An arithmetic site for the rings of integers of algebraic number fields

Abstract

The study of extensions with prescribed ramification plays a central role in algebraic number theory. It is a general hope to benefit from the analogy between number fields and function fields of trancendency degree 1 over finite fields, because for the latter the situation is much better understood. In this paper we introduce a new kind of restriction for the ramification of primes in extensions of algebraic number fields and we study the corresponding covering type over the ring of integers. We will prove that the associated fundamental group has a description similar to that of the 6tale fundamental group of a smooth, projective curve over a finite field. Let K be a number field, p be a prime number and let S be a finite set of places of K containing all archimedian places and all places dividing p. Extensions of K which are unramified outside S correspond to &ale covers of the open subscheme Spec(Ox, s) of Spec(Cx). We denote the maximal pro-p Galois extension of K which is unramified outside S by Ks(p) and we write