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International audienceWe prove that for d ∈ {2, 3, 5, 7, 13} and K a quadratic (or rational) eld of discriminant D and Dirichlet character χ, if a prime p is large enough compared to D, there is a newform f ∈ S2(Γ0(dp 2)) with sign (+1) with respect to the Atkin-Lehner involution w p 2 such that L(f ⊗ χ, 1) = 0. This result is obtained through an estimate of a weighted sum of twists of L-functions which 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. Keywords. Nonvanishing of L-functions of modular forms, Petersson trace formula, rank zero quotients of jacobians. MSC201 11F67 and 14J15

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

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