## Summer School of Mathematics "Berkovich Spaces"## Organisé par : Charles Favre (Institut Mathématique Jussieu) |

*Introduction to Berkovich Analytic Spaces (Lecture 1)*(le 28 juin 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–**Lecture 1:**June 28, 11:00

–*Lecture 2:*June 29, 09:00

–*Lecture 3:*June 29, 11:30

–*Lecture 4:*June 30, 11:30

–*Lecture 5:*July 1, 10h30

–*Lecture 6:*July 2, 09h00

*F^1 Geometry (Lecture 1)*(le 28 juin 2010) —**Vladimir Berkovich**

This minicourse is an introduction to a work in progress on foundations of 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.

–**Lecture 1:**June 28, 14:00

–*Lecture 2:*June 30, 09:00

–*Lecture 3:*July 1, 11:30

–*Lecture 4:*July 2, 10:30

*Dynamics on Berkovich Spaces (Lecture 1)*(le 28 juin 2010) —**Mattias Jonsson**

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.

–**Lecture 1:**June 28, 15:30

–*Lecture 2:*June 29, 10:30

–*Lecture 3:*June 30, 10:30

–*Lecture 4:*July 1, 09:00

–*Lecture 5:*July 2, 11:30

*Introduction to Berkovich Analytic Spaces (Lecture 2)*(le 29 juin 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–*Lecture 1:*June 28, 11:00

–**Lecture 2:**June 29, 09:00

–*Lecture 3:*June 29, 11:30

–*Lecture 4:*June 30, 11:30

–*Lecture 5:*July 1, 10h30

–*Lecture 6:*July 2, 09h00

*Dynamics on Berkovich Spaces (Lecture 2)*(le 29 juin 2010) —**Mattias Jonsson**

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.

–*Lecture 1:*June 28, 15:30

–**Lecture 2:**June 29, 10:30

–*Lecture 3:*June 30, 10:30

–*Lecture 4:*July 1, 09:00

–*Lecture 5:*July 2, 11:30

*Introduction to Berkovich Analytic Spaces (Lecture 3)*(le 29 juin 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–*Lecture 1:*June 28, 11:00

–*Lecture 2:*June 29, 09:00

–**Lecture 3:**June 29, 11:30

–*Lecture 4:*June 30, 11:30

–*Lecture 5:*July 1, 10h30

–*Lecture 6:*July 2, 09h00

*F^1 Geometry (Lecture 2)*(le 30 juin 2010) —**Vladimir Berkovich**

This minicourse is an introduction to a work in progress on foundations of 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.

–*Lecture 1:*June 28, 14:00

–**Lecture 2:**June 30, 09:00

–*Lecture 3:*July 1, 11:30

–*Lecture 4:*July 2, 10:30

*Dynamics on Berkovich Spaces (Lecture 3)*(le 30 juin 2010) —**Mattias Jonsson**

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.

–*Lecture 1:*June 28, 15:30

–*Lecture 2:*June 29, 10:30

–**Lecture 3:**June 30, 10:30

–*Lecture 4:*July 1, 09:00

–*Lecture 5:*July 2, 11:30

*Introduction to Berkovich Analytic Spaces (Lecture 4)*(le 30 juin 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–*Lecture 1:*June 28, 11:00

–*Lecture 2:*June 29, 09:00

–*Lecture 3:*June 29, 11:30

–**Lecture 4:**June 30, 11:30

–*Lecture 5:*July 1, 10h30

–*Lecture 6:*July 2, 09h00

*Dynamics on Berkovich Spaces (Lecture 4)*(le 1^{er}juillet 2010) —**Mattias Jonsson**

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.

–*Lecture 1:*June 28, 15:30

–*Lecture 2:*June 29, 10:30

–*Lecture 3:*June 30, 10:30

–**Lecture 4:**July 1, 09:00

–*Lecture 5:*July 2, 11:30

*Introduction to Berkovich Analytic Spaces (Lecture 5)*(le 1^{er}juillet 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–*Lecture 1:*June 28, 11:00

–*Lecture 2:*June 29, 09:00

–*Lecture 3:*June 29, 11:30

–*Lecture 4:*June 30, 11:30

–**Lecture 5:**July 1, 10h30

–*Lecture 6:*July 2, 09h00

*F^1 Geometry (Lecture 3)*(le 1^{er}juillet 2010) —**Vladimir Berkovich**

This minicourse is an introduction to a work in progress on foundations of 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.

