Moments of zeta functions associated to hyperelliptic curves over finite fields

TL;DR

This paper investigates the moments of zeta functions associated with hyperelliptic curves over finite fields, providing evidence for a conjectured asymptotic formula and deriving exact formulas in some cases.

Contribution

It offers theoretical and numerical support for Andrade and Keating's conjecture on moments with fixed q and large degree d, including exact formulas and remainder term analysis.

Findings

Support for Andrade and Keating's conjecture on moments

Exact formulas for certain cases of moments

Precise estimates of the remainder term in the asymptotic formula

Abstract

Let $q$ be an odd prime power, and $H_{d,q}$ denote the set of square-free monic polynomials $D(x) \in F_q[x]$ of degree $d$. Katz and Sarnak showed that the moments, over $H_{d,q}$, of the zeta functions associated to the curves $y^2=D(x)$, evaluated at the central point, tend, as $q \to \infty$, to the moments of characteristic polynomials, evaluated at the central point, of matrices in $USp(2\lfloor (d-1)/2 \rfloor)$. Using techniques that were originally developed for studying moments of $L$-functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for $q$ fixed and $d \to \infty$. We provide theoretical and numerical evidence in favour of their conjecture. In some cases we are able to work out exact formulas for the moments and use these to precisely determine the size of the remainder term in the predicted moments.