# MathSciDoc: An Archive for Mathematician ∫

#### Number Theorymathscidoc: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.
@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.