On a comparison of Cassels pairings of different elliptic curves
Volume 211 / 2023
Abstract
Let be nonzero integers satisfying e_1+e_2+e_3=0. Let (a,b,c) be a primitive triple of odd integers satisfying e_1a^2+e_2b^2+e_3c^2=0. Consider the elliptic curves E: y^2=x(x-e_1)(x+e_2) and \mathcal E: y^2=x(x-e_1a^2)(x+e_2b^2). Assume that the 2-Selmer groups of E and \mathcal E are minimal. Let n be a positive square-free odd integer, where the prime factors of n are nonzero quadratic residues modulo each odd prime factor of e_1e_2e_3abc. Then under certain conditions, the 2-Selmer group and the Cassels pairing of the quadratic twist E^{(n)} coincide with those of \mathcal E^{(n)}. As a corollary, E^{(n)} has Mordell–Weil rank zero without order 4 element in its Shafarevich–Tate group, if and only if these hold for \mathcal E^{(n)}. We also give some applications for the congruent number elliptic curve.