On a comparison of Cassels pairings of different elliptic curves

Abstract

Let $e_1,e_2,e_3$ be nonzero integers satisfying $e_1+e_2+e_3=0$. Let $(a,b,c)$ be a primitive triple of odd integers satisfying $e_1{a}^2+e_2{b}^2+e_3{c}^2=0$. Denote by $E:y^2=x(x-e_1)(x+e_2)$ and $\mathcal{E}:y^2=x(x-e_1{a}^2)(x+e_2{b}^2)$. 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 $e_1{e}_2{e}_{3}abc$. 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 holds for ${\mathcal{E}}^{(n)}$. We also give some applications for the congruent elliptic curve.

20 pages