p - adic Modular Forms over Shimura Curves over

In this thesis, we set up the basic theory of p-adic modular forms over Shimura curves over Q, parallel to the classical case over modular curves. We define and study the structure of the spaces of p-adic modular forms with respect to certain quaternion algebras over Q. We study the relation of these modular forms with classical quaternionic modular forms. We prove a canonical subgroup theorem for false elliptic curves. That enables us to define the Frobenius morphism of p-adic modular functions. We use rigid analytic geometry to give an alternative description of the p-adic modular forms and their Frobenius morphism. We use this to study the finiteness properties of the Frobenius morphism. We define the U operator of p-adic modular forms and study its continuity properties. We show that the U operator is completely continuous on the space of overconvergent p-adic modular forms. Finally, we use the Fredholm theory of U and rigid geometry to study the dimension of spaces of generalized eigenforms of U of a certain slope. Thesis Supervisor: Steven Kleiman Title: Professor of Mathematics To My Parents Mohammad and Nahid