Edition 2012

*Global Berkovich Spaces*

In this project, we plan to carry out a comprehensive study of global analytic geometry, that is to say the analytic geometry of spaces defined over rings of integers of number fields.

**Foundational results and applications**

In the project, we will approach the global analytic spaces from different points of view. First, we will carry out a local study of those spaces and prove results analogous to those that hold in the complex and p-adic cases. We will also try to understand the cohomology of coherent sheaves and even metrized fiber bundles, in order to find a link with Arakelov theory.

Next, we will study the topology of global analytic spaces. We will first analyse Berkovich’s spectral topology, and then the étale topology (for which a suitable definition is still lacking). We also plan to investigate the associated fundamental groups. We believe that this work will lead to geometric interpretations of some fundamental invariants of number fields. We will also try to shed light on the relation between the étale topology of global analytic spaces and the Weil-étale topology of arithmetic schemes.

Finally, we will explore some applications of the theory of global analytic spaces. We believe that the techniques developed in the first parts of the project will enable us to construct spaces enjoying various nice properties. Such constructions have a wide range of applications, from the inverse Galois problem to information theory. In particular, we plan to use our results in the setting of asymptotic theory of zeta and L-functions, especially in the case of characteristic zero. We believe that our study of global analytic spaces will provide us with new methods to construct asymptotically good families of spaces over number fields, thus yielding new codes and sphere packings.

**A global approach, with analytic and algebraic methods**

Our three main tasks are the following: local and global properties of global analytic spaces, topology of arithmetic spaces and applications. They will have to be dealt with essentially in this order. Still, there will be strong interactions between the three tasks: for example, some property which we might overlook in the first part of the project could be needed for the applications. Hence we will have people in the project working on the three tasks at the same time.

Let us also insist on the fact that even for results that are analogous to those of the complex or p-adic cases, different methods are likely to be needed. As a consequence, we will really need to master the techniques in the known cases first in order to see how they could be improved. To be able to do so, we do not plan to work solely on global analytic spaces, but also on analytic spaces over fields in order to improve the techniques in this area first. This is particularly relevant for subjects that are quite technical and new, like the study of the étale and tempered fundamental groups.

Each time we start working on a new part of the project, we plan to first make some explicit computations (of cohomology groups, fundamental groups, etc.) on simple examples, such as affine spaces, projective spaces, elliptic curves, more general curves or even special kinds of surfaces if we think this is relevant. This will lead to a database of examples that could be useful to other researchers.

We plan to organize regular meetings, one per year, so that every member of the project can be informed of what the others have done. We also believe that it is wise to start working on new topics on these occasions: being all together will make it easier to identify the obstacles and to divide the work to be done.

When the project began, global analytic spaces were only really understood in the case of the line. Since then, Poineau managed to carry out a local study in any dimension. He has proved that the spaces satisfy the expected properties: Noetherianity, excellence, coherence of the structure sheaf, etc.

The new methods that are now available are bound to find many applications and will help investigate the other topics of the project.

Next, we will undertake the global study of global analytic spaces. We will start by investigating the coherent cohomology of the spaces, motivated by explicit computations in simple examples. We will also search for useful criteria for Stein spaces (spaces on which the coherent cohomology vanishes).

Once we manage to understand the basics of global analytic spaces and their coherent cohomology, we will investigate their spectral and étale topology. Our first task will be to formulate a proper definition of étale topology for global analytic spaces, which is currently still lacking.

We will then investigate the topological and étale covers and the fundamental groups of global analytic spaces. We believe that it is possible to carry out this study in the case of curves. For higher-dimensional spaces, though, we will need to introduce new methods as those in the p-adic setting involve formal models, which are not available for global spaces.

We will also analyze the étale cohomology of global analytic spaces. We are deeply interested in the Weil-étale cohomology of arithmetic spaces and will explore the connections with the former.

