On pro-p fundamental groups of marked arithmetic curves

Original · EN

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) of the local field k for all ∈ S∪ S₀, the cup-product H¹(Gₛ∪ ₛ₀ᵗ(k)(p),ₚ) ⊗ H¹(Gₛ∪ ₛ₀ᵗ(k)(p),ₚ) --> H²(Gₛ∪ ₛ₀ᵗ(k)(p),ₚ) 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 Cl(k) and ζₚ∉ k.