Chantal David : On the vanishing of twisted L-functions of elliptic curves over function fields

Posted by Nathan Jones , part of the Number Theory Seminar.

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April 22, 2022, 1 p.m.

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Abstract

Let E be an elliptic curve over Q, and let $\chi$ be a Dirichlet character of order $\ell$ for some prime $\ell \geq 3$. Heuristics based on the distribution of modular symbols and random matrix theory have led to conjectures predicting that the vanishing of the twisted L-functions $L(E, \chi, s)$ at $s = 1$ is a very rare event (David-Fearnley-Kisilevsky and Mazur-Rubin). In particular, it is conjectured that there are only finitely many characters of order $\ell > 5$ such that $L(E, \chi, 1) = 0$ for a fixed curve E. We investigate in this talk the case of elliptic curves over function fields. For Dirichlet L-functions over function fields, Li and Donepudi-Li have shown how to use the geometry to produce infinitely many characters of order $\ell \geq 2$ such that the Dirichlet L-function $L(\chi, s)$ vanishes at s = 1/2, contradicting (the function field analogue of) Chowla’s conjecture. We show that their work can be generalized to isotrivial curves E/Fq(t), and we show that if there is one Dirichlet character $\chi$ of order $\ell$ such that $L(E, \chi, 1) = 0$, then there are infinitely many, leading to some specific examples contradicting (the function field analogue of) the number field conjectures on the vanishing of twisted L-functions. Such a dichotomy does not seem to exists for general (non-isotrivial) curves over Fq(t), and we produce empirical evidence which suggests that the conjectures over number fields also hold over function fields for non-isotrivial E/Fq(t).

Zoom link: https://uic.zoom.us/j/88173268700?pwd=aEhmTGpSOVhidWE4L1VWUnNhNVlvUT09