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

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 $s=1\mathrm{/}2$. 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.