High rank quadratic twists of pairs of elliptic curves

TL;DR

This paper constructs infinite families of elliptic curve pairs over rationals with infinitely many quadratic twists having rank at least 2, extending previous results on positive rank twists.

Contribution

It introduces explicit infinite families of elliptic curve pairs with quadratic twists of rank at least 2, advancing understanding of rank growth in quadratic twists.

Findings

Existence of infinite families with rank ≥ 2 for quadratic twists.

Construction of explicit examples of such elliptic curve pairs.

Extension of previous positive rank results to higher ranks.

Abstract

Given a pair of elliptic curves $E_1$ and $E_2$ over the rational field $\mathbb Q$ whose $j$-invariants are not simultaneously 0 or 1728, Kuwata and Wang proved the existence of infinitely many square-free rationals $d$ such that the $d$-quadratic twists of $E_1$ and $E_2$ are both of positive rank. We construct infinite families of pairs of elliptic curves $E_1$ and $E_2$ over $\mathbb Q$ such that for each pair there exist infinitely many square-free rationals $d$ for which the $d$-quadratic twists of $E_1$ and $E_2$ are both of rank at least 2.