Nonvanishing of Central Values of L-functions of Newforms in S2(Γ0(dp2)) Twisted by Quadratic Characters

Abstract

AbstractWe prove that for d ∊ {2, 3, 5, 7, 13} and K a quadratic (or rational) field of discriminant D and Dirichlet character 𝜒, if a prime p is large enough compared to D, there is a new form f ∊ S2Γ0(dp2)) with sign (+1) with respect to the Atkin–Lehner involution wp2 such that L( f ⊗ 𝜒, •)≠ 0. This result is obtained through an estimate of a weighted sum of twists of L-functions that generalises a result of Ellenberg. It relies on the approximate functional equation for the L-functions L( f ⊗ 𝜒, · ) and a Petersson trace formula restricted to Atkin–Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.