A Stickelberger index for the Tate–Shafarevich group

Abstract

In this paper, we obtain an algebraic description of the critical value of the L–series of a class of elliptic curves over an arbitrary real abelian number field. We do this by construction of a matching Stickelberger–ideal and computation of its index. By inserting the result into the formula of the conjecture of Birch and Swinnerton–Dyer, we can describe the order of the Tate–Shafarevich group by a number of algebraic expressions.