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- W4311653848 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." @default.
- W4311653848 created "2022-12-27" @default.
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- W4311653848 date "2022-07-01" @default.
- W4311653848 modified "2023-09-25" @default.
- W4311653848 title "On the vanishing of twisted $L$-functions of elliptic curves over rational function fields" @default.
- W4311653848 doi "https://doi.org/10.48550/arxiv.2207.00197" @default.
- W4311653848 hasPublicationYear "2022" @default.
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