TITLE: why we need rigid geometry? 
QUESTION [16 upvotes]: Hello, everyone.
I want to ask some questions about rigid geometry.
1.what is the motivation of rigid geometry?
2.what is the applications of rigid geometry for solving arithmetic problems, especially for studying the fundamental groups of algebraic curves? what the beautiful theorems which were first proved by rigid geometry method?
thank you very much

REPLY [3 votes]: It is known that both the Jacquet-Langlands correspondence and the local Langlands correspondence for GLn can be realized in the étale cohomology of a Lubin-Tate tower (or, more precisely, in the étale cohomology of the Berkovich space attached to the rigid analytic space which is the generic fibre of the Lubin-Tate tower).