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Oussama Hamza

Publications and source records attributed to Oussama Hamza.

7 recordsLinked to original sources
arXiv2026

Tame Galois Groups, Linking Numbers and Mildness

Let $p$ be an odd prime and let $S$ be a set of tame primes. We denote by $G_S$ the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside $S$. We prove that for every finite set of tame primes $S_0$ with $|S_0|\geq 2$, there exists a set $S_1$ consisting of two tame primes such that $G_{S_0\cup S_1}$ has cohomological dimension $2$. This refines a result of Labute. More generally, we establish an analogous result for number fields not containing a primitive $p$-th root of unity, under a suitable splitting condition. Our approach answers a question of Labute, from his seminal paper on mild groups, and combines weighted Zassenhaus filtrations, graph-theoretic methods, and Koch-type presentations. As an application, we solve several cohomological Galois inverse problems with prescribed ramification and splitting. We also provide numerical examples and statistics.

math.NT↗
arXiv2026

Filtrations and cohomology on graph products

Let $p$ be a prime. We resolve a question posed by Mináč-Rogelstad-Tân. We relate the Zassenhaus and the lower central series of pro-$p$ groups under a torsion-freeness condition. We also study graph products of (pro-$p$) groups under natural assumptions. In particular, we compute their graded Lie algebras associated with the previous filtrations, as well as their cohomology over $\mathbb{F}_p$. Our approach relies on various filtrations of amalgamated products, as studied in Leoni's PhD thesis. Explicit examples are provided using the Koszul property. As a concrete application, we compute the cohomology over $\mathbb{F}_p$ and the graded Lie algebras associated with the filtrations of graph products of fundamental groups of surfaces. These groups furnish new examples satisfying the torsion-freeness condition, which arises in the question of Mináč-Rogelstad-Tân.

math.GR↗
arXiv2025

Maximal $2$-extensions of Pythagorean fields and Right Angled Artin Groups

In this paper, we describe minimal presentations of maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type. For this purpose, we introduce a new class of pro-$2$ groups: $Δ$-Right Angled Artin groups. We show that maximal pro-$2$ quotients of absolute Galois groups of formally real Pythagorean fields of finite type are $Δ$-Right Angled Artin groups. Conversely, let us assume that a maximal pro-$2$ quotient of an absolute Galois group is a $Δ$-Right Angled Artin group. We then show that the underlying field must be Pythagorean, formally real and of finite type. As an application, we provide an example of a pro-$2$ group which is not a maximal pro-$2$ quotient of an absolute Galois group, although it has Koszul cohomology and satisfies both the Kernel Unipotent and the strong Massey Vanishing properties. We combine tools from group theory, filtrations and associated Lie algebras, profinite version of the Kurosh Theorem on subgroups of free products of groups, as well as several new techniques developed in this work.

math.GR↗
arXiv2025

Massey products and unipotent extensions with restricted ramification

We fix a prime p and construct new cases of pro-p extensions of number fields with restricted ramification and splitting, whose Galois groups decompose as coproducts of pro-p absolute Galois groups of local fields. As a consequence, these pro-p extensions satisfy the strong Massey vanishing property and thus admit large unipotent quotients.

math.NT↗
arXiv2025

On extensions of number fields with given quadratic algebras and cohomology

We introduce a criterion on the presentation of finitely presented pro-$p$ groups which allows us to compute their cohomology groups and infer quotients of mild groups of cohomological dimension strictly larger than two, from (non-free) mild groups. We interpret these groups as Galois groups over $p$-rational fields with prescribed ramification and splitting.

math.GR↗