Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven

Abstract

Let k be a global field, p an odd prime number dierent from char (k) and S, T disjoint, finite sets of primes of k. Let G T(k)(p) = G(k T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S0, which can be chosen disjoint from any given setM of Dirichlet density zero, such that the cohomology of G T[S0 (k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G T[S0 (k)(p) = 2. Furthermore, we can choose S0 in such a way that k T S[S0 (p) realizes the maximal p-extension kp(p) of the local field kp for all p2 S[S0, the cup-product H 1 (G T[S0 (k)(p),Fp) H 1 (G T[S0 (k)(p),Fp)! H 2 (G T[S0 (k)(p),Fp) is surjective and the decomposition groups of the primes in S establish a free product inside G T[S0 (k)(p). This generalizes previous work of the author where similar results were shown in the case T = ? under the restrictive assumption p - #Cl(k) and p / 2 k.