A formula for the central value of certain Hecke L-functions

2003/09/01 by Ariel Pacetti, Pacetti, Ariel
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #math.NT #msc:11G40

paper · pdf · doi:10.48550/arxiv.math/0309023

43 pages

arxiv created 2004/12/22 · arxiv updated 2009/12/01

Abstract

Let N = 1 mod 4 be the negative of a prime, K=Q(sqrtN) and O_K its ring of integers. Let D be a prime ideal in O_K of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters ψ_\D of K with conductor (\D) and infinite type (1,0). Their L-series L(ψ_\D,s) are associated to a CM elliptic curve E(N,\D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψ_\D,s) of the form L(ψ_\D,1) = Ω∑_[\A],I r(\D,[\A],I) m_[\A],I([\D]) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity and [\A] are class group representatives of K. An application of this formula for the case N=-7 will allow us to prove the non-vanishing of a family of L-series of level 7|D| over K.

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