talks

Number theorists have long been interested in the quantitative aspects of the distribution of Galois groups of field extensions. Recently, progress has been realized in counting extensions of function fields over (large enough) finite fields by reducing to characteristic zero, and more specifically to the topology of certain varieties which parametrize extensions. However, these methods apply only to "tame" extensions, where the characteristic does not divide the order of the Galois group.
The "wild" case, when the Galois group is a \( p \)-group and \( p \) is the characteristic of the base field, is very mysterious. In recent work with Fabian Gundlach, we have related extensions of the local function field \( \mathbb F_q(\!(T)\!) \) to the solutions to certain equations over the ring \( W(\mathbb F_q) \) of Witt vectors. These equations involve the absolute Frobenius automorphism \( \sigma \colon x \mapsto x^p \), making them difference equations. Counting extensions (including questions like reduction to characteristic \( 0 \), uniformity in the prime \( p \), etc.) is then related to counting points on difference schemes and to the "asymptotic behavior" of the absolute Frobenius automorphism \( \sigma \) as \( p \) grows, thus connecting the initial problem to Hrushovski—Lang—Weil-type estimates.
In this talk, I will present these connections and our current results. Efforts will be made towards rephrasing some of our questions in model-theoretic language, including open questions which may benefit from a model-theoretic perspective.

The asymptotic distribution of field extensions with fixed Galois group \(G\) is a well-studied topic, aiming to make the predictions of inverse Galois theory quantitative. Over number fields and for tame extensions of function fields, conjectures and results give a clear idea of the situation. In contrast, the case of non-abelian wild extensions is a terra incognita.
In this talk, we fix a prime \( p > 2 \), and we focus on \( p \)-groups \( G \) of nilpotency class \( 2 \). We explain how to parametrize \( G \)-extensions in characteristic \( p \), and we reduce the problem of counting them into a "typical" arithmetic geometry problem: counting solutions to polynomial equations over finite fields. Moreover, we mention instances where this counting can be carried out.
This work is a joint collaboration with Fabian Gundlach.

The asymptotic distribution of field extensions (often counted by discriminant) is a active topic that aims to make the predictions of inverse Galois theory quantitative. Over number fields, precise conjectures and partial results give a clear idea of the situation, and a similar picture emerges for tamely ramified extensions of function fields. In contrast, the wildly ramified case remains a terra incognita: the case of abelian extensions was only recently (partially) solved using class field theory. In this talk, we fix a prime \(p > 2\), and we focus on \( p \)-groups \( G \) of nilpotency class \( 2 \) (the "most abelian" among non-abelian \( p \)-groups). We will explain how to parametrize \( G \)-extensions of fields of characteristic \( p \). Work by Abrashkin describes the ramification filtration of these extensions, and reduces the counting problem (over local fields) to a "typical" problem of arithmetic geometry: counting solutions to polynomial equations over finite fields. We present several instances where this local counting can be carried out. If time permits, we will also discuss results over global function fields, which follow from the local case via a local-global principle.
This work is a joint collaboration with Fabian Gundlach.

We shall prove the following theorem: if \( K \) is a field of characteristic zero, any \( K \)-linear isomorphism \( f \colon K[X] \to K[X] \) for which any polynomial \( P \) such that \( f(P) \) is squarefree is itself squarefree is of the form \( P \mapsto c \cdot P(aX+b) \) for some constants \( a,c \in K^\times \), \( b \in K \). If we have additional time, we shall discuss generalizations.

Cet exposé présente des résultats au sujet de la densité asymptotique des discriminants des extensions (intérieures comme extérieures) d'une algèbre centrale simple sur un corps de nombres. Ce projet généralise la question de la distribution des corps de nombres au cas des algèbres simples (ou des corps non-commutatifs).
Les travaux présentés sont le fruit d'une collaboration avec Fabian Gundlach.

The asymptotic distribution of field extensions (often counted by discriminant) is a well-studied topic that aims to make the predictions of inverse Galois theory quantitative. Over number fields, precise conjectures (and partial results) give a clear idea of the situation (Wright, Malle, Bhargava, ...). A similar picture emerges for tamely ramified extensions of function fields (Ellenberg, Tran, Venkatesh, and Westerland). In contrast, the wildly ramified case remains a terra incognita: the case of abelian extensions was only recently solved (Lagemann, Klüners-Müller, Potthast, Gundlach) using class field theory.
In this talk, we fix a prime \( p > 2 \), and we focus on \( p \)-groups \( G \) of nilpotency class \( 2 \) (the "most abelian" among non-abelian \( p \)-groups). We explain how to parametrize \( G \)-extensions of fields of characteristic \( p \) using Lie algebras and Witt vectors. Work by Abrashkin describes the ramification filtration of these extensions and reduces the counting problem (over local fields) to a "typical" arithmetic geometry problem: counting solutions to polynomial equations over finite fields. We present several instances where this local counting can be carried out. If time permits, we will also discuss results over global function fields, which follow from the local case via a local-global principle.
This work is a joint collaboration with Fabian Gundlach.

