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

Abstract

We give a geometric criterion for Dirichlet L-functions associated to cyclic characters over the rational function field 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 F q . Using this geometric criterion, we obtain a lower bound on the number of cubic characters over F q (t) whose L-functions vanish at the central point where q = p 4n for any rational prime p ≡ 2 mod 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.