Vanishing of Dirichlet L-functions at the central point over function fields

TL;DR

This paper establishes a geometric criterion for when Dirichlet L-functions over function fields vanish at the central point, linking algebraic geometry with number theory and providing bounds for specific character families.

Contribution

It introduces a new geometric criterion for L-function vanishing at the central point and derives bounds for cubic characters over function fields.

Findings

Lower bound on the number of cubic characters with vanishing L-functions

Connection between curve maps and L-function zeros

Implications for characters of other orders using supersingular curves

Abstract

We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over $\mathbb{F}_q(t)$ whose $L$-functions vanish at the central point where $q=p^{4n}$ for any rational prime $p \equiv 2 \bmod 3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$-functions of Dirichlet characters of other orders.