| Let E be an elliptic curve over Q and ℓ be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ℓ. Under certain assumptions, we prove the vanishing of the Galois cohomology group H1(Gal(K(E[ℓ i])/K),E[ℓ i]) for all i ≥ 1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ℓ-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture. |
