Journal of the Ramanujan Mathematical Society

Volume 31, Issue 1, March 2016 pp. 63–77.

Pell surfaces and elliptic curves

Authors: K. J. Manasa and B. R. Shankar
Author institution:Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Surathkal

Summary: Let E m be the elliptic curve y 2 = x 3 - 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 3: y 2 + mz 2 = x3 be the Pell surface. We~show that the collection of primitive integral points on S 3 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.

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