Workshop: Non-Archimedean Geometry and Algebraic Groups

November 3 - 5, 2016,   Frankfurt am Main

Organised by: Bertrand Rémy, Jakob Stix, Annette Werner
Partially supported by the ANR project GeoLie.

Speakers:

Anna Cadoret (Polytechnique) Jean-Louis Colliot-Thélène (CNRS, Université Paris-Sud)
Hélène Esnault (FU Berlin) Philippe Gille (CNRS, Lyon)
Steffen Müller (Oldenburg) Tobias Schmidt (Rennes)
Amaury Thuillier (Lyon) Torsten Wedhorn (Darmstadt)
Stefan Wewers (Ulm) Olivier Wittenberg (CNRS, ENS)

Photo of Participants, Frankfurt November 3, 2016

Practical Information:


Schedule:

All talks will take place in lecture room H 14 in the lecture building "Jügelhaus". Access to the lecture building is from Gräfstrasse 50-54, see number 15 on the map, just opposite Hotel West and next to the insitut for mathematics in Robert-Mayer Strasse 6-10.

  Thursday Friday Saturday
09:30-10:30 Wewers
Computing semistable reduction, and non-archimedian structure of covers of curves
Thuillier
Tropicalization of analytic spaces
Schmidt
Arithmetic D-modules and non-archimedean localization
10:30-11:00 coffee/tea break coffee/tea break coffee/tea break
11:00-12:00 Gille
Survey on recent results on maximal
tori of algebraic groups
Müller
Quadratic Chabauty
Esnault
A Lefschetz theorem for overconvergent isocrystals with Frobenius structure
15:30-16:30 Cadoret
Geometric monodromy:
semisimplicity and maximality
Wedhorn
Spherical Spaces
 
16:30-17:00 coffee/tea break coffee/tea break
17:00-18:00 Colliot-Thélène
Lokal-global Prinzip für homogene Räume linearer algebraischer Gruppen über einem Funktionenkörper einer Variablen über einem lokalen Körper
Wittenberg
Fourfold Massey products over number fields
 

 

Coffee and tea breaks will take place in the foyer next to the lecture hall.


Abstracts:

Anna Cadoret (Polytechnique), Thursday, 15:30-16:30
Geometric monodromy - semisimplicity and maximality

Let X be a connected scheme, smooth and separated over an algebraically closed field k of characteristic p, let f:Y → X be a smooth proper morphism and x a geometric point on X. We show the geometric variant of the Tate semi simplicity conjecture with F-coefficients, namely that the action of the étale fundamental group π1(X,x) on the étale cohomology group H*(Yx,F) is semi simple for ℓ large enough. We also show that this is equivalent to the fact that the image of π1(X,x) acting on H*(Yx,Z) is as large as possible (what we call `almost hyperspecial'). Both results are based on the fact that the tensor invariants of π1(X,x) acting on H*(Yx,F) are the reduction modulo-ℓ of the tensor invariants of π1(X,x) acting on H*(Yx,Z) for ℓ large enough.
Another application of the geometric Tate semi simplicity conjecture with F-coefficients is that the usual (i.e. with Q-coefficients) arithmetic Tate conjectures (semisimplicity and fullness) imply the arithmetic Tate conjectures with F-coefficients for ℓ>> 0.
This is a joint work with Chun-Yin Hui and Akio Tamagawa.

Jean-Louis Colliot-Thélène (CNRS, Université Paris-Sud), Thursday, 17:00-18:00
Lokal-global Prinzip für homogene Räume linearer algebraischer Gruppen über einem Funktionenkörper einer Variablen über einem lokalen Körper

Sei k ein komplett diskret bewerteter Körper und K ein Funktionenkörper in einer Variablen über k. Sei ΩK, bzw. ΩK/k, die Menge der diskreten, einrangigen Bewertungen auf K, bzw. die Menge derjenigen, die trivial auf k sind. Sei Kv die Komplettierung von K an der Stelle v.
Sei X eine K-Varietät. Wenn X(Kv) ≠ ∅ für alle v ∈ ΩK, bzw. für alle v ∈ ΩK/k, folgt daraus daß die Menge X(K) der rationalen Punkten nicht leer ist?
Ergebnisse im Falle X homogener Raum einer linearen algebraischen Gruppe G wurden erreicht. Die bejahende Antwort im Falle G = PGLn, X prinzipieller homogener Raum und ΩK/k geht auf Lichtenbaum (1969) zurück, der den Dualitätssatz von Tate für abelsche Varietäten benutzt. Wenn k ein p-adischer Körper ist, dann gibt es Beziehungen mit höherer Klassenkörpertheorie. In dieser Richtung hat man Ergebnisse von Parimala und Suresh, und von Harari, Scheiderer, Szamuely und Izquierdo. Wenn der Restklassenkörper von k beliebig ist, wurden bemerkenswerte neue Ergebnisse von Harbater, Hartmann und Krashen (2009) erzielt mittels einer neuen “Patching” Methode (Harbater, Hartmann). Es folgte eine Reihe von Arbeiten dieser Autoren und von Parimala, Suresh, dem Sprecher und Y. Hu. Höhere unverzweigte Kohomologie spielt in beiden Fällen eine Rolle.
Ich werde versuchen, ein globales Bild von den Methoden und Ergebnissen zu geben.

