Conference "Fundamental Groups in Arithmetic Geometry"
26 May - 3 June 2016
Talks: titles and abstracts
- Fabrizio Andreatta (Milan) : Anabelian p-adic Hodge theory for curves
(joint work with A. Iovita and M. Kim): Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present comparison results in the spirit of Fontaine theory for its unipotent fundamental groups (p-adic étale, de Rham, crystalline). As an application I will provide a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.
- Joseph Ayoub (Zürich) : Motivic fundamental groups
After a quick overview on Voevodsky's triangulated motives, I will explain the construction of a motivic Galois group attached to the Betti realisation. Then, I'll state a few key results and discuss the link with periods of algebraic varieties.
- Fedor Bogomolov (Moscow, New-York) : Rational version of almost abelian section conjecture for fields.
The statement of almost abelian section conjecture for functional fields can be vaguely stated as the existence of a correspondence between geometric sections and special maps between abelian Galois groups of the closures of the corresponding fields. The latter contain a system of so called liftable subgroups, i.e. the subgroups which lift to abelian subgroups of the nilpotent quotient of level 2 and the maps between the abelian Galois groups must map liftable subgroups into the liftable once in order for the corresponding geometric section to exist. The conjecture states that a version of a converse statement also holds.
In my talk I will discuss the proof of the rational version of the dual statement which holds for a much broader class fo fields and amounts to the following result relating noninjective homomorphisms of multiplicative groups of fields ( denoted further by K and L) and valuations. I consider homomorphisms ψ :K^{*}→ L^{*} which respect algebraic dependence over some ground subfields in K and L. Then under mild additional conditions on such a homomorphism ( existence of at least two algebraically independent elements) there is a nonarchemedian valuation with a property that ψ on the subgroup of invertible elements in the ring of integers is essentially a residue map. The valuation which appears corresponds to the section for a surjective map between two algebraic varieties.
This is a joint work with Yuri Tschinkel and Marat Rovinsky
- Ishai Dan-Cohen (Essen) : A factorization of Kim's conjecture via motivic rational homotopy theory
I will report on an ongoing project, based on conversations with Marc Levine and Tomer Schlank. Work of David White and Cisinski-Déglise enables us to place the unipotent fundamental group of Deligne-Goncharov in its natural context, namely, that of rational motivic complexes with cup product. This allows us to divide Kim's conjecture for the punctured line (which says that integral points are completely characterized by obstructions coming from the unipotent fundamental group) into a union of four conjectures with a homotopical flavor.
- Hélène Esnault (Berlin) : Connections in characteristic 0 and in positive characteristic
The p-curvature conjecture predicts a sort of Ceboarev like theorem for the differential Galois group of an integrable connection. We make much stronger assumptions in order to be able to derive some properties of the connection.
Work in progress with Mark Kisin.
- Yuichiro Hoshi (Kyoto) : Introduction to Mono-anabelian Geometry
In this mini-course, I will give an introduction to mono-anabelian geometry. In a classical study of anabelian geometry, one discusses a comparison between two geometric objects via their arithmetic fundamental groups. By contrast, mono-anabelian geometry centers around the task of establishing "group-theoretic algorithms" that require as input data only the arithmetic fundamental group of a single geometric object. This mini-course will concentrate on mono-anabelian geometry related to p-adic local fields, as well as hyperbolic curves over p-adic local fields.
- Lars Kindler (Harvard) : Lefschetz theorems for the tame fundamental group and applications
We will present two Lefschetz theorems for tamely ramified coverings recently obtained jointly with Hélène Esnault, based on work of Grothendieck-Murre and Drinfeld, and then discuss a few applications.
- Emmanuel Lepage (Paris) : Non-archimedean fundamental groups.
We'll survey two fundamental groups built from Berkovich étale topology for non-archimedean analytic spaces : de Jong's étale fundamental group and André's tempered fundamental group. We'll discuss some anabelian properties.
- Sophie Morel (Princeton) : Travaux de V. Lafforgue sur la correspondance de Langlands globale
Le but est d'énoncer les résultats récents de V. Lafforgue sur la correspondance de Langlands globale pour les corps de fonctions et d'en présenter la démonstration, en particulier les idées nouvelles inspirées par la correspondance de Langlands géométrique.
- Stefan Patrikis (Salt Lake City) : Deformations of Galois representations and exceptional monodromy
I will explain the construction of geometric l-adic Galois representations of the Galois group of Q having algebraic monodromy groups equal to the various exceptional groups. The key ingredient is a generalization to any Dynkin type of Ravi Ramakrishna's technique for deforming two-dimensional mod l Galois representations to characteristic zero.
- Michel Raynaud (Orsay) : The fundamental group in algebraic geoemetry
We will review fundamental results about the theory of étale fundamental groups, with a special emphasis on the case of curves. In particular, we will discuss- Comparison with the topological fundamental group in the complex case.
- Finitness conditions. The tame fundamental group for affine curves in char. p>0.
- Specialization of the étale fundamental group on a complete dvr.
- Examples.
- The abelian étale fundamental group.
- Tamás Szamuely (Budapest) : Fundamental groups: an overview
We survey various incarnations of the fundamental group and the relations between them.
- Alberto Vezzani (Paris) : The rigid realisation and motivic Galois groups
We recall the construction of the motivic tilting equivalence and the overconvergent de Rham realisation functor. We also show how they can be used to generalise results of Ayoub on motivic Galois groups from the (equal) characteristic zero case to the positive and mixed characteristic case.
- Stefan Wewers (Ulm) : Chabauty-Kim method
I will try to give an introduction to Minhyong Kim's nonabelian generalization of Chabauty's method and summarize the conjectures and results obtained so far.