Moments of Dirichlet $L$-functions with prime conductors over function fields

TL;DR

This paper calculates moments of quadratic Dirichlet L-functions with prime conductors over function fields, revealing lower order terms and implications for the ranks of twisted elliptic curves.

Contribution

It provides the first asymptotic formulas for second moments with prime conductors and the mean values of derivatives of L-functions over function fields.

Findings

Second moment asymptotics with lower order terms

Existence of a twist with analytic rank one

Asymptotic formula for derivatives of L-functions

Abstract

We compute the second moment in the family of quadratic Dirichlet $L$-functions with prime conductors over $\mathbb{F}_q[x]$ when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of $L$-functions associated to quadratic twists of a fixed elliptic curve over $\mathbb{F}_q(t)$ by monic irreducible polynomials, which allows us to show that there exists a monic irreducible polynomial such that the analytic rank of the corresponding twisted elliptic curve is equal to $1$.