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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk

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Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven

Alexander Schmidt1

1NWF I—Mathematik, Universität Regensburg, 93040 Regensburg, Deutschland. e-mail:

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 640, Pages 203–235, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2010.025, February 2010

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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 ∈ SS 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 ζpk.

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