Imaginary Quadratic Fields, Elliptic Curves and Pell surfaces

Abstract

This thesis consists of two parts: rst part (Chapters 1, 2 and 3 ) deals with the Cubic Pell's equation and Units of Pure Cubic Fields. We study an algorithm given by Barbeau to compute solutions of a cubic analogue of Pell's equation, x3 + my3 + m2z3 − 3mxyz = 1: For a pure cubic eld K = Q(p3 m) with ring of algebraic integers as OK; the above equation arises naturally in connection with the study of units in OK: Comparisons with other methods like the Jacobi-Perron algorithm are also done. Extensive computations using Python have been carried out and the tables are compared to those obtained by Wada. In the second part (Chater 4 & 5 ) we have related elliptic curves, imaginary quadratic elds, and Pell surfaces. Let Em be the elliptic curve y2 = x3 − m; where m > 0 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 eld K = Q(p−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 show that our homomorphism κ; the homomorphism of Hambleton and Lemmermeyer and the homomorphism φ of Soleng are related.