Arithmetic Applications:- The Elliptic Case

Abstract

In this final chapter, we will discuss two ways in which our algebraic machinery gives connections between certain L-functions and the Galois module structure of extensions arising from elliptic curves with complex multiplication. The first of these, which we will describe only briefly, is analogous to the cyclotomic result of the previous chapter, and the second concerns the implications of the conjecture of Birch and Swinnerton-Dyer for Galois module structure. Much, but not all, of the content of this chapter can be extended to abelian varieties with complex multiplication.