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A Classical Family of Elliptic Curves having Rank One and the 2-Primary Part of their Tate-Shafarevich Group Non-Trivial

Yukako Kezuka Fakultät für Mathematik, Universität Regensburg, Germany Yongxiong Li Yau Mathematical Science Center, Tsinghua University, Beijing, China

Number Theory mathscidoc:2204.24004

Documenta Math, 25, 2115-2147, 2020.11
We study elliptic curves of the form x^3+y^3=2^p and x^3+y^3=2p^2 where p is any odd prime satisfying p ≡ 2 mod 9 or p ≡ 5 mod 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 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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@inproceedings{yukako2020a,
  title={A Classical Family of Elliptic Curves having Rank One and the 2-Primary Part of their Tate-Shafarevich Group Non-Trivial},
  author={Yukako Kezuka, and Yongxiong Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422144311963603104},
  booktitle={Documenta Math},
  volume={25},
  pages={2115-2147},
  year={2020},
}
Yukako Kezuka, and Yongxiong Li. A Classical Family of Elliptic Curves having Rank One and the 2-Primary Part of their Tate-Shafarevich Group Non-Trivial. 2020. Vol. 25. In Documenta Math. pp.2115-2147. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220422144311963603104.
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