## The central value of Dirichlet L-functions over function fields and related topics by Wanlin Li

A Dirichlet character over F_q(t) corresponds to a curve over F_q.
Using this connection to geometry, we construct families of characters whose L-functions vanish (resp. does not vanish) at the central point. The existence of infinitely many vanishing L-functions is in contrast with the situation over the rational numbers, where a conjecture of Chowla predicts there should be no such. Towards Chowla's conjecture, for each fixed q, we present an explicit upper bound on the number of such quadratic characters which decreases as q grows and it goes to 0 percent as q goes to infinity. In this talk, I will also discuss phenomena and interesting questions related to this problem. Some results in this talk are from projects joint with Ravi Donepudi, Jordan Ellenberg and Mark Shusterman.