## An explicit Gross–Zagier formula related to the Sylvester conjecture

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- by Yueke Hu, Jie Shu and Hongbo Yin PDF
- Trans. Amer. Math. Soc.
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## Abstract:

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.## References

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## Additional Information

**Yueke Hu**- Affiliation: Department of Mathematics, ETH, Zurich, Switzerland
- MR Author ID: 1187394
- 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
- MR Author ID: 1005637
- Email: yhb2004@mail.sdu.edu.cn
- 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 6905-6925 - MSC (2010): Primary 11G05
- DOI: https://doi.org/10.1090/tran/7760
- MathSciNet review: 4024542