A nonzero rational number is called a cube sum if it is of the form a³+b³, with a,bÎ. To determine whether a rational number  is a cube sum is closely related to the arithmetic of the corresponding elliptic curve . A famous conjecture concerning the cube sums is the so-called Sylvester conjecture: Any prime congruent to 4,7,8 mod 9 is a cube sum. Dasgupta and Voight proved certain primes 4, 7 mod 9 are cube sums by establishing the nontriviality of certain related Heegner points. Based on the work of Dasgupta and Voight, we prove the Birch and Swinnerton-Dyer conjecture for the related elliptic curves by establishing the explicit Gross-Zagier formulae of the related Heegner points. This is a joint work with Hongbo Yin.

海报