IUMJ

Title: Parallelopipeds of positive rank twists of elliptic curves

Authors: Bo-Hae Im and Michael Larsen

Issue: Volume 60 (2011), Issue 1, 311-318

Abstract:

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.