La distribution asymptotique des extensions de corps (souvent comptés par discriminant) est un sujet actif, dont le but est de rendre quantitatives les prédictions de la théorie de Galois inverse. Sur les corps de nombres, des conjectures précises (et des résultats partiels) donnent une bonne idée de ce qui se passe (Wright, Malle, Bhargava, …), et la situation est à peu près identique pour les extensions modérément ramifiées des corps de fonctions (Ellenberg, Tran, Venkatesh et Westerland). Le cas sauvagement ramifié est, par contraste, une terra incognita : le cas des extensions abéliennes n'a été résolu que très récemment (Lagemann, Klüners—Müller, Potthast, Gundlach) en utilisant la théorie du corps de classes.
Dans cet exposé, on fixe un premier \( p > 2 \), et on s'intéresse aux \( p \)-groupes \( G \) "les plus abéliens parmi les non-abéliens" : ceux de classe de nilpotence \( 2 \). On explique comment paramétrer les \( G \)-extensions d'un corps de caractéristique \( p \) (en utilisant des algèbres de Lie et des vecteurs de Witt). Se pose alors la question du comptage. Des travaux d'Abrashkin décrivent la filtration de ramification, et permettent de transformer le problème (sur les corps locaux) en un problème « typique » de géométrie arithmétique : on doit compter les solutions d'équations polynomiales sur des corps finis. Dans le cas local, on présentera plusieurs situations où ce comptage peut être réalisé. Si le temps le permet, nous parlerons des résultats dans le cas global, qui découlent du cas local via un principe local-global.
Tous les travaux présentés sont le fruit d'une collaboration avec Fabian Gundlach.

In this talk, we review techniques used for parametrizing extensions of fields of characteristic \( p \), and we show how these techniques specialize to known theories (Artin—Schreier—Witt theory, \( \varphi \)-modules, ...). We then review the Lazard correspondence, which relates \( p \)-groups of nilpotency class smaller than \( p \) with Lie algebras. By combining these two ingredients, we obtain a glimpse of Abrashkin's nilpotent Artin-Schreier theory. All required facts concerning Lie algebras will be recalled.

This talk is centered around results concerning the asymptotic density of discriminants of (inner and outer) extensions of a given simple algebra over a number field. This project is a noncommutative generalization of the question of the distribution of number fields. This is joint work with Fabian Gundlach.

Using the language and the tools of rigid analytic geometry, Harbater (1987) has defined a "patching operation" which can be used to solve the inverse Galois problem over fields like \( \mathbb Q_p(T) \) or \( \mathbb F_p(\!(X)\!)(T) \). Later, Haran and Völklein (1996) rephrased this construction in a purely algebraic language, replacing all geometric arguments with (almost entirely) explicit constructions. Our goal is to present their proof.

We present and explain the proof of results concerning the asymptotic density of discriminants of extensions of a given division algebra over a number field. This is an extension of the question of the distribution of number fields to the case of non-commutative fields. We explain what happens both in the case of "inner Galois extensions" (analogous to central simple algebras over a commutative field) and "outer Galois extensions" (analogous to ordinary Galois extensions of a commutative field). This is joint work with Fabian Gundlach.

Celebrated bridges between analytic geometry and algebraic geometry lead to an equivalence of categories between finite extensions of \( \mathbb C(T) \) and finite ramified covers of the Riemann sphere (i.e., the complex projective line). These covers are well-understood, and this correspondence directly implies a positive answer to the inverse Galois problem over \( \mathbb C(T) \), as well as a classification of the corresponding extensions. The regular inverse Galois problem is the question of whether every finite group can be realized as the Galois group of a finite extension of \( \mathbb Q(T) \) which is regular (i.e., with no non-rational elements algebraic over \( \mathbb Q \)). This question can be reframed in terms of covers of the line: can one find geometrically connected Galois generically étale covers of the projective line over \( \mathbb Q \) which have a given automorphism group? Equivalently: among all connected Galois ramified covers of the complex projective line with a specific Galois group, are there any whose field of definition is \( \mathbb Q \)? Few methods exist to study this descent question. However, if \( G \) is a group with trivial center, one such method is the so-called "rigidity criterion": one can deduce that a Galois connected cover of the projective line with Galois group \( G \) is defined over \( \mathbb Q \) from the fact that it is uniquely determined by a few elementary geometric invariants. This criterion can, in turn, be checked group-theoretically using character theory. This method, and variants thereof, are used in the proof that 25 of the 26 sporadic finite simple groups are Galois groups over \( \mathbb Q(T) \) (and, incidentally, over \( \mathbb Q \)). In this talk, we will explain the geometric ideas leading to the rigidity criterion and give applications.

Nous introduisons la propriété suivante sur un anneau : il est fadélien lorsque pour tous éléments non nuls \( a \) et \( x \), l’élément \( x \) peut s’écrire comme \( ab + ca \). Cette propriété était intervenue naturellement, dans un cadre plus spécifique, dans des travaux antérieurs sur les V-domaines. Passées les premières propriétés qui découlent de la définition, le défi est de construire des exemples non-triviaux pour cerner le caractère restrictif de la définition. En utilisant la notion d’extension d’Ore, nous construisons des anneaux fadéliens qui ne sont pas des corps gauches, puis nous les rendons non-noethériens (et même moins). Nous mentionnerons quelques questions ouvertes.
Tous les travaux présentés sont le fruit d'une collaboration avec Robin Khanfir.

Le problème inverse de Galois est un problème historique de la théorie des nombres, déjà étudié par Hilbert, et toujours ouvert. Étant donné un groupe fini \( G \), on demande s'il existe une extension galoisienne du corps des nombres rationnels dont \( G \) est le groupe de Galois. Au cours du vingtième siècle, il est apparu que cette question admettait une interprétation en termes de revêtements : on cherche en fait des points rationnels sur les « espaces de modules de Hurwitz », qui classifient les \( G \)-revêtements. Des méthodes géométriques, topologiques et combinatoires donnent alors un point de vue nouveau sur le problème et permettent de réaliser des groupes. Le but de l'exposé est de donner un aperçu de cette intervention de la géométrie dans une question d'arithmétique, de façon introductive et panoramique.