Introduction to Elliptic Curves

Abstract

The aim of this chapter is to give a brief survey of results, essentially without proofs, about elliptic curves, complex multiplication and their relations to class groups of imaginary quadratic fields. A few algorithms will be given (in Section 7.4, so as not to interrupt the flow of the presentation), but, unlike other chapters, the main emphasis will be on the theory (some of which will be needed in the next chapters). We also describe the superb landscape that is emerging in this theory, although much remains conjectural. It is worth noting that many of the recent advances on the subject (in particular the Birch and Swinnerton-Dyer conjecture) were direct consequences of number-theoretical experiments. This lends further support to the claim that number theory, even in its sophisticated areas, is an experimental as well as a theoretical science.