Conjectures for the integral moments and ratios of L-functions over function fields

TL;DR

This paper extends heuristic models for moments and ratios of L-functions from number fields to function fields, specifically for hyperelliptic curves over finite fields, revealing similarities with random matrix theory and analyzing zero distributions.

Contribution

It introduces a heuristic for moments and ratios of L-functions over function fields, paralleling number field models, and applies it to compute zero density for hyperelliptic curve L-functions.

Findings

Heuristic formulas for moments and ratios over function fields.

Resemblance to characteristic polynomials of random matrices.

Calculation of the one-level density of zeros.

Abstract

We extend to the function field setting the heuristic previously developed, by Conrey, Farmer, Keating, Rubinstein and Snaith, for the integral moments and ratios of $L$-functions defined over number fields. Specifically, we give a heuristic for the moments and ratios of a family of $L$-functions associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_{q}$ in the limit as $g\rightarrow\infty$. Like in the number field case, there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, we calculate the one-level density for the zeros of these $L$-functions.