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

Abstract

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 χ of G c . Our main result is that if L K ( E , χ , 1 ) ≠ 0 then Sel p ( E / H c ) ⊗ χ 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 χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-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.