  Let E be a rational 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 be the ring class field of K of conductor c prime to ND with Galois
group G over K. Fix a complex character \chi of G. Our main result is that if
the special value of the \chi-twisted L-function of E/K is non-zero then the
tensor product (with respect to \chi) of the p-Selmer group of E/H with W over
Z[G] is 0 for all but finitely many primes p, where 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 (Bertolini and Darmon) and of the vanishing of the
p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank
zero.
