Pell Surfaces and Elliptic Curves

Let E_m 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 S₃ : y² + mz² = x³ be the Pell surface. We show that the collection of primitive integral points on S3 coming from the elliptic curve E_m 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 E_m to Cl(K)[3] using 3-descent on E_m, whose kernel contains 3E_m(Q). We also explain how our homomorphism κ, the homomorphism ψ of Hambleton and Lemmermeyer and the homomorphism φ of Soleng are related.