We are quite confident that our study of étale covers of global spaces will enable us to construct spaces with prescribed properties. This would lead to new results for the inverse Galois problem over rings of arithmetic power series, especially in the higher-dimensional case, which is completely open. We also hope that it will give us new methods to construct asymptotically good families, in the sense of information theory.

We plan to present our work at international conferences and seminars and publish it in international journals, thus making it available to the mathematical community. We believe that it can open a way to new research, both theoretical (development and applications of global analytic geometry, Weil-étale cohomology, etc.) and technological (through the applications to information theory). The subject of global analytic spaces bears an analytic flavor but may also be of interest to other people, for instance to number theorists. We aim to show that global analytic spaces are tractable objects and to provide a toolbox so that people from other areas may start using them successfully.

We will make special efforts to make the theory of global analytic geometry and its applications available to young researchers. We plan to devote an important part of the budget to the organization of workshops, conferences and a summer school with introductory lectures aimed at researchers from various domains who want to enter the subject. We also plan to write survey articles with the same purpose. We strongly believe that the events we plan to organize on various new and active topics related to the project will be beneficial to the communities of number theorists and arithmetic geometers, and especially to Phd students and young post-docs.

At the moment (T0 + 6 months), we already have:

- articles in international journals;

- talks in international conferences;

- introductory mini-courses online (http://ag.hse.ru/ducros-poineau);

- a first workshop in preparation (to take place in January 2014).

IRMA Institut de Recherche Mathématique Avancée

Laboratoire public

ANR grant: **130 000 euros**

Beginning and duration: **mars 2013 - 48 mois**

In this project, we plan to carry out a comprehensive study of global analytic geometry, that is to say the analytic geometry of spaces defined over rings of integers of number fields.

Analytic geometry was first developed over the field of complex numbers and proved very successful. More recently, the work of Tate, followed by others (Berkovich, Huber, etc.), laid the foundations of analytic geometry over any complete valued field (in particular over the p-adic fields). This theory is now well developed and has given rise to numerous applications (Langlands program, dynamics, motivic integration, etc.).

Berkovich's definition of analytic spaces actually makes it possible to go even further and define analytic spaces over rings of integers of number fields. These "global" spaces are naturally fibered into complex and p-adic analytic spaces and form a bridge between archimedean and non-archimedean geometry. Although very promising, the study of the geometry of global analytic spaces is still in its infancy. The aim of our project is to develop it in a systematic way and to apply it to various problems in algebraic and arithmetic geometry.

In the project, we will approach the global analytic spaces from different points of view. First, we will carry out a local study of those spaces and prove results analogous to those that hold in the complex and p-adic cases. We will also try to understand the cohomology of coherent sheaves and even metrized fiber bundles, in order to find a link with Arakelov theory.

Next, we will study the topology of global analytic spaces. We will first analyse Berkovich’s spectral topology, and then the étale topology (for which a suitable definition is still lacking). We also plan to investigate the associated fundamental groups. We believe that this work will lead to geometric interpretations of some fundamental invariants of number fields. We will also try to shed light on the relation between the étale topology of global analytic spaces and the Weil-étale topology of arithmetic schemes.

Finally, we will explore some applications of the theory of global analytic spaces. We believe that the techniques developed in the first parts of the project will enable us to construct spaces enjoying various nice properties. Such constructions have a wide range of applications, from the inverse Galois problem to information theory. In particular, we plan to use our results in the setting of asymptotic theory of zeta and L-functions, especially in the case of characteristic zero. We believe that our study of global analytic spaces will provide us with new methods to construct asymptotically good families of spaces over number fields, thus yielding new codes and sphere packings.

ANR Programme: JCJC - SIMI 1 - Mathématiques et interactions (JCJC SIMI 1) 2012

Project ID: ANR-12-JS01-0007

Project coordinator:**Monsieur POINEAU Jérôme (Université de Caen)**

Project web site: http://www-irma.u-strasbg.fr/~poineau/globes/

The project coordinator is the author of this abstract and is therefore responsible for the content of the summary. The ANR disclaims all responsibility in connection with its content.