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

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 $\ge 5$) for twists by Dirichlet characters of prime order $\ell \ge 3$, 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.