Pell Surfaces and Elliptic Curves

Abstract

Let Em be the elliptic curve y2 = x3 a�� m, where m is a squarefree positive integer and a��m a�� 2, 3 (mod 4). Let Cl(K)[3] denote the 3-torsion subgroup of the ideal class group of the quadratic field K = Q( a�� a��m). Let S3 - y2 + mz2 = x3 be the Pell surface. We show that the collection of primitive integral points on S3 coming from the elliptic curve Em do not form a group with respect to the binary operation given by Hambleton and Lemmermeyer. We also show that there is a group homomorphism κ from rational points of Em to Cl(K)[3] using 3-descent on Em, whose kernel contains 3Em(Q). We also explain how our homomorphism κ, the homomorphism ψ of Hambleton and Lemmermeyer and the homomorphism φ of Soleng are related.