Geodesic
A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial
Documenta mathematica, Tome 25 (2020), pp. 2115-2147
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We study elliptic curves of the form x3+y3=2p and x3+y3=2p2 where p is any odd prime satisfying p≡2mod9 or p≡5mod9. 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(3p​) to obtain some examples of elliptic curves with rank one and non-trivial 2-part of the Tate-Shafarevich group.
DOI : 10.4171/dm/795
Classification : 11G05, 11R29
Mots-clés : elliptic curves, complex multiplication, Tate-Shafarevich group, ideal class group 
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     author = {Yukako Kezuka and Yongxiong Li},
     title = {A {Classical} {Family} of {Elliptic} {Curves} having {Rank} {One} and the $2${-Primary} {Part} of their {Tate-Shafarevich} {Group} {Non-Trivial}},
     journal = {Documenta mathematica},
     pages = {2115--2147},
     year = {2020},
     volume = {25},
     doi = {10.4171/dm/795},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/795/}
}
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Yukako Kezuka; Yongxiong Li. A Classical Family of Elliptic Curves having Rank One and the $2$-Primary Part of their Tate-Shafarevich Group Non-Trivial. Documenta mathematica, Tome 25 (2020), pp. 2115-2147. doi: 10.4171/dm/795

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