Alexander Schmidt: On pro-p fundamental groups of marked arithmetic curves

       Author: Alexander Schmidt
       Title: On pro-p fundamental groups of marked arithmetic curves
       Pages: 33
       in: J. reine u. angew. Math. 640 (2010) 203-235 (German version)

	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 S₀, 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 S₀ 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 ∪ S₀, the cup-product H¹(G_{S∪ S₀}^T(k)(p),F_p) &otimes H¹(G_{S∪ S₀}^T(k)(p),F_p) → H²(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.

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 S₀, 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 S₀ 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 ∪ S₀, the cup-product
H¹(G_{S∪ S₀}^T(k)(p),F_p) &otimes H¹(G_{S∪ S₀}^T(k)(p),F_p) → H²(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

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