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
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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