Quandles associated to Galois covers of arithmetic schemes

Abstract

Let $X$ be a normal, separated and integral scheme of finite type over $\mathbb{Z}$ and $\mathcal{M}$ a set of closed points of $X$. To a Galois cover $\tilde{X}$ of $X$ unramified over $\mathcal{M}$, we associate a quandle whose underlying set consists of points of $\tilde{X}$ lying over $\mathcal{M}$. As the limit of such quandles over all \'etale Galois covers and all \'etale abelian covers, we define topological quandles $Q(X, \mathcal{M})$ and $Q^\mathrm{ab}(X, \mathcal{M})$, respectively. Then we study the problem of reconstruction. Let $K$ be $\mathbb{Q}$ or a quadratic field, $\mathcal{O}_K$ its ring of integers, $X=\mathrm{Spec} \mathcal{O}_K\setminus\{\mathfrak{p}\}$ the complement of a closed point such that $\pi_1(X)^\mathrm{ab}$ is infinite, and $\mathcal{M}$ a set of maximal ideals with density $1$. Using results from $p$-adic transcendental number theory, we show that $K$, $\mathfrak{p}$ and the projection $\mathcal{M}\to\mathrm{Spec} \mathbb{Z}$ can be recovered from the topological quandle $Q(X, \mathcal{M})$ or $Q^\mathrm{ab}(X, \mathcal{M})$.