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arXiv · 1412.1950

Cube Sum Problem and an Explicit Gross-Zagier Formula

Li Cai·Jie Shu·Ye Tian
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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.

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Li Cai, Jie Shu, Ye Tian. 2014-12-05. Cube Sum Problem and an Explicit Gross-Zagier Formula. https://arxiv.org/abs/1412.1950

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