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

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 $K=\mathbb{Q}(\sqrt{m^2{p}^{2r}+1})$. We first show that for $m<C$, where constant $C$ depends on $p$, the fundamental unit $\epsilon$ of $K$ satisfies the congruence ${\epsilon }^{p-1}\equiv 1\mathrm{mod}{p}^2$, 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 $K$.

16 pages, comments are welcome