Sato-Tate Distributions

TL;DR

This paper explores the theoretical framework connecting Galois representations, motivic L-functions, and Sato-Tate groups, providing explicit formulations and classifications for abelian varieties up to dimension 3.

Contribution

It offers an explicit formulation of the Sato-Tate conjecture for abelian varieties and classifies Sato-Tate groups for dimensions up to 3, including trace distribution computations.

Findings

Explicit Sato-Tate conjecture formulation for abelian varieties.

Classification of Sato-Tate groups for dimensions g <= 3.

Computed trace distributions for specific cases.

Abstract

In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g <= 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry.