  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.
