The one-variable main conjecture for elliptic curves with complex multiplication  pp. 353-372

The one-variable main conjecture for elliptic curves with complex multiplication

By Karl Rubin

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In a forthcoming paper [12] we will present a proof of the one- and two-variable “main conjectures” of Iwasawa theory for imaginary quadratic fields. This proof uses the marvelous recent methods of Kolyvagin [6], combined with ideas from [9] and [11] and a great deal of technical Iwasawa theory. Because it deals with the two-variable situation, with primes of degree two as well as those of degree one, and with all imaginary quadratic fields, the proof in [12] will necessarily be quite complicated and, at least at first glance, rather unintelligible.

The purpose of this paper is to present a proof of the one-variable main conjecture in the simplest setting (see §1 for the precise statement). That is, we consider only imaginary quadratic fields K of class number one, elliptic curves E defined over K with complex multiplication by K, and only primes of good reduction which split in K. This is the setting in which Coates and Wiles worked in [1] and [2]. These restrictions make it possible to simplify the proof considerably. However, the important ideas of the general proof do appear here, and even with these restrictions there are powerful applications (see Theorem 1.2).