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 F q (t) (where F q is a finite field of q elements and characteristic ≥ 5) for twists by Dirichlet characters of prime order ℓ ≥ 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 [Li18, DL21] who proved vanishing at s = 1/2 for infinitely many Dirichlet L-functions over F q (t) based on the existence of one, and we can prove that if there is one χ 0 such that L(E, χ 0 , 1) = 0, then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over F q (t) behave differently than the general ones.