Session S26 - Finite fields and applications

Tuesday, July 20, 16:00 ~ 16:50 UTC-3

Nonvanishing for cubic $L$-functions over function fields

Matilde Lalín

Université de Montréal, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 Lalín. 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 in the function field case. 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 Radziwill, Harper, and Radziwill - Soundararajan. Our results are on function fields, but with additional work they could be extended to number fields, assuming the Generalized Riemann Hypothesis.

Joint work with Chantal David (Concordia University) and Alexandra Florea (Columbia University).

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