Journal of the Ramanujan Mathematical Society

Volume 28, Issue 2, June 2013 pp. 233–245.

Non-vanishing of Artin-twisted L-functions of elliptic curves

Authors: Thomas Ward
Author institution: Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL

Summary: Let E be an elliptic curve and ρ an Artin representation, both defined over Q. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters χ, ramified only at p, such that the Artin-twisted L-values L(E,ρ \otimes χ, β) are non-zero when β 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 ρ.

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