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Cube sum problem and an explicit Gross-Zagier formula
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 139, Number 3, June 2017
- pp. 785-816
- 10.1353/ajm.2017.0021
- Article
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A nonzero rational number is called a {\it cube sum} if it is of the form $a^3+b^3$ with $a,b\in{\Bbb Q}^\times$. 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 present also a general construction of Heegner points and obtain an explicit Gross-Zagier formula which is used to prove the Birch and Swinnerton-Dyer conjecture for certain elliptic curves related to the cube sum problem.