On certain root number $1$ cases of the cube sum problem
Shamik Das, Somnath Jha
TL;DR
This work investigates which integers in root-number-1 families for the cube sum problem can be written as sums of two rational cubes by linking Mordell-curve rank questions to arithmetic invariants of cubic fields. The authors develop explicit necessary conditions expressed through the $2$-part and $3$-part of class groups of carefully chosen cubic fields, utilizing the $2$-Selmer and $3$-isogeny Selmer groups of the curve $E_{-432n^2}$ and cubic Hilbert symbols, together with parity conjectures. Theorem A and Theorem D provide concrete criteria for primes in classes modulo $9$ (notably $\ell\equiv7\pmod{9}$ and $\ell\equiv4\pmod{9}$) that link cube-sum status of $3\ell$ and $3\ell^2$ to the structure of cubic-field class groups, with corollaries giving explicit subgroup containment such as $\mathbb{Z}/6\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}$. The results include density statements showing a positive proportion of primes in these classes yield non-cube sums. Overall, the paper connects Diophantine cube-sum questions to deep arithmetic of cubic fields, offering verifiable criteria and density insights anchored in Selmer-group-parity techniques.
Abstract
We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}: Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of two rational cubes. We give explicit criteria in terms of the $2$-parts and $3$-parts of the ideal class groups of certain cubic number fields to determine whether such an $n$ is a cube sum. In particular, we study integers $n$ divisible by $3$ such that the global root number of $E_{-432n^2}$ is $1$. For example, for a prime $\ell \equiv 7 \pmod{9}$, we show that for $3\ell$ to be a sum of two rational cubes, it is necessary that the ideal class group of $\Q(\sqrt[3]{12\ell})$ contains $\frac{\Z}{6\Z}\oplus \frac{\Z}{3\Z}$ as a subgroup. Moreover, for a positive proportion of primes $\ell \equiv 7 \pmod{9}$, $3\ell$ can not be a sum of two rational cubes. A key ingredient in the proof is to explore the relation between the $2$-Selmer group and the $3$-isogeny Selmer group of $E_{-432n^2}$ with the ideal class groups of appropriate cubic number fields.