Infinite Descent on Elliptic Curves with Complex Multiplication

Abstract

It is a pleasure to dedicate this paper to I. R. Safarevic, in recognition of his important work on the arithmetic of elliptic curves. As anyone who has worked on the arithmetic of elliptic curves is acutely aware, it is still dominated today, despite its long and rich history, by a wealth of tantilizing conjectures, which are convincingly supported by numerical evidence. The most important amongst these conjectures, at least from the point of view of diophantine equations, is the conjecture of Birch and Swinnerton-Dyer, which grew out of the attempt to apply to elliptic curves the quantitative local to global principles employed by Siegel in his celebrated work on quadratic forms. This conjecture is so well known that there is no need to repeat its precise statement here. However, we do wish to point out that no-one has yet found a direct and natural link between the existence of rational points of infinite order on an elliptic curve defined over a number field and the behaviour of its Hasse-Weil L-Series at the point s = 1 in the complex plane, as is predicted by the conjecture of Birch and Swinnerton-Dyer. Guided by Artin and Tate’s [15] success with the geometric analogue, most recent work has attempted to establish such a connexion indirectly by p-adic techniques which combine the classical infinite descent of Mordell and Weil with ideas from Iwasawa’s theory of Z _p -extensions of number fields. The first results in this direction were found by Mazur [10], who studied descent theory on abelian varieties over an arbitrary Z _p -extension of the base field, assuming only that the abelian variety has good ordinary reduction at every ramified prime for the Z _p -extension (we shall describe this by simply saying that the abelian variety is ordinary for the Z _p -extension). The present paper, which was motivated by earlier work of Wiles and myself [3], pursues the study of this descent theory in a very special case, which nevertheless contains important families of elliptic curves, whose arithmetic properties are presumably typical of all elliptic curves. Namely, we consider elliptic curves with complex multiplication, and a certain non-cyclotomic Z _p -extension of the base field, whose existence is closely associated with the hypothesis of complex multiplication, and for which the elliptic curve is ordinary (sec §2 for the precise definition). Much of the material in this article is not recent work and was presented in the Hermann Weyl Lectures at the Institute for Advanced Study, Princeton, in 1979, and will eventually form part of a more detailed set of notes on these lectures. However, the crucial results of §3, which are due to Bernadette Perrin-Riou [12], were only proven after these lectures took place. Finally, there have been two important recent developments on the problems discussed in this paper. Firstly, P. Schneider (see [13], [14], and a manuscript in preparation) has now established the equality of the analytic height and algebraic height for arbitrary abelian varieties and Z _p -extensions of the base field, for which the abelian variety is ordinary. Secondly, Mazur and Tate (see their article in this volume) have found some striking new descriptions of the analytic height attached to an elliptic curve over a Z _p -extension, subject always to the hypothesis that the elliptic curve is ordinary for the Z _p -extension.