$\mathcal P$-adic modular forms over Shimura curves over totally real fields

Abstract

We set up the basic theory of $\mathcal P$ -adic modular forms over certain unitary PEL Shimura curves M ′ K ′ . For any PEL abelian scheme classified by M ′ K ′ , which is not ‘too supersingular’, we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p . Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent $\mathcal P$ -adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.