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 (J Number Theory 191:85–103, 2018) and Donepudi and Li (Rocky Mountain J Math 51(5):1615–1628, 2021) 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.