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arXiv · 1610.07967

High rank quadratic twists of pairs of elliptic curves

Mohammad Sadek·Mohamed Alaa
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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.

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Mohammad Sadek, Mohamed Alaa. 2016-10-25. High rank quadratic twists of pairs of elliptic curves. https://arxiv.org/abs/1610.07967

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