Traces, high powers and one level density for families of curves over finite fields

TL;DR

This paper develops a new technique to compute expected traces of Frobenius matrices for families of algebraic curves over finite fields, extending previous work and applying it to various curve families to analyze their one-level density.

Contribution

It introduces a novel method for calculating expected trace values of Frobenius matrices for large genus curves over finite fields, generalizing prior approaches.

Findings

Computed expected trace values for various curve families.

Extended explicit formulas for dependence on ramification places.

Provided an explicit proof of the Lindelöf bound in the function field setting.

Abstract

The zeta function of a curve $C$ over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix $\Theta_C$. We develop and present a new technique to compute the expected value of $\mathrm{Tr}(\Theta_C^n)$ for various moduli spaces of curves of genus $g$ over a fixed finite field in the limit as $g$ is large, generalizing and extending the work of Rudnick and Chinis. This is achieved by using function field zeta functions, explicit formulae, and the densities of prime polynomials with prescribed ramification types at certain places as given by Bucur, David, Feigon, Kaplan, Lal\'in and Wood [BDF$^+$16] and by Zhao. We extend [BDF$^+$16] by describing explicit dependence on the place and give an explicit proof of the Lindel\"{o}f bound for function field Dirichlet $L$-functions $L(1/2 + it, \chi)$. As applications, we compute the one-level density for hyperelliptic curves, cyclic $\ell$-covers, and cubic non-Galois covers.