Cohomology of absolute Galois groups

TL;DR

This thesis investigates the structure of absolute Galois groups by introducing new classes of pro-p groups, utilizing Galois cohomology, and exploring their Lie algebraic properties to better understand their realizability and characteristics.

Contribution

It introduces Bloch-Kato pro-p groups and cyclotomic orientation, providing new insights into the structure of maximal pro-p Galois groups and their cohomological properties.

Findings

Defined Bloch-Kato pro-p groups satisfying Bloch-Kato conjecture conditions.

Revealed structural properties of θ-abelian pro-p groups as bounds for Galois groups.

Analyzed the Lie algebra and universal envelope of Galois groups via Zassenhaus filtration.

Abstract

The main problem this thesis deals with is the characterization of profinite groups which are realizable as absolute Galois groups of fields: this is currently one of the major problems in Galois theory. Usually one reduces the problem to the pro-$p$ case, i.e., one would like to know which pro-$p$ groups occur as maximal pro-$p$ Galois groups, i.e., maximal pro-$p$ quotients of absolute Galois groups. Indeed, pro-$p$ groups are easier to deal with than general profinite groups, yet they carry a lot of information on the whole absolute Galois group. We define a new class of pro-$p$ groups, called Bloch-Kato pro-$p$ group, whose Galois cohomology satisfies the consequences of the Bloch-Kato conjecture. Also we introduce the notion of cyclotomic orientation for a pro-$p$ group. With this approach, we are able to recover new substantial information about the structure of maximal pro-$p$ Galois groups, and in particular on $\theta$-abelian pro-$p$ groups, which represent the "upper bound" of such groups. Also, we study the restricted Lie algebra and the universal envelope induced by the Zassenhaus filtration of a maximal pro-$p$ Galois group, and their relations with Galois cohomology via Koszul duality. Altogether, this thesis provides a rather new approach to maximal pro-$p$ Galois groups, besides new substantial results.