Principe local-global pour les espaces homogènes de groupes algébriques linéaires sur les corps de fonc- tions d’une variable sur un corps local

Soient k le corps des fractions d’un anneau de valuation discrète complet et K un corps de fonctions d’une variable sur k. Soit ΩK, resp. ΩK/k, l’ensemble des valuations discrètes de rang 1 sur K, resp. l’ensemble de celles qui sont triviales sur k.
Si une K-variété algébrique X a des points rationnels dans tous les complétés Kv pour v dans ΩK, resp. ΩK/k, X admet-elle un point rationnel sur K ?
On a des résultats lorsque X est un espace homogène d’un groupe algébrique G linéaire connexe. Pour un espace principal homogène de G = PGLn, k un corps p-adique et ΩK/k, une réponse positive remonte à un théorème de Lichtenbaum (1969) reposant sur la dualité de Tate pour les variétés abéliennes sur les corps p-adiques. Pour k un corps p-adique, il y a des liens avec la théorie du corps de classes supérieur. Il y a dans cette direction des travaux de Parimala et Suresh, et de Harari, Scheiderer, Szamuely, Izquierdo. Pour k un corps local à corps résiduel quelconque, une percée fut faite par Harbater, Hartmann, Krashen (2009), via une nouvelle technique de “recollement” (Harbater et Hartmann). Depuis, plusieurs résultats ont été obtenus par ces auteurs, Parimala, Suresh, l’orateur, Y. Hu. Dans les deux cas, la cohomologie non ramifié supérieure joue un rôle.
J’essaierai de brosser un portrait d’ensemble de la situation.

Hélène Esnault (FU Berlin), Saturday, 11:00-12:00
A Lefschetz theorem for overconvergent isocrystals with Frobenius structure

We show a Lefschetz theorem for irreducible overconvergent F-isocrystals on smooth varieties defined over a finite field. We shall recall how it inserts in the framework of existing Lefschetz theorems for ℓ-adic sheaves, only briefly as it would then not be a very ‘non-archimedean’ lecture. We shall also mention how it inserts into Deligne’s program of construction of crystalline ‘companions’. (joint with Tomoyuki Abe)

Philippe Gille (CNRS, Lyon), Thursday, 11:00-12:00
Survey on recent results on maximal tori of algebraic groups

Prasad and Rapinchuk investigated the isopectrality problem for certain Riemannian varieties by analysing to what extent a semisimple algebraic group defined over a number field is determined by its maximal tori. We shall report advances on this topic by Chernousov/Rapinchuk/Rapinchuk, Bayer-Fluckiger/Lee/Parimala and others by discussing the case of non-archimedean fields and local-global principles.

Steffen Müller (Oldenburg), Friday, 11:00-12:00
Quadratic Chabauty

I will discuss how to p-adically approximate rational or integral points on hyperelliptic curves defined over number fields using iterated Coleman integrals, as predicted by Kim's non-abelian Chabauty program. Our approach extends the method of Chabauty and Coleman and can be used to provably compute all rational or integral points.
This is joint work with Amnon Besser, Jennifer Balakrishnan and Netan Dogra.

Tobias Schmidt (Université Rennes 1), Saturday, 09:30-10:30
Arithmetic D-modules and non-archimedean localization

The classical Beilinson-Bernstein localization theorem relates representations of a complex semisimple Lie algebra to differential equations on its flag manifold. It implies, among other things, strong classification results for the representation theory of real analytic reductive groups. In this talk, I will report on a corresponding emerging non-archimedean picture which links analytic representations of a p-adic reductive group to arithmetic D-modules on non-archimedean flag varieties. If time permits, I will also discuss some first applications.
This is joint work with Christine Huyghe, Deepam Patel and Matthias Strauch.

Amaury Thuillier (Lyon), Friday, 09:30-10:30
Tropicalization of analytic spaces

Let k be a non-archimedean field and let X be a closed analytic subspace of the n-dimensional unit ball Bn over k, i.e., the vanishing locus of some ideal in the Tate algebra k{T1,...,Tn}. It is well-known (Berkovich, Gubler, Ducros, Martin) that the image of X under the tropicalization map (|T1|,...,|Tn|) : Bn → [0,1]n is a piecewise monomial subspace of pure dimension dim(X). I will present of new approach of this result based on an extension of the theory of Gröbner basis to Tate algebras.

Torsten Wedhorn (Darmstadt), Friday, 15:30-16:30
Spherical Spaces

Spherical varieties form an important class of varieties that includes flag varieties, toric varieties, and symmetric spaces. In this talk I explain a systematic approach to define families of spherical varieties. A rigidity result will be explained and I explain how to generalize known classification results for spherical varieties over algebraically closed fields to spherical spaces over more general bases.

Stefan Wewers (Ulm), Thursday, 09:30-10:30
Computing semistable reduction, and non-archimedian structure of covers of curves

The motivating problem for my talk is the explicit computation of semistable reduction of curves over a p-adic number field K. I will focus on the case of cyclic covers of degree p of the projective line, which is simply given by an equation of the form

yp = f(x)

(and where the prime p is also the characteristic of the residue field of the ground field K). Building on earlier work of Coleman and Matignon/Lehr I will present a general solution to the problem in this special case. The formulation of this solution benefits from the language of nonarchimedian analytic geometry, in the following way. If
φ : Y → X = P1K

is the cover corresponding to the above equation, then the essential information regarding the semistable reduction of the curve Y is encoded by a certain affinoid subdomain Xet of Xan, which I call the etale locus. This affinoid can be described in terms of explicit inequalities. The main step in computing the semistable reduction of Y is then achieved by explictly determining the Shilov boundary of Xet.
The resulting algorithm is being implemented in Sage (this is joint work with Julian Rueth).

Olivier Wittenberg (CNRS, ENS), Friday, 17:00-18:00
Fourfold Massey products over number fields

Massey products in the Galois cohomology of an arbitrary field are conjectured to always vanish. This conjecture is known to hold for triple Massey products. We establish it for fourfold Massey products in the Galois cohomology of number fields with coefficients in Z/2Z.
(Joint work with Pierre Guillot, Ján Mináč, Nguyễn Duy Tân, Adam Topaz.)