Parallelopipeds of Positive Rank Twists of Elliptic Curves

TL;DR

This paper demonstrates that for any dimension n, there exists an elliptic curve over rationals with an n-dimensional subspace where all quadratic twists have positive rank, revealing a rich structure of elliptic curves and their twists.

Contribution

It constructs explicit examples of elliptic curves with large-dimensional subspaces of twists having positive rank, advancing understanding of rank distribution in quadratic twists.

Findings

Existence of elliptic curves with n-dimensional subspaces of positive rank twists for all n

Construction of explicit examples of such elliptic curves

Insight into the structure of quadratic twists and their ranks

Abstract

For every n there exists an elliptic curve E over the rational numbers and an n-dimensional subspace V of non-zero rationals modulo squares such that for all v in V, the quadratic twist of E by v has positive rank.