arXiv subjects

Julian Feuerpfeil

Publications and source records attributed to Julian Feuerpfeil.

5 recordsLinked to original sources

arXiv2026

Bockstein Spectral Sequences and Applications to the Tame Fontaine Mazur Conjecture

For a number field $K$, a prime $p$ and a finite set of tame places $S$ we consider the groups $G_{K,S}$ - the Galois groups of the maximal pro-$p$ extension of $K$ unramified outside $S$. The tame Fontaine--Mazur Conjecture predicts that these groups have no nontrivial uniformly powerful pro-$p$ quotients. In this paper we develop a new approach to this problem using Bockstein spectral sequences and Lie-theoretic tools. This allows us to extend and refine an earlier method due to J.~Labute, who was only able to consider the case where $p^2\nmid N(\mathfrak{q})-1$ for each $\mathfrak{q}\in S$. Based on this we develop a method to verify the uniform Fontaine--Mazur property for many $G_{K,S}$ with $|S|=3$ arbitrary. Under mild conditions on $K$ we show that for infinitely many triples $S$, the groups $G_{K,S}$ have no nontrivial uniform quotients. We also exhibit a large class of these groups, which are infinite. Finally, we present numerical evidence indicating that the criteria developed here detect the uniform Fontaine--Mazur property with very high probability for $|S|=3$.

math.NT↗

arXiv2026

A cohomological translation of the Kaplansky radical for profinite groups

The Kaplansky radical of a field consists of the nonzero elements represented by every norm quadratic form in two variables. D. Kijima and M. Nishi conjectured that, for quadratic extensions, the Kaplansky radicals are related by the norm map in a manner analogous to Hilbert's Theorem 90. Although this H-conjecture was disproved by K.J. Becher and D.B. Leep, it is known to hold for several important classes of fields. We introduce a cohomological analogue of the Kaplansky radical for arbitrary profinite groups and primes $p$, defined as the orthogonal of $\mathrm{H}^1(G,\mathbb{F}_p)$ with respect to the cup product with itself. For absolute Galois groups, this recovers the classical Kaplansky radical when $p=2$ and the $p$-radical of Dario-Engler for arbitrary p. We also formulate a group-theoretic analogue of the H-conjecture, proving that, for fields, it is equivalent to the original conjectural property and depends only on the maximal pro-$2$ quotient of the absolute Galois group. We establish this property for broad classes of fields, including local and global fields, rational function fields, and all fields whose maximal pro-$p$ Galois group is of elementary type. Beyond its arithmetic origins, we investigate the property for general pro-$p$ groups, proving its stability under several natural group-theoretic constructions and obtaining new examples, including generalized right-angled Artin pro-$p$ groups and fundamental pro-$p$ groups of suitable graphs of groups, many of which cannot occur as maximal pro-$p$ Galois groups.

math.NT↗

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↗

arXiv2025

A Hilbert 90 Property for S-Class Groups and Applications to the Gross--Kuz'min Conjecture

Let $L/K$ be a cyclic extension of number fields, and let $S$ be a finite set of places of $K$ containing the ramified and Archimedean ones. We say that $L/K$ has the $\mathbf{cl}^S$-Hilbert 90 property if, for any generator $\sigma \in \mathrm{Gal}(L/K)$, the kernel of the arithmetic norm map $\mathrm{cl}^S(L) \to \mathrm{cl}^S(K)$ coincides with $(1 - \sigma)\mathrm{cl}^S(L)$. We establish a criterion for the $\mathbf{cl}^S$-Hilbert 90 property that depends only on arithmetic data of the base field and does not require any knowledge of the class group of $L$. We then show that, for $\mathbb{Z}_p$-extensions, the $\mathbf{cl}^{S_p}$-Hilbert 90 property at finite layers already implies the finiteness of the coinvariants of the associated Kuz'min-Tate module, providing a new finite-level criterion for an Iwasawa-theoretic property related to the Gross--Kuz'min conjecture. This criterion can be expressed in terms of a local map closely related to Fermat quotients, making the criterion amenable to explicit computation. Motivated by extensive numerical experiments, we formulate a conjecture predicting that this criterion is satisfied for all but finitely many primes in the totally real case and present a random matrix heuristic supporting this prediction.

math.NT↗

arXiv2025

On the Bogomolov-Positselski Conjecture

Let $p$ be a prime. An oriented pro-$p$ group $(G,\theta)$ is said to have the Bogomolov--Positselski property if it is Kummerian and if $I_\theta(G)$ is a free pro-$p$ group. In this paper, we provide a new criterion for an oriented pro-$p$ group to satisfy the Bogomolov--Positselski property. This criterion builds on earlier work of Positselski (arXiv:1405.0965) and Quadrelli--Weigel (arXiv:2103.12438), relates their approaches, and answers a question raised in (arXiv:2103.12438). Under additional assumptions, we obtain two further sufficient criteria. The first is analogous to a Merkurjev--Suslin type statement. The second allows one to weaken the hypotheses appearing in Positselski's criterion (arXiv:1405.0965 Theorem 2). Finally, we show that the stronger conditions are satisfied by pro-$p$ groups of elementary type. As a consequence, the Elementary Type Conjecture implies Positselski's ``Module Koszulity Conjecture 1'' (arXiv:1008.0095) for fields with finitely generated maximal pro-$p$ Galois group.

math.GR↗