–*Lecture 1:*June 28, 14:00

–*Lecture 2:*June 30, 09:00

–**Lecture 3:**July 1, 11:30

–*Lecture 4:*July 2, 10:30

*Introduction to Berkovich Analytic Spaces (Lecture 6)*(le 2 juillet 2010) —**Michael Temkin**

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 will be divided into 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 a formal scheme, §5 analytic curves.

–*Lecture 1:*June 28, 11:00

–*Lecture 2:*June 29, 09:00

–*Lecture 3:*June 29, 11:30

–*Lecture 4:*June 30, 11:30

–*Lecture 5:*July 1, 10h30

–**Lecture 6:**July 2, 09h00

*F^1 Geometry (Lecture 4)*(le 2 juillet 2010) —**Vladimir Berkovich**

This minicourse is an introduction to a work in progress on foundations of 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.

–*Lecture 1:*June 28, 14:00

–*Lecture 2:*June 30, 09:00

–*Lecture 3:*July 1, 11:30

–**Lecture 4:**July 2, 10:30

*Dynamics on Berkovich Spaces (Lecture 5)*(le 2 juillet 2010) —**Mattias Jonsson**

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.

–*Lecture 1:*June 28, 15:30

–*Lecture 2:*June 29, 10:30

–*Lecture 3:*June 30, 10:30

–*Lecture 4:*July 1, 09:00

–**Lecture 5:**July 2, 11:30

*Étale Cohomology (Lecture 1)*(le 5 juillet 2010) —**Antoine Ducros**

Étale cohomology was introduced in the scheme-theoretic context by 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 Weil conjectures. For other deep arithmetic reasons (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 series of lectures, I plan, after having given some general motivations, to spend some time 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 forth. My purpose is not to give detailed proofs, which are for most of them highly technical. I will rather insist on examples, trying to show how étale cohomology can at the same time be quite close to the classical topological intuition, and deal in a completely natural manner with deep-field arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.

–**Lecture 1:**July 5, 09:00

–*Lecture 2:*July 6, 11:30

–*Lecture 3:*July 7, 10:30

–*Lecture 4:*July 9, 09:00

*Model Theory and Analytic Geometry (Lecture 1)*(le 5 juillet 2010) —**François Loeser**

In this course, an approach of Berkovich’s theory is given based on model theory. It gives new insights into the topology of these spaces.

–**Lecture 1:**July 5, 10:30

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:*July 6, 09:00

–*Lecture 3:*July 8, 09:00

–*Lecture 4:*July 9, 11:30

*Bruhat-Tits Buildings and Analytic Geometry (Lecture 1)*(le 5 juillet 2010) —**Bertrand Rémy**et**Amaury Thuillier**

Let G be a reductive algebraic group defined over a non-Archimedean local field k. During the 60’s and 70’s, F. Bruhat and J. Tits have been working on a fine description of groups of rational points like G(k). The achievement of this work is a combinatorial description that can be stated in geometric terms, i.e., using the Euclidean building of G over k. The latter space, which is both a complete metric space and a simplicial complex, can be seen in many ways as a (singular) analogue of the Riemannian symmetric space of a semi-simple real Lie group. During the 80’s, V. Berkovich developed an approach to analytic geometry over non-Archimedean complete fields, thus enriching the classical theory due to Tate-Raynaud. From the beginning he mentionned the possibility to combine his theory with Bruhat-Tits’ one. In these talks, we intend to present a joint work with A. Werner, in which we develop and extend Berkovich’s ideas. We show in particular that they enable one to define and describe natural compactifications of the Bruhat-Tits building of a group G over k as above. These compactifications can also be obtained—again by means of Berkovich geometry—by procedures which very much look like Satake’s initial ideas for symmetric spaces. The necessary material concerning Bruhat-Tits theory will be presented during the first lecture.

–**Lecture 1:**July 5, 11:30

–*Lecture 2:*July 6, 10:30

–*Lecture 3:*July 7, 09:00

–*Lecture 4:*July 8, 10:30

*Model Theory and Analytic Geometry (Lecture 2)*(le 6 juillet 2010) —**François Loeser**

In this course, an approach of Berkovich’s theory is given based on model theory. It gives new insights into the topology of these spaces.

