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 F_q(t) to vanish at the central point s = ₂¹. 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 = p⁴ⁿ 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.