Cube Sum Problem and an Explicit Gross-Zagier Formula

TL;DR

This paper investigates the cube sum problem, constructs Heegner points, and derives an explicit Gross-Zagier formula to prove cases of the Birch and Swinnerton-Dyer conjecture for specific elliptic curves.

Contribution

It provides a new explicit Gross-Zagier formula and demonstrates its application to the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to cube sums.

Findings

Existence of infinitely many cube-free integers with specified prime factors where 2n is a cube sum or not.

Construction of Heegner points for the problem.

An explicit Gross-Zagier formula used to verify the BSD conjecture in specific cases.

Abstract

A nonzero rational number is called a cube sum if it is of form $a^3+b^3$ with $a,b\in \mathbb{Q}^\times$. In this paper, we prove that for any odd integer $k\geq 1$, there exist infinitely many cube-free odd integers $n$ with exactly $k$ distinct prime factors such that $2n$ is a cube sum (resp. not a cube sum). We give also a general construction of Heegner point and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curve related to the cube sum problem.