Sato-Tate groups of genus 2 curves

TL;DR

This paper discusses the Sato-Tate conjecture for genus 2 curves, classifies the associated groups for abelian surfaces, and connects the distribution of zeta functions over finite fields to group-theoretic structures.

Contribution

It defines the Sato-Tate group for abelian varieties and classifies these groups specifically for genus 2 curves, extending the understanding of their distribution properties.

Findings

Classification of Sato-Tate groups for abelian surfaces

Connection between zeta function distributions and group theory

Extension of the Sato-Tate conjecture to genus 2 curves

Abstract

We describe the analogue of the Sato-Tate conjecture for an abelian variety over a number field; this predicts that the zeta functions of the reductions over various finite fields, when properly normalized, have a limiting distribution predicted by a certain group-theoretic construction related to Hodge theory, Galois images, and endomorphisms. After making precise the definition of the "Sato-Tate group" appearing in this conjecture, we describe the classification of Sato-Tate groups of abelian surfaces due to Fite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture series presented at the NATO Advanced Study Institute "Arithmetic of Hyperelliptic Curves" held in Ohrid (Macedonia) August 25-September 5, 2014, and are expected to appear in a proceedings volume.)