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

TL;DR

This paper proves that for primes p congruent to 2 or 5 modulo 9, at least one of 3p or 3p^2 is a cube sum by constructing Heegner points and establishing explicit Gross-Zagier formulas.

Contribution

It introduces new methods to verify cube sums of specific forms using Heegner points and provides explicit formulas related to the Birch and Swinnerton-Dyer conjecture.

Findings

At least one of 3p or 3p^2 is a cube sum for primes p ≡ 2,5 mod 9

Explicit Gross-Zagier formulas for the constructed Heegner points

Variants of the BSD conjecture for related elliptic curves

Abstract

Let $p\equiv 2,5\mod 9$ 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.