Edited by J. Coates
Edited by M. J. Taylor
Publisher: Cambridge University Press
Print Publication Year: 1991
Online Publication Date:December 2009
Online ISBN:9780511526053
Paperback ISBN:9780521386197
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511526053.015
Subjects: Number Theory
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INTRODUCTION
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).
pp. i-iv
pp. v-vi
pp. vii-vii
pp. viii-viii
Lectures on automorphic L-functions : Read PDF
pp. 1-60
Gauss sums and local constants for GL(N) : Read PDF
pp. 61-74
L-functions and Galois modules : Read PDF
pp. 75-140
Motivic p-adic L-functions : Read PDF
pp. 141-172
The Beilinson conjectures : Read PDF
pp. 173-210
Iwasawa theory for motives : Read PDF
pp. 211-234
Kolyvagin's work for modular elliptic curves : Read PDF
pp. 235-256
Index theory, potential theory, and the Riemann hypothesis : Read PDF
pp. 257-270
Katz p-adic L-functions, congruence modules and deformation of Galois representations : Read PDF
pp. 271-294
Kolyvagin's work on Shafarevich-Tate groups : Read PDF
pp. 295-316
Arithmetic of diagonal quartic surfaces I : Read PDF
pp. 317-338
On certain Artin L-Series : Read PDF
pp. 339-352
The one-variable main conjecture for elliptic curves with complex multiplication : Read PDF
pp. 353-372
Remarks on special values of L-functions : Read PDF
pp. 373-392