–*Lecture 1:*July 5, 10:30

–**Lecture 2:**July 6, 09:00

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:*July 8, 09:00

–*Lecture 4:*July 9, 11:30

*Bruhat-Tits Buildings and Analytic Geometry (Lecture 2)*(le 6 juillet 2010) —**Bertrand Rémy**et**Amaury Thuillier**

Let G be a reductive algebraic group defined over a non-Archimedean local field k. During the 60’s and 70’s, F. Bruhat and J. Tits have been working on a fine description of groups of rational points like G(k). The achievement of this work is a combinatorial description that can be stated in geometric terms, i.e., using the Euclidean building of G over k. The latter space, which is both a complete metric space and a simplicial complex, can be seen in many ways as a (singular) analogue of the Riemannian symmetric space of a semi-simple real Lie group. During the 80’s, V. Berkovich developed an approach to analytic geometry over non-Archimedean complete fields, thus enriching the classical theory due to Tate-Raynaud. From the beginning he mentionned the possibility to combine his theory with Bruhat-Tits’ one. In these talks, we intend to present a joint work with A. Werner, in which we develop and extend Berkovich’s ideas. We show in particular that they enable one to define and describe natural compactifications of the Bruhat-Tits building of a group G over k as above. These compactifications can also be obtained—again by means of Berkovich geometry—by procedures which very much look like Satake’s initial ideas for symmetric spaces. The necessary material concerning Bruhat-Tits theory will be presented during the first lecture.

–*Lecture 1:*July 5, 11:30

–**Lecture 2:**July 6, 10:30

–*Lecture 3:*July 7, 09:00

–*Lecture 4:*July 8, 10:30

*Étale Cohomology (Lecture 2)*(le 6 juillet 2010) —**Antoine Ducros**

Étale cohomology was introduced in the scheme-theoretic context by 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 Weil conjectures. For other deep arithmetic reasons (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 series of lectures, I plan, after having given some general motivations, to spend some time 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 forth. My purpose is not to give detailed proofs, which are for most of them highly technical. I will rather insist on examples, trying to show how étale cohomology can at the same time be quite close to the classical topological intuition, and deal in a completely natural manner with deep-field arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.

–*Lecture 1:*July 5, 09:00

–**Lecture 2:**July 6, 11:30

–*Lecture 3:*July 7, 10:30

–*Lecture 4:*July 9, 09:00

*Bruhat-Tits Buildings and Analytic Geometry (Lecture 3)*(le 7 juillet 2010) —**Bertrand Rémy**et**Amaury Thuillier**

Let G be a reductive algebraic group defined over a non-Archimedean local field k. During the 60’s and 70’s, F. Bruhat and J. Tits have been working on a fine description of groups of rational points like G(k). The achievement of this work is a combinatorial description that can be stated in geometric terms, i.e., using the Euclidean building of G over k. The latter space, which is both a complete metric space and a simplicial complex, can be seen in many ways as a (singular) analogue of the Riemannian symmetric space of a semi-simple real Lie group. During the 80’s, V. Berkovich developed an approach to analytic geometry over non-Archimedean complete fields, thus enriching the classical theory due to Tate-Raynaud. From the beginning he mentionned the possibility to combine his theory with Bruhat-Tits’ one. In these talks, we intend to present a joint work with A. Werner, in which we develop and extend Berkovich’s ideas. We show in particular that they enable one to define and describe natural compactifications of the Bruhat-Tits building of a group G over k as above. These compactifications can also be obtained—again by means of Berkovich geometry—by procedures which very much look like Satake’s initial ideas for symmetric spaces. The necessary material concerning Bruhat-Tits theory will be presented during the first lecture.

–*Lecture 1:*July 5, 11:30

–*Lecture 2:*July 6, 10:30

–**Lecture 3:**July 7, 09:00

–*Lecture 4:*July 8, 10:30

*Étale Cohomology (Lecture 3)*(le 7 juillet 2010) —**Antoine Ducros**

Étale cohomology was introduced in the scheme-theoretic context by 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 Weil conjectures. For other deep arithmetic reasons (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 series of lectures, I plan, after having given some general motivations, to spend some time 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 forth. My purpose is not to give detailed proofs, which are for most of them highly technical. I will rather insist on examples, trying to show how étale cohomology can at the same time be quite close to the classical topological intuition, and deal in a completely natural manner with deep-field arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.

