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 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 GSS0T(k)(p) coincides with the étale cohomology of the associated marked arithmetic curve. In particular, cd GSS0T(k)(p)=2.
Furthermore, we can choose S0 in such a way that kSS0T(p) realizes the maximal p-extension k(p) of the local field k for all ℘ ∈ SS0, the cup-product
   H1(GSS0T(k)(p),Fp) &otimes H1(GSS0T(k)(p),Fp) → H2(GSS0T(k)(p),Fp)
is surjective and the decomposition groups of the primes in S establish a free product inside GSS0T(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 ζpk.

       preprint pdf-file marked.pdf     Version in deutscher Sprache

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