Some new infinite families of non-$p$-rational real quadratic fields

Abstract

Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real quadratic fields, unramified above any of the $p_j$. Alternatively these may be described as infinite sequences of instances of $\mathbb{Q}(\sqrt{D})$, for varying $D$, where every $p_j$ is a $k$-Wall-Sun-Sun prime, or equivalently a generalised Fibonacci-Wieferich prime. One feature of these techniques is that they may be used to yield fields $K=\mathbb{Q}(\sqrt{D})$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $p^a$ for $a\geq1$ arbitrarily large.