Zusammenfassung
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) = G(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, ...
Zusammenfassung
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) = G(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 boolean OR S0)(T)(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G(S boolean OR S0)(T)(k) (p) = 2. Furthermore, we can choose So in such a way that k(S boolean OR S0)(T)(p) realizes the maximal p-extension k(p)(p) of the local field k(p) for all p is an element of S boolean OR S(0). the cup-product H(1)(G(S boolean OR S0)(T)(k)(p), F(p)) circle times H(1) (G(S boolean OR S0)(T)(k)(p),F(p)) -> H(2) (G(S boolean OR S0)(T)(k)(p),F(p)) is surjective and the decomposition groups of the primes in S establish a free product inside G(S boolean OR S0)(T)(k)(p). This generalizes previous work of the author where similar results were shown in the case T = 0 under the restrictive assumption p inverted iota l #C1(k) and zeta(p) is not an element of k.
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