GLOBES

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.