talks

Dénombrement des extensions d’une algèbre centrale simple sur un corps de nombres

LMNO, Caen

• Journées galoisiennes d'hiver (alt)

• Abstract

  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.

Asymptotic distribution of wildly ramified extensions of function fields

ISTA, Klosterneuburg

• Algebraic Geometry and Number Theory Seminar

• Abstract

  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.

Distribution asymptotique des extensions sauvagement ramifiées des corps de fonctions

Université Sorbonne Paris Nord

• Séminaire de l'équipe "Arithmétique et Géométrie Algébrique"

• Abstract

  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.

Nilpotent Artin­-Schreier theory [notes]

Universität Paderborn

• Oberseminar Number Theory and Arithmetical Statistics

• Abstract

  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, φ-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.

Dénombrement des extensions d'une algèbre centrale simple sur un corps de nombres

CIRM, Luminy

• École de Théorie des Nombres (EThéN)

• Abstract

  This talk is centered around results concerning the asymptotical 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.

Algebraic patching for beginners [notes]

Universität Paderborn

• Oberseminar Number Theory and Arithmetical Statistics

• Abstract

  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 Qₚ(T) or Fₚ((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.

Counting extensions of division algebras over number fields

Universität Paderborn

• Oberseminar Number Theory and Arithmetical Statistics

• Abstract

  We present and explain the proof of results concerning the asymptotical 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.

Covers and rigidity in inverse Galois theory [notes]

Universität Paderborn

• Oberseminar Number Theory and Arithmetical Statistics

• Abstract

  Celebrated bridges between analytic geometry and algebraic geometry lead to an equivalence of categories between finite extensions of 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 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 ℚ(T) which is regular (i.e., with no non-rational elements algebraic over 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 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 ones whose field of definition is 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 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 ℚ(T) (and, incidentally, over ℚ). In this talk, we will explain the geometric ideas leading to the rigidity criterion and give applications.

Covers of ℙ¹ and their moduli: where arithmetic, geometry and combinatorics meet [notes]

MFO Oberwolfach

• MFO-RIMS Tandem Workshop - Arithmetic Homotopy and Galois Theory [full report]

• Abstract

  Covers of the Riemann sphere may be described in many ways: geometrically as covering maps, algebraically as generically étale morphisms, as field extensions of function fields, as tuples of permutations, as subgroups of (profinite) free groups, etc. The fact that we can go back and forth between these descriptions is incredibly powerful: for example, purely group-theoretic criteria are shown to imply cases of the inverse Galois problem. More precise descriptions are of use in the study of statistics of function fields in the tamely ramified case. In this talk, we will review some aspects of this theory in an informal way. We will give a glimpse of Hurwitz moduli spaces and of some ideas from Grothendieck-Teichmüller theory. If we have time, we will talk about the difficulties concerning the use of these methods in the wildly ramified case.

Geometry and arithmetic of components of Hurwitz spaces [slides]

Université de Lille

• PhD Defense (read the manuscript)

Une forme faible de divisibilité dans les anneaux non-commutatifs

Université d'Artois, Lens

• Séminaire d'Algèbre de la faculté Jean Perrin

• Abstract

  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.

Raconte-moi le problème de Galois inverse !

École Normale Supérieure, Paris

• Séminaire Raconte-moi

• Abstract

  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 G-revêtements : on cherche en fait des points rationnels sur les "espaces de modules de Hurwitz", qui classifient les 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.

Geometric methods for inverse Galois theory [slides]

RIMS Kyōto (remote)

• 4th Kyōto-Nanjing Workshop on Arithmetic and Geometry

• Abstract

  Inverse Galois theory belongs to a family of more general counting problems for field extensions over some base field K. These questions are related with the count of covers defined over K, and incidentally with the count of K-points on a moduli space of covers, called a Hurwitz space. The ring of components is a tool introduced by Ellenberg, Venkatesh and Westerland to study the homology of these moduli spaces. We will discuss variations on their idea. We first look at the ring of components of covers of the complex projective line, which is commutative, and relate the geometry of its spectrum to group-theoretic and combinatorial quantities. We also define a version of their ring which is aware of arithmetic phenomena.

Espaces de Hurwitz et problème inverse de Galois

École Normale Supérieure, Paris

• Journée de rentrée du Département de Mathématiques et Applications

Covers and the Inverse Galois Problem

Institut Fourier, Grenoble

• Séminaire Compréhensible

• Abstract

  Since Antiquity, mathematicians have been trying to find rational solutions to systems of polynomials equations (sometimes in a less precise language). Such questions are called Diophantine equations. During the 19th century, it became clear that the key to solving these equations lay in the study of fields and especially in a central tool: the Galois group, which measures how "symmetric" a field is. A natural yet unsolved question is the inverse Galois problem: can every finite group be realized as the Galois group of a finite extension of the field of rational numbers?
  In this talk, we will tell the story of how our viewpoint on this problem has shifted in the last century. We will first see that the problem is actually related to geometry: are there certain covers of the projective line which are defined by equations with rational coefficients? Then, we will see that covers may themselves be seen as points on a bigger geometrical space: the Hurwitz space. The inverse Galois problem is then asking about the existence of rational points on this space: this is itself a Diophantine equation! We will mention some of the achievements obtained using this paradigm, for example the realization of the Monster group as a Galois group.

Rings of Components of Hurwitz Spaces

Université de Lille

• Conference Polynomials and Arithmetic

• Abstract

  Understanding G-covers is crucial to the Inverse Galois Problem, an important open question of Galois theory. In this view, we study the geometric structure of a space of G-covers, called a Hurwitz space. As part of their work concerning the Cohen-Lenstra conjecture, Ellenberg, Venkatesh and Westerland introduced new tools to study Hurwitz spaces. One of their tools is the ring of components. Under strong group-theoretic hypotheses, they used that ring to show that the homology of Hurwitz spaces stabilizes when the number of branch points increases, which they in turn used to prove field-theoretic estimates. I will present the core objects of their work and some of their results. Then, I will describe some of the generalizations I obtained. Namely, I relate geometric properties of the ring of components to group-theoretic and combinatorial invariants, and define a version of their ring that is “aware of arithmetic phenomena”. This opens the way to new methods for studying the Inverse Galois Problem.