Mordell-Weil ranks of quadratic twists of pairs of elliptic curves.
Motivated by a conjecture of Mazur about the topological closure of the set of rational points on algebraic varieties over \(\mathbb Q\), \textit{M. Kuwata} and \textit{L. Wang} [Int. Math. Res. Not. 4, No. 4, 113--123 (1993; Zbl 0804.14008)] proved that for elliptic curves \(E^1\) and \(E^2\) over \(\mathbb Q\) whose \(j\)-invariant are not simultaneously \(0\) or \(1728\), there are infinitely many square-free integers \(d\) such that the rank of the Mordell-Weil group of the \(d\)-quadratic twists \(E^i_d\) are both \(>0\). In this paper the authors consider the problem of the existence of infinitely many square-free integers \(d\) such that the rank of the Mordell-Weil group of the \(d\)-quadratic twists \(E^1_d\) and \(E^2_d\) are (i) both \(0\), or (ii) precisely one \(0\), for \(E^1\) and \(E^2\) two non-isogenous elliptic curves over \(\mathbb Q\). In the direction of question (i), they prove that there exist fundamental discriminants \(D_1\), \(D_2\) and a set of primes \(T\) of positive density such that the rank of \(E^1_{dD_1}\) and the rank of \(E^2_{dD_2}\) are \(0\), where \(d\) is any product of an even number of primes in \(T\), if the elliptic curves \(E^1\) and \(E^2\) are without rational \(2\)-torsion points. Results concerning question (ii) for twists of the modular curve \(X_0(19)\) and for some elliptic curves related to special ternary quadratic forms are obtained. Assuming the Birch and Swinnerton-Dyer conjecture, the authors can give an affirmative answer to question (ii) for general elliptic curves.
scientific article; zbMATH DE number 1886726