The fourth moment of quadratic Dirichlet $L$--functions over function fields

TL;DR

This paper derives an asymptotic formula for the fourth moment of quadratic Dirichlet L-functions over function fields, confirming conjectured leading terms as the genus grows.

Contribution

It provides the first proof of the asymptotic formula for the fourth moment, including the leading three terms, aligning with existing conjectures.

Findings

Asymptotic formula matches conjectured polynomial of degree 10

Leading three terms of the fourth moment are explicitly computed

Results support the conjectured behavior of L-functions over function fields

Abstract

We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet $L$--functions over $\mathbb{F}_q[x]$, as the base field $\mathbb{F}_q$ is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to $q^{2g+1} P(2g+1)$ up to an error of size $o(q^{2g+1})$, where $P$ is a polynomial of degree $10$ with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.