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

Abstract

In this paper, we show 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.