Traces of high powers of the Frobenius class in the hyperelliptic ensemble

TL;DR

This paper analyzes the behavior of high powers of the Frobenius class in hyperelliptic curves over finite fields, comparing theoretical predictions with random matrix models and exploring implications for zero distributions.

Contribution

It computes expected traces of high powers of Frobenius classes for hyperelliptic curves and compares them with symplectic group averages, extending understanding of zero statistics.

Findings

Expected trace matches random matrix predictions for large powers

Agreement with symplectic statistics for low-lying zeros

Partial confirmation of Katz-Sarnak conjecture on zero distribution

Abstract

The zeta function of a curve over a finite field may be expressed in terms of the characteristic polynomial of a unitary symplectic matrix, called the Frobenius class of the curve. We compute the expected value of the trace of the n-th power of the Frobenius class for an ensemble of hyperelliptic curves of genus g over a fixed finite field in the limit of large genus, and compare the results to the corresponding averages over the unitary symplectic group USp(2g). We are able to compute the averages for powers n almost up to 4g, finding agreement with the Random Matrix results except for small n and for n=2g. As an application we compute the one-level density of zeros of the zeta function of the curves, including lower-order terms, for test functions whose Fourier transform is supported in (-2,2). The results confirm in part a conjecture of Katz and Sarnak, that to leading order the low-lying zeros for this ensemble have symplectic statistics.