asymptotic descriptions of Frobenius traces where the conjecture remains unproven.

Findings

Proved new cases of the Sato-Tate conjecture for abelian surfaces.

Described asymptotic behavior of Frobenius traces in unproven cases.

Established optimality of these asymptotics with current knowledge.

Abstract

We prove a few new cases of the Sato-Tate conjecture for abelian surfaces, using a new automorphy theorem of Allen et al. Then in the unproven cases, we use partial results to describe nontrivial asymptotics on the trace of Frobenius, and prove their optimality given current knowledge.