Vanishing of some cohomology groups and bounds for the Shafarevich–Tate groups of elliptic curves

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[ℓi])/K),E[ℓi]) 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.