An explicit Gross-Zagier formula related to the Sylvester Conjecture

Abstract

Let $p\equiv 4,7\mathrm{mod}9$ be a rational prime number such that $3\mathrm{mod}p$ is not a cubic residue. In this paper we prove the 3-part of the product of the full BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier formula, 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.

This is a new version of our last paper with the same title. We make a substantial revision based the referee's comment. In particular, collaborating with Yueke Hu, we give a theoretical computation of the local period integral which is technical and complicated in the last version