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
G_{S}^{T}(k)(p)=Gal(k_{S}^{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
S_{0}, which can be chosen disjoint from any given set M of
Dirichlet density zero, such that the cohomology of G_{S∪ S0}^{T}(k)(p)
coincides with the étale cohomology of the
associated marked arithmetic curve. In particular, cd G_{S∪ S0}^{T}(k)(p)=2.
Furthermore, we can choose S_{0} in such a way that k_{S∪ S0}^{T}(p) realizes the maximal p-extension k_{℘}(p) of the local field k_{℘} for all ℘ ∈ S ∪ S_{0}, the cup-product H^{1}(G_{S∪ S0}^{T}(k)(p),F_{p}) &otimes H^{1}(G_{S∪ S0}^{T}(k)(p),F_{p}) → H^{2}(G_{S∪ S0}^{T}(k)(p),F_{p}) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S∪ S0}^{T}(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))=1 and ζ_{p}∉ k. |
preprint pdf-file marked.pdf Version in deutscher Sprache markiert.de.pdf