Number Theory Seminar
Chantal David
Concordia University
On the vanishing of twisted L-functions of elliptic curves over function fields
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
Friday April 22, 2022 at 1:00 PM in Zoom