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

Understand

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.