Quadratic twists of pairs of elliptic curves

Abstract

Given two elliptic curves defined over a number field K, not both with j-invariant zero, we show that there are infinitely many $D\in K^{\times }$ with pairwise distinct image in $K^{\times }/{K^{\times }}^2$, such that the quadratic twist of both curves by D have positive Mordell-Weil rank. The proof depends on relating the values of pairs of cubic polynomials to rational points on another elliptic curve, and on a fiber product construction.