\"Uber Pro-p-Fundamentalgruppen markierter arithmetischer Kurven

TL;DR

This paper proves the existence of a finite set of primes S_0 such that the Galois group of a maximal p-extension over a global field has cohomological dimension 2 and matches the etale cohomology of a related arithmetic curve, generalizing previous results.

Contribution

It introduces a method to select primes S_0 ensuring the Galois group's cohomology aligns with etale cohomology, extending prior work to more general settings.

Findings

G_{S∪S_0}^T(k)(p) has cohomological dimension 2.

The Galois extension realizes maximal p-extensions locally.

The cup-product map is surjective, and decomposition groups form a free product.

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 G_S^T(k)(p)=Gal(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\cup S_0}^T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G_{S\cup S_0}^T(k)(p)=2. Furthermore, we can choose S_0 in such a way that k_{S\cup S_0}^T(p) realizes the maximal p-extension k_\p(p) of the local field k_\p for all \p\in S\cup S_0, the cup-product H^1(G_{S\cup S_0}^T(k)(p),\F_p) \otimes H^1(G_{S\cup S_0}^T(k)(p),\F_p) --> H^2(G_{S\cup S_0}^T(k)(p),\F_p) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S\cup S_0}^T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=\emptyset under the restrictive assumption p\nmid Cl(k) and \zeta_p\notin k.