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An explicit Gross-Zagier formula related to the Sylvester Conjecture

Yueke Hu Department of Mathematics, ETH, Zurich, Switzerland Jie Shu School of Mathematical Sciences, Tongji University, Shanghai 200092, People’s Republic of China Hongbo Yin School of Mathematics, Shandong University, Jinan 250100, People’s Republic of China

Number Theory mathscidoc:2206.24011

Transactions of the American Mathematical Society, 372, (10), 6905-6925, 2019.1
Let $p\equiv 4,7\ \mathrm {mod}\ 9$ be a rational prime number such that $3\ \mathrm {mod}\ p$ is not a cube. In this paper, we prove the $3$-part of $|\textrm {III}(E_p)|\cdot |\textrm {III}(E_{3p^2})|$ 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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@inproceedings{yueke2019an,
  title={An explicit Gross-Zagier formula related to the Sylvester Conjecture},
  author={Yueke Hu, Jie Shu, and Hongbo Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629150446339173487},
  booktitle={Transactions of the American Mathematical Society},
  volume={372},
  number={10},
  pages={6905-6925},
  year={2019},
}
Yueke Hu, Jie Shu, and Hongbo Yin. An explicit Gross-Zagier formula related to the Sylvester Conjecture. 2019. Vol. 372. In Transactions of the American Mathematical Society. pp.6905-6925. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220629150446339173487.
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