Cubic residuacity of real quadratic integers

TL;DR

This paper characterizes primes for which a specific real quadratic integer with cubic norm is a cubic residue, extending a conjecture from a 2025 paper and linking class group classes to residue properties.

Contribution

It identifies all classes in a form class group that correspond to primes where a given quadratic integer is a cubic residue, generalizing a prior conjecture.

Findings

Classifies primes based on their relation to quadratic integer residues.

Extends a conjecture from previous research to a broader setting.

Provides a complete characterization of residue classes for specific quadratic integers.

Abstract

Given a real quadratic integer $u=A+B\sqrt{D}$ with cubic norm, we identify all the classes in a related form class group that represent primes $p$ for which $u$ is a cubic residue mod $p$. A special case of this result was conjectured in a 2025 paper of Evans, Lemmermeyer, Sun, and Van Veen.