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arXiv · 2207.00197

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

Antoine Comeau-Lapointe·Chantal David·Matilde Lalin·Wanlin Li
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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.

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Antoine Comeau-Lapointe, Chantal David, Matilde Lalin, Wanlin Li. 2022-07-01. On the vanishing of twisted $L$-functions of elliptic curves over rational function fields. https://arxiv.org/abs/2207.00197

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