Title: Nonvanishing for cubic L-functions over function fields
Speaker: Matilde Lalin - University of Montreal
Abstract: Chowla's conjecture predicts that $L (1/2, \chi)$ does not vanish for Dirichlet $L$-functions associated with primitive characters $\chi$. It was first conjectured for the case of $\chi$ quadratic. For that case, Soundararajan proved that at least 87.5\% of the values $L (1/2, \chi)$ do not vanish, by calculating the first mollified moments. For cubic characters, the first moment has been calculated by Baier and Young (on $\mathbb{Q}$), by Luo (for a restricted family on $\mathbb{Q} (\sqrt{-3})$), and on function fields by David, Florea, and Lal\'in. In this talk we prove that there is a positive proportion of cubic Dirichlet characters for which the corresponding $L$-function at the central value does not vanish. We arrive at this result by computing the first mollified moment using techniques that we previously developed in our work on the first moment of cubic $L$-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwi\l\l, Harper, and Radziwi\l\l - Soundararajan. Our results are on function fields, but with additional work they could be extended to number fields, assuming GRH. This is joint work with Chantal David and Alexandra Florea.
