QUANDLES ASSOCIATED TO GALOIS COVERS OF ARITHMETIC SCHEMES

Abstract

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 \ {p} the complement of a closed point such that π1(X)ab is infinite, and M a set of primes with Dirichlet density one. Using results from p-adic transcendental number theory, we show that K, p and the projection M → Spec Z can be recovered from the topological quandle Q(X, M) or Qab(X, M).