Cube sums of form $3p$ and $3p^2$

Abstract

Let $p\equiv 2,5\mathrm{mod}9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit Gross-Zagier formulae for these Heegner points and give variants of the Birch and Swinnerton-Dyer conjecture of the related elliptic curves.

arXiv admin note: text overlap with arXiv:1708.05266