The p - adic upper half plane Course and Project

This space was first introduced by Mumford, where it plays a key role in the generalization to higher genus of Tate’s theory of p-adic uniformization of elliptic curves with semistable reduction. Slightly later, Drinfeld and Cerednik showed that appropriate quotients of this space by discrete arithmetic subgroups of PGL2(K) coming from quaternion algebras yield Shimura curves. Since that time, through work of Morita, Schneider, Bertolini, Darmon, Iovita, the authors, and many others, this space and its relationships to arithmetic have been the subject of intensive study. In this course, we will study some aspects of this recent work.