Cubic residuacity of real quadratic integers

TL;DR

This work addresses identifying primes $p ≡ 1 mod 3$ for which a real quadratic integer $u=A+B sqrt(D)$ with cube norm is a cubic residue modulo $p$. By constructing the set $S_D$, the invariant $C(u)$, and the discriminant $4d$ with the form class group $H=H(4d)$, the authors prove a key equivalence between cubic residuacity and a cubic residue symbol via the Eisenstein integers, enabling a criterion $L(u)=1$ for residuacity. They then define a homomorphism $J$ from $H$ to the cube roots of unity, restrict to the subgroup $H_2$ of classes representing primes $p ≡ 1 mod 3$, and use Cebotarev density to show the kernel $G$ has index $6$ in $H$, giving a precise subgroup description of primes where $u$ is a cubic residue. In the principal-class case, this verifies a conjecture of Evans, Lemmermeyer, Sun, and Van Veen about the field generated by the real cube root of $u$, linking binary quadratic forms, class field theory, and cubic reciprocity in a density-and-structure framework.

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.

Theorems & Definitions (11)

• Theorem 1.1
• Example 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof