On the vanishing of twisted $L$-functions of elliptic curves over rational function fields

TL;DR

This paper studies the conditions under which twisted L-functions of elliptic curves over rational function fields vanish at s=1, combining theoretical insights with numerical data, and distinguishes behaviors between constant and non-constant curves.

Contribution

It extends understanding of L-function vanishing over function fields, adapting techniques for constant curves, and provides numerical evidence supporting rarity of vanishing for non-constant curves.

Findings

Vanishing at s=1 is rare for non-constant curves.

If one twist causes vanishing, infinitely many do.

Constant curves exhibit different vanishing behavior.

Abstract

We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt the techniques of Li and Donepudi--Li who proved vanishing at $s=1/2$ for infinitely many Dirichlet $L$-functions over $\mathbb{F}_q(t)$ based on the existence of one, and we can prove that if there is one $\chi_0$ such that $L(E, \chi_0, 1)=0$, then there are infinitely many. Finally, we provide some examples which show that twisted $L$-functions of constant elliptic curves over $\mathbb{F}_q(t)$ behave differently than the general ones.