Rank Growth over Cubic Radical Extensions

Definitions and mathematical statement

Let $E/{\mathbb{Q}}$ be the particular elliptic curve intended in the source, and for $m\in{\mathbb{Z}}$ put \[ K_m={\mathbb{Q}}(\sqrt[3]{m}). \] A literal meaning of acquiring new points is strict enlargement under base change: \[ E(K_m)\setminus E({\mathbb{Q}})\ne\varnothing. \] The classification problem is to determine \[ \mathcal N_E=\{m\in{\mathbb{Z}}:E(K_m)\ne E({\mathbb{Q}})\}. \] Here rational points mean points rational over $K_m$, and the inclusion of $E({\mathbb{Q}})$ is the natural one. The curve's equation, conductor, or any other identifying data are not supplied, so $\mathcal N_E$ cannot be specialized to a definite curve.

The title and assessment instead emphasize rank growth. That would ask for the potentially different set \[ \mathcal R_E=\{m\in{\mathbb{Z}}:\operatorname{rank}E(K_m)>\operatorname{rank}E({\mathbb{Q}})\}, \qquad \operatorname{rank}E(K)=\dim_{{\mathbb{Q}}}(E(K)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}). \] Rank growth implies new points, but strict point-group enlargement could concern torsion or finite-index enlargement rather than rank. Thus the two targets must not be identified without a source-specific argument. No claim here asserts that either phenomenon occurs for the unspecified curve.

If $m$ is an integer cube, including zero, then $K_m={\mathbb{Q}}$; for noncubes the extension has degree three. Different integers can define the same field, so an integer classification and a classification of distinct field extensions have different indexing. The source imposes no cube-free, positivity, or nonzero restriction on $m$, and none is added.

The supplied assessment checked 2026-08-21 says that the passage is too truncated to identify a solved finite instance and that the corresponding general rank-growth question remains open. The corrected-verification field calls the curve explicit, but the supplied text names it only as $E$; this missing identity is an unresolved formulation issue.

Short English statement

For the intended elliptic curve, classify cubic radical fields that add points, or, if intended, increase its Mordell--Weil rank. The curve itself is not identified, and adding points is not automatically the same as increasing rank.

Sources

• Local UnsolvedMath registry, permanent ID 30000276, catalogue entry, Rank Growth over Cubic Radical Extensions.
• Explicit Methods in Number Theory (2006), Oberwolfach Reports, pp. 1799--1866, DOI https://doi.org/10.4171/owr/2005/32.
• Supplied reference labeled "Dokchitser noncommutative Iwasawa computations": https://arxiv.org/abs/math/0509286.
• Supplied literature triage checked 2026-08-21; no defining equation for $E$, rank theorem, or classification criterion is reproduced.