Parallelopipeds of Positive Rank Twists of Elliptic Curves

Abstract

Let E be an elliptic curve over Q for which the set of quadratic twists with positive rank has positive density. Then for every n epsilon N there exists a w epsilon Q(x)/Q(x2) and an n-dimensional subspace V of Q(x)/Q(x2) such that for all v epsilon V, the quadratic twist E-vw has positive rank