HOMOMORPHISMS FROM THE GROUP OF RATIONAL-POINTS ON ELLIPTIC-CURVES TO CLASS-GROUPS OF QUADRATIC NUMBER-FIELDS

The main purpose of this paper is to prove that there is a homomorphism from the group of primitive points on an elliptic curve given by an equation Y2 = X3 + a2X2 + a4X + a6 to the ideal class group of the order Z + Z square-root a6. Two applications are given. First we prove a conjecture concerning the order of ideals coming from rational points of infinite order on the curve. Then we describe how to construct families of quadratic number fields containing a subgroup of the ideal class group isomorphic to the torsion group of the curve. (C) 1994 Academic Press, Inc.