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

✍ Scribed by Ariel Pacetti

• Publisher: Elsevier Science
• Year: 2005
• Tongue: English
• Weight: 471 KB
• Volume: 113
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2004.12.004

✦ Synopsis

Let N ≡ 1 mod 4 be the negative of a prime, K = Q( √ N) and O K its ring of integers. Let D be a prime ideal in O K of prime norm congruent to 3 mod 4. Under these assumptions, there exists Hecke characters D of K with conductor (D) and infinite type (1, 0). Their Lseries L( D , 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( D , s) of the form L( D , 1) = [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.