The Galois module structure of certain arithmetic principal homogeneous spaces

Abstract

In this paper we obtain Galois module results for rings of integers of certain abelian extensions of cyclotomic extensions and of division fields for elliptic curves with complex multiplication. In the cyclotomic case we consider the rings of integers of non-ramified extensions, and in the elliptic case we are concerned with extensions arising from the Selmer group of the curve. In each case we can describe those rings of integers which are free Galois modules, by means of L-function congruences. This then is a further instance of how Galois module structure is dominated by L-functions, with the classic case being the tame theory (see [F], [T3]), where it is the root numbers of the sympletic characters of the Galois group which determine the Galois module structure. In broad terms, the results of this paper arise from marrying the algebraic results in [Tl] and certain powerful results from Iwasawa theory (both cyclotomic and elliptic). For aesthetic reasons, as well as for the sake of brevity, we introduce notation which will simultaneously cover both the cyclotomic and the elliptic case. We begin by defining the notation for