Galois theory for the Selmer group of an Abelian variety

This paper concerns the Galois theoretic behavior of the p-primary subgroup Sel(A)(F)(p) of the Selmer group for an Abelian variety A defined over a number field F in an extension K/F such that the Galois group Gd(K/F) is a p-adic Lie group. Here p is any prime such that A has potentially good, ordinary reduction at all primes of F lying above p. The principal results concern the kernel and the cokernel of the natural map s(K/F): Sel(A)(F')(p) --> Sel(A)(K)(p)(Gd(K/F')) where F' is any finite extension of F contained in K. Under various hypotheses on the extension K/F, it is proved that the kernel and cokernel are finite. More precise results about their structure are also obtained. The results are generalizations of theorems of B. Mazur and M. Harris.