ON THE VANISHING OF COHOMOLOGIES OF p-ADIC GALOIS REPRESENTATIONS ASSOCIATED WITH ELLIPTIC CURVES

Abstract

Let K be a p-adic field and let E be an elliptic curve over K with potential good reduction. For some large Galois extension L of K containing all p-power roots of unity, we show the vanishing of certain Galois cohomology groups of L with values in the p-adic representation associated with E. We use these to prove analogous results in the global case. This generalizes some results of Coates, Sujatha and Wintenberger.