Abstract:One of the monumental examples of Langlands program is the Jacquet-Langlands correspondence which is a correspondence between automorphic forms on ${\rm GL}(2)$ and its twisted forms. The correspondence between division algebras of dimension $n^2$ and ${\rm GL}(n)$ was proved by Jacquet and Langlands in both the local and global settings, hence the name. In 1983, Rogawski extended the local Jacquet-Langlands correspondence to division algebras of higher dimension in characteristic 0. Deligne, Kazhdan and Vigneras carried out the case of a general inner form of ${\rm GL}(n,F)$ in characteristic 0 in 1984, and Badulescu in characteristic $p$ in 2002. Each of these cases was accomplished by embedding the local problem into a global one and then applying Selberg trace formula methods. In this talk, we study the geometry of the special fibers of certain Shimura curves and give a direct proof of global-to-local Jacquet-Langlands compactibility by Cerednik-Drinfeld uniformizations theorem.