A Classical Family of Elliptic Curves having Rank One and the -Primary Part of their Tate-Shafarevich Group Non-Trivial
Yukako Kezuka
Fakultät für Mathematik, Universität Regensburg, 93049 Regensburg, GermanyYongxiong Li
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
Abstract
We study elliptic curves of the form and where is any odd prime satisfying or . We first show that the -part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their -Selmer group to the -rank of the ideal class group of to obtain some examples of elliptic curves with rank one and non-trivial -part of the Tate-Shafarevich group.
Cite this article
Yukako Kezuka, Yongxiong Li, A Classical Family of Elliptic Curves having Rank One and the -Primary Part of their Tate-Shafarevich Group Non-Trivial. Doc. Math. 25 (2020), pp. 2115–2147
DOI 10.4171/DM/795