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

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 $Θ_C$. We develop and present a new technique to compute the expected value of $\mathrm{Tr}(Θ_C^n)$ 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, Lalín 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 Lindelöf bound for function field Dirichlet $L$-functions $L(1/2 + it, χ)$. As applications, we compute the one-level density for hyperelliptic curves, cyclic $\ell$-covers, and cubic non-Galois covers.