On certain root number $1$ cases of the cube sum problem

TL;DR

This paper investigates specific integer families where the elliptic curve's global root number is 1, but the integers may or may not be sums of two rational cubes, providing explicit criteria based on class groups of cubic fields.

Contribution

It introduces explicit criteria linking ideal class groups of cubic fields to the cube sum problem for integers with root number 1, especially those divisible by 3.

Findings

For primes ulfilling certain congruences, 3ulfilling certain congruences, 3ails to be a sum of two rational cubes.

The ideal class group structure of ields determines the cube sum property.

A positive proportion of primes ail to make 3ulfilling certain congruences, 3ails to be a sum of two rational cubes.

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.