$\mathcal P$-adic modular forms over Shimura curves over totally real fields
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.(Received January 3 2002)
(Accepted March 7 2003)
Key Words: p-adic modular forms; quaternionic and Hilbert modular forms; Shimura curves; the U-operator.
11F85 (primary); 11F55; 11F33; 14G35 (secondary).
Dedicated to S. Shahshahani
p1 Department of Mathematics, McGill University, Montreal, Quebec H3A 2K6, Canada (e-mail: email@example.com)