On the distribution of the trace in the unitary symplectic group and the distribution of Frobenius

TL;DR

This paper investigates the distribution of the trace in the unitary symplectic group, linking it to Frobenius eigenvalues and point counts of curves, providing explicit formulas for genus 2 cases.

Contribution

It offers new explicit formulas for the trace distribution in the symplectic group, especially for genus 2, and connects these to Frobenius eigenvalues and point distributions.

Findings

Derived four formulas for the trace distribution at genus 2

Connected trace distribution to Frobenius eigenvalues in abelian varieties

Provided an elementary symmetric functions expression for the trace distribution

Abstract

The purpose of this article is to study the distribution of the trace on the unitary symplectic group. We recall its relevance to equidistribution results for the eigenvalues of the Frobenius in families of abelian varieties over finite fields, and to the limiting distribution of the number of points of curves. We give four expressions of the trace distribution if g = 2, in terms of special functions, and also an expression of the distribution of the trace in terms of elementary symmetric functions. In an appendix, we prove a formula for the trace of the exterior power of the identity representation.