Locally Associated Orders in Real Quadratic Number Fields

TL;DR

The paper analyzes locally associated orders in real quadratic number fields, focusing on $K=\mathbb Q[\sqrt{p}]$ with prime $p$. It leverages unit-theoretic properties, the multiplicative $L(n,d)$ function, and a locality criterion tied to the minimal power of the fundamental unit in $R_n$ to derive concrete checks for when prime-power index orders are locally associated. A central result connects local associativity to explicit congruence-based conditions and Pell-type equations, notably showing that $R_p$ is locally associated iff the Pell equation $x^2-y^2p=1$ has a solution with $p\nmid y$ (for odd $p$), with broader implications for higher powers via coprime $L$-values. The work also identifies undetermined cases linked to Pell-type behavior and outlines a conjecture supported by extensive data, guiding future investigations into these arithmetic phenomena.

Abstract

In 2025, the concept of an order in a number field being associated, ideal-preserving, or locally associated was introduced in order to tackle problems in factorization. In this paper, we explore locally associated orders in real quadratic number fields of the form $\mathbb{Q}[\sqrt{p}]$, with $p\in\mathbb{N}$ prime. In particular, we develop strategies and produce results which make determining when a given order in such a number field is (or is not) locally associated much easier. We also highlight the relatively few cases which defy simple characterization, leading to a conjecture on the solutions to Pell's equations of the form $x^2-y^2p=1$.

Theorems & Definitions (44)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3