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

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 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 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.