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

TL;DR

This paper proves the nonvanishing of certain Hecke L-functions over cyclotomic fields, using eigenfunction computations and Weil representation, with applications to ranks of Jacobians of specific algebraic curves.

Contribution

It introduces new methods for nonvanishing results of Hecke L-functions on cyclotomic fields, including explicit eigenfunction calculations and applications to algebraic curves.

Findings

Infinitely many twists of Jacobians have zero analytic rank.

Nonvanishing of Hecke L-functions for specific Hecke characters.

Applications to ranks of Jacobians of Fermat and hyperelliptic curves.

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.