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

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2003

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Abstract

Let N ≡ 1 mod 4 be the negative of a prime, K = Q( √ N) and OK its ring of integers. Let D be a prime ideal in OK of prime norm congruent to 3 modulo 4. Under these assumptions, there exists Hecke characters ψD of K with conductor D and infinite type (1, 0). Their L-series L(ψD, s) are associated to a CM elliptic curve A(N, D) defined over the Hilbert class field of K. We will prove a Waldspurger-type formula for L(ψD, s) of the form L(ψD, 1) = Ω P I r(D, I)m[D] (I) where the sum is over class ideal representatives I of a maximal order in the quaternion algebra ramified at |N| and infinity. 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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