Vanishing of Dirichlet Lfunctions at the central point over function fields
Abstract
We give a geometric criterion for Dirichlet $L$functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the central point can be interpreted as the existence of a map from the projective curve associated to the character to some abelian variety over $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on the number of cubic characters over $\mathbb{F}_q(t)$ whose $L$functions vanish at the central point where $q=p^{4n}$ for any rational prime $p \equiv 2 \bmod 3$. We also use recent results about the existence of supersingular superelliptic curves to deduce consequences for the $L$functions of Dirichlet characters of other orders.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.15319
 Bibcode:
 2020arXiv201215319D
 Keywords:

 Mathematics  Number Theory
 EPrint:
 14 pages