Quandles associated to Galois covers of arithmetic schemes

TL;DR

This paper introduces topological quandles associated with Galois covers of arithmetic schemes and demonstrates how key number-theoretic data can be reconstructed from these quandles, linking algebraic geometry and number theory.

Contribution

It defines new topological quandles from Galois covers of schemes and shows they encode enough information to recover fundamental number-theoretic data.

Findings

Reconstruction of number fields and primes from quandles

Introduction of topological quandles for Galois covers

Connection between quandles and arithmetic scheme invariants

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 $Q^\mathrm{ab}(X, \mathcal{M})$.