Elliptic curves and class field theory

2003/04/16 by Barry Mazur, Karl Rubin
Mathematics · #math.NT #msc:11G05 #msc:11R23

paper · pdf

published as Proceedings of the ICM, Beijing 2002, vol. 2, 185--196

arxiv created 2003/04/16 · arxiv updated 2009/11/30

Abstract

Suppose E is an elliptic curve defined over \Q. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on E over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of Mordell-Weil groups of E over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.

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