On the Non p-Rationality and Iwasawa Invariants of Certain Real Quadratic Fields

Let \(p\) be an odd prime, and \(m,r \in \mathbb{Z}^+\) with \(m\) coprime to \(p\). In this paper we investigate the real quadratic fields \(K = \mathbb{Q}(\sqrt{m^2p^{2r} + 1})\). We first show that for \(m < C\), where constant \(C\) depends on \(p\), the fundamental unit \(\varepsilon\) of \(...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-08
Main Authors: Peikai Qi, Stokes, Matt
Format: Article
Language:English
Subjects:
Congruences
Fields (mathematics)
Invariants
Number theory
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let \(p\) be an odd prime, and \(m,r \in \mathbb{Z}^+\) with \(m\) coprime to \(p\). In this paper we investigate the real quadratic fields \(K = \mathbb{Q}(\sqrt{m^2p^{2r} + 1})\). We first show that for \(m < C\), where constant \(C\) depends on \(p\), the fundamental unit \(\varepsilon\) of \(K\) satisfies the congruence \(\varepsilon^{p-1} \equiv 1 \mod{p^2}\), which implies that \(K\) is a non \(p\)-rational field. Varying \(r\) then gives an infinite family of non \(p\)-rational fields. When \(m = 1\) and \(p\) is a non-Wieferich prime, we use a criterion of Fukuda and Komatsu to show that if \(p\) does not divide the class number of \(K\), then the Iwasawa invariants for cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) vanish. We conjecture that there are infinitely many \(r\) such that \(p\) does not divide the class number of \(K\).
ISSN:2331-8422