On pro-p fundamental groups of marked arithmetic curves

TL;DR

This paper proves the existence of a finite set of primes that, when added to a given set, aligns the Galois group's cohomology with etale cohomology of an associated arithmetic curve, revealing structural properties of maximal p-extensions.

Contribution

It generalizes previous results by establishing conditions under which the Galois group of maximal p-extensions matches etale cohomology, including cases with additional splitting conditions and without restrictive assumptions.

Findings

Cohomology of G_{S∪S_0}^T(k)(p) matches etale cohomology of the associated curve.

The Galois group has cohomological dimension 2.

Decomposition groups form a free product inside the Galois group.

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.