# Overview

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p-adic uniformization arose initially as an analogue of complex uniformization. The central example of complex uniformization goes back to Riemann, Fricke, Klein, Picard, Fuchs and Poincaré among others and had started with Riemann's famous classification of simply connected Riemann surfaces, in particular proving that every compact Riemann surface of genus 2 or higher is uniformized by the upper half plane. The study of the nature of the uniformization and the fundamental groups involved as well as connections to differential equations and function theory were initiated by the authors named above and continues to the present day.

The development of (rigid) analytic geometry over p-adic fields and its connection with the theory of formal schemes opened the way for the construction of an analogue of such uniformization, where the upper half plane is replaced by the p-adic (Drinfeld) upper half plane. Seminal contributions were made by Drinfeld, who introduced the moduli spaces of formal OB-modules and Mumford, who considered connections with very strongly degenerating families of curves and with Schottky groups. Mumford also developed a uniformization of strongly degenerating abelian varieties, putting Tate's ad hoc discussion of degenerating elliptic curves (the Tate curve") in a broader perspective. Mumford's work has been critical in understanding the boundary of Shimura varieties and constitutes a very important ingredient of the work of Chai-Faltings. Boutot and Carayol provided many details for Drinfeld's construction and in particular explained very methodically the connection to moduli spaces of p-divisible groups. Such groups play a role in studying abelian varieties mod p analogous to the role played by lattices in the study of complex abelian varieties; that is to say, they are all-important.

With the work of Rapoport and Zink, the subject of p-adic uniformization came to the forefront of arith- metic geometry. In ny" moduli space of abelian varieties, once one chooses a prime p, there is a nerve running through the moduli space, the so called basic stratum. The formal completion of the moduli space along this nerve is the analogue of a contractible neighborhood of a _ber in complex geometry. Namely, it is a zone where one can study vanishing cycles and local cohomology questions. In particular, we have learned that this construction is intimately related with the Langlands program and other questions dealing with modular forms. In retrospect of nearly a century, some of this theory can be seen in Deuring's work on elliptic curves.

Rapoport-Zink spaces o_er examples of p-adic uniformization extending beyond Drinfeld spaces. There are many interesting problem concerning them that are the subject of current research. Scholze's recent ground breaking work has provided a new perspective on Rapoport-Zink spaces, while at the same time achieving astounding applications to classical geometric questions such as Deligne's weight-monodromy conjecture, construction of Galois representations associated to torsion classes in cohomology of Shimura varieties and more.

The workshop on p-adic uniformization aims to start at the bottom and touch of some of the most recent developments. It will provide an overview of this theory, beginning with background on rigid analytic geometry and formal schemes, the work of Drinfeld and Boutot-Carayol, and finishing with an exposition of some of the results of Rapoport-Zink, Scholze, and Kudla-Rapoport.

In collaboration with the Centre universitaire en calcul mathématique algébrique (CICMA).