Pell surfaces and elliptic curves

Abstract

Let E<inf>m</inf> be the elliptic curve y2 = x3 - m, where m is a squarefree positive integer and - m = 2,3 (mod 4). Let Cl(K)[3] denote the 3-torsion subgroup of the ideal class group of the quadratic field K = Q ?-m). Let S<inf>3</inf>: y2 + mz2 = x3 be the Pell surface. We show that the collection of primitive integral points on S<inf>3</inf> coming from the elliptic curve E<inf>m</inf> 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 ? fromrational points of E<inf>m</inf> to Cl(K)[3] using 3-descent on E<inf>m</inf>, whose kernel contains 3E<inf>m</inf>(Q). We also explain how our homomorphism ?, the homomorphism ? of Hambleton and Lemmermeyer and the homomorphism ? of Soleng are related.