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Transactions of the American Mathematical Society

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An explicit Gross-Zagier formula related to the Sylvester conjecture


Authors: Yueke Hu, Jie Shu and Hongbo Yin
Journal: Trans. Amer. Math. Soc. 372 (2019), 6905-6925
MSC (2010): Primary 11G05
DOI: https://doi.org/10.1090/tran/7760
Published electronically: January 16, 2019
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Abstract: Let $ p\equiv 4,7\mod 9$ be a rational prime number such that $ 3\mod p$ is not a cube. In this paper, we prove the $ 3$-part of $ \vert{\rm III}(E_p)\vert\cdot \vert{\rm III}(E_{3p^2})\vert$ is as predicted by the Birch and Swinnerton-Dyer conjecture, where $ E_p: x^3+y^3=p$ and $ E_{3p^2}: x^3+y^3=3p^2$ are the elliptic curves related to the Sylvester conjecture and cube sum problems.


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Additional Information

Yueke Hu
Affiliation: Department of Mathematics, ETH, Zurich, Switzerland
Email: huyueke2012@gmail.com

Jie Shu
Affiliation: School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s Republic of China
Email: shujie@tongji.edu.cn

Hongbo Yin
Affiliation: School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China
Email: yhb2004@mail.sdu.edu.cn

DOI: https://doi.org/10.1090/tran/7760
Received by editor(s): October 25, 2018
Published electronically: January 16, 2019
Additional Notes: The first author was supported by SNF-169247
The second author was supported by NSFC-11701092
The third author was partially supported by NSFC-11701548 and The Fundamental Research Funds of Shandong University.
Article copyright: © Copyright 2019 American Mathematical Society