2015/08/17 by Nobuyoshi Takahashi, Takahashi, Nobuyoshi · 1 citation
Computer Science · Mathematics · #14G32 (Primary) #20N02 #57M27 (Secondary) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT) #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1508.03937
openalex publication_date 2015/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a normal, separated and integral scheme of finite type over ℤ and M a set of closed points of X. To a Galois cover X of X unramified over M, we associate a quandle whose underlying set consists of points of X lying over M. As the limit of such quandles over all étale Galois covers and all étale abelian covers, we define topological quandles Q(X, M) and Qab(X, M), respectively. Then we study the problem of reconstruction. Let K be ℚ or a quadratic field, OK its ring of integers, X=Spec OK∖\\mathfrakp\ the complement of a closed point such that π1(X)ab is infinite, and M a set of maximal ideals with density 1. Using results from p-adic transcendental number theory, we show that K, \mathfrakp and the projection M\toSpec ℤ can be recovered from the topological quandle Q(X, M) or Qab(X, M).