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

TL;DR

This paper studies the distribution of 3-Selmer groups in cubic twist families of elliptic curves with j-invariant 0, improving bounds on the density of sums of two rational cubes by developing new analytic techniques.

Contribution

It determines the distribution of 3-Selmer groups in these families and introduces a trilinear large sieve to handle large -Selmer groups, advancing understanding of rational cube sums.

Findings

Improved upper bound on the density of integers as sums of two rational cubes.

Conditional improvement of the lower bound on this density.

Development of a trilinear large sieve for generalized Redei symbols.

Abstract

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.