2008/06/04 by Alexander Schmidt, Schmidt, Alexander · 3 citations
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Historical Studies and Socio-cultural Analysis #Advanced Algebra and Geometry
paper · doi:10.48550/arxiv.0806.0772
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 GST(k)(p)=Gal(kST(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 GS∪ S0T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd GS∪ S0T(k)(p)=2. Furthermore, we can choose S0 in such a way that kS∪ S0T(p) realizes the maximal p-extension k_\p(p) of the local field k_\p for all \p∈ S∪ S0, the cup-product H1(GS∪ S0T(k)(p),\Fp) ⊗ H1(GS∪ S0T(k)(p),\Fp) --> H2(GS∪ S0T(k)(p),\Fp) is surjective and the decomposition groups of the primes in S establish a free product inside GS∪ S0T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=∅ under the restrictive assumption p\nmid Cl(k) and ζp∉ k.