Vanishing of hyperelliptic L-functions at the central point

We obtain a lower bound on the number of quadratic Dirichlet L-functions over the rational function field which vanish at the central point s=1/2. This is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such L-functions. The approach is based on the observation that vanishing at the central point can be interpreted geometrically, as the existence of a map to a fixed abelian variety from the hyperelliptic curve associated to the character.