An explicit Gross–Zagier formula related to the Sylvester conjecture

Hu, Yueke; Shu, Jie; Yin, Hongbo

doi:10.1090/tran/7760

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. 372 (2019), 6905-6925 Request permission

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.

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