Galois Theory for the Selmer Group of an Abelian Variety

Abstract

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 G ( 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 G ( 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. Mazurand M. Harris.