





Pell Surfaces and Elliptic Curves
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Let Em 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 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.
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