Sums of rational cubes and the $3$-Selmer group

Recently, Alp\"oge-Bhargava-Shnidman determined the average size of the $2$-Selmer group in the cubic twist family of any elliptic curve over $\mathbb{Q}$ with $j$-invariant $0$. We obtain the distribution of the $3$-Selmer groups in the same family. As a consequence, we improve their upper bound on the density of integers expressible as a sum of two rational cubes. Assuming a $3$-converse theorem, we also improve their lower bound on this density. The $\sqrt{-3}$-Selmer group in this cubic twist family is well-known to be large, which poses significant challenges to the methods previously developed by the second author. We overcome this problem by strengthening the analytic core of these methods. Specifically, we prove a "trilinear large sieve" for an appropriate generalization of the classical R\'edei symbol, then use this to control the restriction of the Cassels-Tate pairing to the $\sqrt{-3}$-Selmer groups in these twist families.