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arXiv · 2211.00305

Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields

Keunyoung Jeong·Yeong-Wook Kwon·Junyeong Park
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Abstract

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$. As an application, we show that for each isogeny factor of the Jacobian of the $p$-th Fermat curve where $2$ is a quadratic residue modulo $p$, there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the $11$-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero.

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Keunyoung Jeong, Yeong-Wook Kwon, Junyeong Park. 2022-11-01. Nonvanishing of $L$-function of some Hecke characters on cyclotomic fields. https://arxiv.org/abs/2211.00305

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