Parallelopipeds of positive rank twists of elliptic curves Bo-Hae ImMichael Larsen 11G05elliptic curveMordell-Weil grouppositive rankquadratic twist Let $E$ be an elliptic curve over $\mathbb{Q}$ for which the set of quadratic twists with positive rank has positive density. Then for every $n\in\mathbb{N}$ there exists a $w\in\mathbb{Q}^{\times}/{\mathbb{Q}^{\times}}^2$ and an $n$-dimensional subspace $V$ of $\mathbb{Q}^{\times}/{\mathbb{Q}^{\times}}^2$ such that for all $v\in V$, the quadratic twist $E_{vw}$ has positive rank. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4398 10.1512/iumj.2011.60.4398 en Indiana Univ. Math. J. 60 (2011) 311 - 318 state-of-the-art mathematics http://iumj.org/access/