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arXiv · 1911.04532

A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial

Yukako Kezuka·Yongxiong Li
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Abstract

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.

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Yukako Kezuka, Yongxiong Li. 2019-11-11. A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial. https://doi.org/10.25537/dm.2020v25.2115-2147

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