Mordell–Weil Ranks of Quadratic Twists of Pairs of Elliptic Curves Original Research Article

Motivated by a conjecture of Mazur, Kuwata and Wang proved that for elliptic curves E1 and E2 whose j-invariants are not simultaneously 0 or 1728, there exist infinitely many square-free integers d for which the rank of the Mordell–Weil group of the d-quadratic twists of E1 and E2 satisfy: rk(Ed1, image)>0 and rk(Ed2, image)>0. Here we present results for the related questions: Are there infinitely many square-free integers d for which: rk(Ed1, image)=0 and rk(Ed2, image)=0? And, are there infinitely many square-free integers d for which: rk(Ed1, image)=0 and rk(Ed2, image)>0?