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

2022/07/01 by Comeau-Lapointe, Antoine, David, Chantal, Lalin, Matilde +1
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.00197

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 \mathbbF_q(t) (where \mathbbF_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 Li and Donepudi--Li who proved vanishing at s=1/2 for infinitely many Dirichlet L-functions over \mathbbF_q(t) based on the existence of one, and we can prove that if there is one χ₀ such that L(E, χ₀, 1)=0, then there are infinitely many. Finally, we provide some examples which show that twisted L-functions of constant elliptic curves over \mathbbF_q(t) behave differently than the general ones.

Related

• Vanishing of L-functions of elliptic curves over number fields
• Non-vanishing of Artin-twisted L-functions of Elliptic Curves
• Vanishing of Dirichlet L-functions at the central point over function fields
• L-values of elliptic curves twisted by cubic characters
• On Twists of A Family of Elliptic Curves and Their L-Function