On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E-/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H-c be the ring class field of K of conductor c prime to ND with Galois group G(c) over K. Fix a complex character chi of G(c). Our main result is that if L-K (E, chi, 1) not equal 0 then Sel(p)(E/H-c) circle times(chi) W = 0 for all but finitely many primes p, where Sel(p)(E/H-c) is the p-Selmer group of E over H-c and W is a Suitable finite extension of Z(p) containing the values of chi. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a chi-twisted version of the Birch and Swinnerton-Dyer conjecture for E over H-c (Bertolini and Darmon) and of the vanishing of Sel(p)(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero. (C) 2009 Elsevier Inc. All rights reserved.