  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.
