Title: Some new infinite families of non-$p$-rational real quadratic fields Show Chinese title

Title: 一些新的非$p$-有理实二次域的无限族

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. Show Chinese abstract

Abstract: 固定一个不包含$2$或$3$的有限素数集合$\{ p_j \}$。通过一些从尝试解决量子信息中的 SIC-POVM 问题中得出的观察，我们给出了一种简单的方法，用于构造一个无限族同时非$p_j$有理的实二次域，在任何$p_j$上都不分歧。 或者这些可以描述为无限序列的$\mathbb{Q}(\sqrt{D})$实例，对于不同的$D$，其中每个$p_j$是一个$k$-Wall-Sun-Sun 素数，或者等价地，是一个广义的 Fibonacci-Wieferich 素数。 这些技术的一个特点是，它们可以用来产生域$K=\mathbb{Q}(\sqrt{D})$，其中伽罗瓦群的挠群的$p$次幂循环部分，对于$K$的最大阿贝尔反$p$扩张在高于$p$的素数上不分裂，其大小为$p^a$，对于$a\geq1$可以任意大。