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

TL;DR

This paper studies certain real quadratic fields, proving their non p-rationality for small parameters and showing conditions under which their Iwasawa invariants vanish, with conjectures on infinite class number divisibility.

Contribution

It establishes non p-rationality for a family of real quadratic fields and links class number divisibility to vanishing Iwasawa invariants, extending understanding of these fields.

Findings

For small m, fundamental units satisfy a specific congruence implying non p-rationality.

An infinite family of non p-rational fields is constructed by varying r.

Under certain conditions, Iwasawa invariants for these fields vanish.

Abstract

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 $K$.