Auto-correlation functions of Sato-Tate distributions and identities of symplectic characters

TL;DR

This paper explicitly computes auto-correlation functions of Sato-Tate distributions for genus 2 curves, revealing new identities involving symplectic group characters, advancing understanding of their statistical properties.

Contribution

It provides explicit formulas for auto-correlation functions of Sato-Tate distributions and uncovers new identities of symplectic characters for all ranks.

Findings

Explicit formulas for auto-correlation functions

New identities of symplectic group characters

Connections between Sato-Tate distributions and symplectic representations

Abstract

The Sato-Tate distributions for genus 2 curves (conjecturally) describe the statistics of numbers of rational points on the curves. In this paper, we explicitly compute the auto-correlation functions of Sato-Tate distributions for genus 2 curves as sums of irreducible characters of symplectic groups. Our computations bring about families of identities involving irreducible characters of symplectic groups $Sp(2m)$ for all $m \in \mathbb Z_{\ge 1}$, which have interest in their own rights.