–*Lecture 1:*July 5, 09:00

–*Lecture 2:*July 6, 11:30

–**Lecture 3:**July 7, 10:30

–*Lecture 4:*July 9, 09:00

*Berkovich Spaces over Z (Lecture 1)*(le 7 juillet 2010) —**Jérôme Poineau**

This course is divided 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.

–**Lecture 1:**July 7, 11:30

–*Lecture 2:*July 8, 11:30

–*Lecture 3:*July 9, 10:30

*Model Theory and Analytic Geometry (Lecture 3)*(le 8 juillet 2010) —**François Loeser**

In this course, an approach of Berkovich’s theory is given based on model theory. It gives new insights into the topology of these spaces.

–*Lecture 1:*July 5, 10:30

–*Lecture 2:*July 6, 09:00

–**Lecture 3:**July 8, 09:00

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 U^{an}.

–*Lecture 4:*July 9, 11:30

*Bruhat-Tits Buildings and Analytic Geometry (Lecture 4)*(le 8 juillet 2010) —**Bertrand Rémy**et**Amaury Thuillier**

Let G be a reductive algebraic group defined over a non-Archimedean local field k. During the 60’s and 70’s, F. Bruhat and J. Tits have been working on a fine description of groups of rational points like G(k). The achievement of this work is a combinatorial description that can be stated in geometric terms, i.e., using the Euclidean building of G over k. The latter space, which is both a complete metric space and a simplicial complex, can be seen in many ways as a (singular) analogue of the Riemannian symmetric space of a semi-simple real Lie group. During the 80’s, V. Berkovich developed an approach to analytic geometry over non-Archimedean complete fields, thus enriching the classical theory due to Tate-Raynaud. From the beginning he mentionned the possibility to combine his theory with Bruhat-Tits’ one. In these talks, we intend to present a joint work with A. Werner, in which we develop and extend Berkovich’s ideas. We show in particular that they enable one to define and describe natural compactifications of the Bruhat-Tits building of a group G over k as above. These compactifications can also be obtained—again by means of Berkovich geometry—by procedures which very much look like Satake’s initial ideas for symmetric spaces. The necessary material concerning Bruhat-Tits theory will be presented during the first lecture.

–*Lecture 1:*July 5, 11:30

–*Lecture 2:*July 6, 10:30

–*Lecture 3:*July 7, 09:00

–**Lecture 4:**July 8, 10:30

*Berkovich Spaces over Z (Lecture 2)*(le 8 juillet 2010) —**Jérôme Poineau**

This course is divided 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.

–*Lecture 1:*July 7, 11:30

–**Lecture 2:**July 8, 11:30

–*Lecture 3:*July 9, 10:30

*Étale Cohomology (Lecture 4)*(le 9 juillet 2010) —**Antoine Ducros**

Étale cohomology was introduced in the scheme-theoretic context by 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 Weil conjectures. For other deep arithmetic reasons (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 series of lectures, I plan, after having given some general motivations, to spend some time 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 forth. My purpose is not to give detailed proofs, which are for most of them highly technical. I will rather insist on examples, trying to show how étale cohomology can at the same time be quite close to the classical topological intuition, and deal in a completely natural manner with deep-field arithmetic phenomena (such as Galois theory), which allows sometimes to think to the latter in a purely geometrical way.

–*Lecture 1:*July 5, 09:00

–*Lecture 2:*July 6, 11:30

–*Lecture 3:*July 7, 10:30

–**Lecture 4:**July 9, 09:00

*Berkovich Spaces over Z (Lecture 3)*(le 9 juillet 2010) —**Jérôme Poineau**

This course is divided 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.

–*Lecture 1:*July 7, 11:30

–*Lecture 2:*July 8, 11:30

–**Lecture 3:**July 9, 10:30

*Model Theory and Analytic Geometry (Lecture 4)*(le 9 juillet 2010) —**François Loeser**

In this course, an approach of Berkovich’s theory is given based on model theory. It gives new insights into the topology of these spaces.

–*Lecture 1:*July 5, 10:30

–*Lecture 2:*July 6, 09:00

–*Lecture 3:*July 8, 09:00

–**Lecture 4:**July 9, 11:30

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).

Charles Favre (Institut Mathématique Jussieu) |

Ecole d’été Langues et langage : compréhension, traduction, argumentation