The program consists of 7 mini-courses of 4 to 6 hours.
(Weizmann Institute) : F1 Geometry
Antoine DUCROS (Université Paris 6) : Etale cohomology
François LOESER (Ecole Normale Supérieure, Paris) : Model theory and analytic geometry
Jérome POINEAU (Université de Strasbourg) : Berkovich spaces over Z
(Weizmann Institute) : F1 Geometry
The minicourse is an introduction to work in progress on foundations
algebraic and analytic geometry over the field of one element. This work
originates in non-Archimedean analytic geometry as a result of a search
for appropriate framework for so called skeletons of analytic spaces and
formal schemes, and is related to logarithmic and tropical geometry.
(University Paris 6) : Etale
Étale cohomology was introduced in the scheme-theoretic
Grothendieck in the 50's and 60's in order to provide a purely algebraic
cohomology theory, satisfying the same fundamental properties as the
singular cohomology of complex varieties, which was needed for proving the
For other deep arithmetic reasons (that times, related to Langlands
program) it appeared later that it should also be worthwhile developing
such a theory in the p-adic analytic context ; this was done by Berkovich
in the early 90's.
In this serie of lectures, I plan, after having given some general
motivations, to spend some times about the notion of a Grothendieck
topology and its associated cohomology theory; then I will explain the
basic ideas and properties of both scheme-theoretic and
Berkovich-theoretic étale cohomology theories (which are closely related
to each other), and the fundamental results like various comparison
theorems, Poincaré duality, purity and so on.
My purpose won't be to give detailed proofs, which are for most of them
highly technically involved; I will rather insist on examples, trying to
show how étale cohomology can at the same time be quite closed to the
classical topological intuition, and deal in a completely natural manner
with deep field arithmetic phenomena (as an example, it encodes Galois
theory).... which allows sometimes to think to the latter in a purely
The foundations of scheme-theoretic étale cohomology
(definition of étale morphisms, the fundamental group, the
notion of a Grothendieck topology and the étale cohomology
itself) have been written in SGA 1 (LNM 224, new edition by the SMF now
available) and SGA IV (LNM 269, 270 and 305), but both those books are
highly technical, and not easily readable by somebody who feels not
very comfortable with scheme theory. The book SGA 4 1/2 (LNM 569) which
presents the results without proofs, is more accessible ; it has the
advantage to give some motivations and comparisons with the classical
(i.e. topological) situation.
I would however suggest to the reader who is a little bit afraid by the SGA's to refer to Milne's book Étale cohomology , which is certainly simpler -- though it also requires some familiarity with schemes.
I mention a beautiful historical text about étale cohomology by Illusie, Grothendieck et la cohomologie étale.
As far as Berkovich's étale cohomology is concerned, the only reference I know at the moment is Berkovich's foundational article Étale cohomology for non-Archimedean analytic spaces
in IHES 78 (1993).
In these lectures I will present two instances where dynamics on
Berkovich spaces appear naturally.
The first case is in the context of iterations of selfmaps of the (standard) projective line over a non-archimedean field such as the p-adic numbers. When trying to extend results from
the archimedean setting (over the complex numbers), it turns out be both natural and fruitful
to study the induced dynamics on the associated Berkovich projective line.
The second case concerns iterations of (germs of) holomorphic selfmaps
f:C2->C2 fixing the origin, f(0)=0. When the fixed point is superattracting, that is, the differential df(0) is identically zero,
the dynamics can be analyzed by studying the induced action on the Berkovich affine plane over the field C equipped with the trivial valuation.
Beyond the subject of dynamics, these lectures will provide a
"hands-on" introduction to Berkovich spaces in relatively concrete settings, where the topological structure is essentially
that of an R-tree. In studying the second instance above, we will
also have the opportunity to explore the link between Berkovich
spaces and the algebro-geometric study of valuations, going back to Zariski.
If time permits, I will also other dynamical situations, such as
the behavior at infinity of iterates of two-dimensional polynomial selfmaps. I may also briefly discuss the higher-dimensional case.
François LOESER (Ecole normale supérieure) : Model theory and analytic geometry
Lecture 1 : A quick review of Model Theory.
1) Basic notions: Languages, structures, definable sets, types.
2) An example : o-minimal structures as a paradigm for tame topology.
Lecture 2 : Model Theory of valued fields.
1) An application of Model Theory in the 60's: the Ax-Kochen-Ersov theorem.
2) Quantifier elimination for algebraically closed valued fields.
3) Elimination of imaginaries.
Lecture 3 : The basic object of study.
Given an algebraic variety U over a non-archimedean valued field, we introduce
the space Â of stably dominated types on it and prove that it is strictly pro-definable.
We also explain how it compares with Uan.
Lecture 4 : Some topological properties of Û
1) Definable compactness and its relation with properness
2) Γ-internal subsets of Û are topologically tame
3) More advanced material (if times allows).
Jérome POINEAU (Strasbourg's university) : Berkovich spaces over Z
We shall split the course into two parts.
First we shall recall Berkovich's general construction of
analytic spaces over Z with a particular emphasis on the affine line. It
is a remarkable fact that in many respects this space behaves like usual
analytic spaces: it is for instance Hausdorff, locally compact and locally
path connected, and its local rings are Henselian, Noetherian, and regular.
Second, we shall turn our attention to the study of Stein subsets of the
affine line over Z. These subsets are defined in terms of the vanishing of
coherent cohomology. We shall derive some applications of this study to
the construction of convergent power series with integral coefficients
having prescribed poles, and to the inverse Galois problem.
In this mini-course we will introduce Berkovich analytic spaces over
a non- archimedean field
and will study their basic properties. A familiarity with algebraic geometry and commutative algebra
is the main prerequisite for the course. Some familiarity with field valuations and formal schemes
may also be helpful, though I will mention briefly the facts we will need about them. In order to cover
the large amount of material we will concentrate on describing definitions and constructions and
formulating the main results of the theory, although in some cases main ideas of the proofs will be outlined.
The course can be divided to five parts as follows: §1 valuations, non-archimedean fields and Banach algebras,
§2 affinoid algebras and spaces, §3 analytic spaces, §4 connection to other categories: analytification of algebraic
varieties and generic fiber of formal schemes, §5 analytic curves.