A proof of Sylvester's conjecture

Abstract

We prove Sylvester's conjecture, originating in his 1879 study of ternary cubic equations, that every prime $p\equiv 4,7,8(\mathrm{mod}9)$ is a sum of two rational cubes. Elkies announced a proof for the classes $4$ and $7$ in 1994, and Yin recently supplied a complete proof. For the remaining class $p\equiv 8(\mathrm{mod}9)$, we prove that the elliptic curve $E_p:y^2=x^3+p^2/4$ has analytic rank one, as predicted by the Birch and Swinnerton-Dyer conjecture, and so $p$ is a sum of two rational cubes. The proof begins by adapting the auxiliary Rankin--Selberg construction from the authors' work on the rank one converse for CM elliptic curves. The Rankin--Selberg $L$-function factors as the $L$-function of $E_p$ times a complementary $L$-function. Chan's $3$-isogeny descent and the rank zero converse show that the complementary central $L$-value is non-zero, and so it suffices to prove that a cubic component of the associated Heegner point is non-torsion. A basic difficulty is that the unweighted Hecke trace of the underlying CM orbit vanishes. Our decisive idea is to take $\lambda$-division before taking the trace, where $\lambda =1-\omega$ and $\omega$ is a primitive cube root of unity. We prove that the resulting division boundary is non-zero by analysing Frobenius at $p$. The Galois action on the CM orbit and ramification theory then transfer this non-vanishing to the cubic component.