Non-vanishing of Artin-twisted L-functions of Elliptic Curves

TL;DR

This paper proves the existence of infinitely many Dirichlet characters, ramified only at one prime, that produce non-zero Artin-twisted L-values of elliptic curves within a certain critical region, assuming conjectural continuations.

Contribution

It demonstrates that the characters can be chosen to be ramified at only one prime, even if that prime divides the conductor of the Artin representation, which is a novel result.

Findings

Existence of infinitely many such characters

Non-vanishing of L-values in the critical strip

Characters can be ramified at a single prime dividing the conductor

Abstract

Let E be an elliptic curve and \rho an Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters \chi, ramified only at p, such that the Artin-twisted L-values L(E,\rho \chi,\beta) are non-zero when \beta lies in a specified region in the critical strip (assuming the conjectural continuations and functional equations for these L-functions). The new contribution of our paper is that we may choose our characters to be ramified only at one prime, which may divide the conductor of \rho.