Statistics for products of traces of high powers of the frobenius class of hyperelliptic curves

TL;DR

This paper investigates the statistical behavior of traces of high powers of Frobenius classes for hyperelliptic curves, revealing their convergence to random matrix theory predictions and Gaussian fluctuations in eigenphase statistics.

Contribution

It establishes the limiting distribution of product traces of Frobenius classes matching the unitary symplectic group and analyzes Gaussian behavior in eigenphase linear statistics.

Findings

Limiting expectation matches USp(2g) ensemble

First moments of eigenphase statistics are Gaussian

Convergence of product traces to random matrix predictions

Abstract

We study the averages of products of traces of high powers of the Frobenius class of hyperelliptic curves of genus g over a fixed finite field. We show that for increasing genus g, the limiting expectation of these products equals to the expectation when the curve varies over the unitary symplectic group USp(2g). We also consider the scaling limit of linear statistics for eigenphases of the Frobenius class of hyperelliptic curves, and show that their first few moments are Gaussian.