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We give a geometric criterion for Dirichlet $L$-functions associated tocyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanishat the central point $s=1/2$. The idea is based on the observation thatvanishing at the central point can be interpreted as the existence of a mapfrom the projective curve associated to the character to some abelian varietyover $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound onthe number of cubic characters over $\mathbb{F}_q(t)$ whose $L$-functionsvanish 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 supersingularsuperelliptic curves to deduce consequences for the $L$-functions of Dirichletcharacters of other orders.

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