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

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))\otimes 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.