Arithmetic on elliptic curves with complex multiplication. II (Q1070305)

This paper is a continuation of the study of the arithmetic of elliptic curves with complex multiplication by \(\mathbb Q(\sqrt{-p})\) which was initiated by \textit{B. H. Gross} [``Arithmetic on elliptic curves with complex multiplication'', Lect. Notes Math. 776 (1980; Zbl 0433.14032)]. In general, the discussion concerns elliptic curves defined over the Hilbert class field, \(H\), of \(\mathbb Q(\sqrt{-p})\) with \(j\) invariant that of the ring of integers \({\mathcal O}\). Such curves admit complex multiplication by \({\mathcal O}\), and those which are isogenous over \(H\) to all their Galois conjugates are called \(\mathbb Q\)-curves. Various results on the eigenspaces of certain Selmer groups are obtained, and in chapter III there is a refinement of the Birch and Swinnerton-Dyer conjecture in the case when the \(\mathbb Q\)-curve has finitely many \(H\)-rational points. The final chapter presents and discusses some computations in support of the conjecture.