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arXiv · 2303.05058

On the quadratic twist of elliptic curves with full $2$-torsion

Zhangjie Wang·Shenxing Zhang
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Abstract

Let $E: y^2=x(x-a^2)(x+b^2)$ be an elliptic curve with full $2$-torsion group, where $a$ and $b$ are coprime integers and $2(a^2+b^2)$ is a square. Assume that the $2$-Selmer group of $E$ has rank two. We characterize all quadratic twists of $E$ with Mordell-Weil rank zero and $2$-primary Shafarevich-Tate groups $(\mathbb Z/2\mathbb Z)^2$, under certain conditions. We also obtain a distribution result of these elliptic curves.

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Zhangjie Wang, Shenxing Zhang. 2023-03-09. On the quadratic twist of elliptic curves with full $2$-torsion. https://arxiv.org/abs/2303.05058

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