## Abstract

Let *k* be a global field, *p* an odd prime number different from char(*k*) and *S*, *T* disjoint, finite sets of primes of *k*. Let 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 *S*
_{0}, which can be chosen disjoint from any given set ℳ of Dirichlet density zero, such that the cohomology of coincides with the étale cohomology of the associated marked arithmetic curve. In particular, . Furthermore, we can choose *S*
_{0} in such a way that realizes the maximal *p*-extension *k*
_{𝔭}(*p*) of the local field *k*
_{𝔭} for all 𝔭 ∈ *S* ∪ *S*
_{0}, the cup-product is surjective and the decomposition groups of the primes in *S* establish a free product inside . This generalizes previous work of the author where similar results were shown in the case *T* = ∅ under the restrictive assumption *p* ∤ #Cl(*k*) and ζ_{p} ∉*k*.

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