A Classical Family of Elliptic Curves having Rank One and the 22-Primary Part of their Tate-Shafarevich Group Non-Trivial

  • Yukako Kezuka

    Fakultät für Mathematik, Universität Regensburg, 93049 Regensburg, Germany
  • Yongxiong Li

    Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial cover
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Abstract

We study elliptic curves of the form x3+y3=2px^3+y^3=2p and x3+y3=2p2x^3+y^3=2p^2 where pp is any odd prime satisfying p2mod9p\equiv 2 \mod 9 or p5mod9p\equiv 5 \mod 9. We first show that the 33-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their 22-Selmer group to the 22-rank of the ideal class group of Q(p3)\mathbb{Q}(\sqrt[3]{p}) to obtain some examples of elliptic curves with rank one and non-trivial 22-part of the Tate-Shafarevich group.

Cite this article

Yukako Kezuka, Yongxiong Li, A Classical Family of Elliptic Curves having Rank One and the 22-Primary Part of their Tate-Shafarevich Group Non-Trivial. Doc. Math. 25 (2020), pp. 2115–2147

DOI 10.4171/DM/795