Compositio Mathematica

Compositio Mathematica (2004), 140:2:359-395 London Mathematical Society
Copyright © Foundation Compositio Mathematica 2004
doi:10.1112/S0010437X03000150

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


Payman L Kassaei a1p1
a1 Department of Mathematics, Brandeis University, MS 050, P.O. Box 9110, Waltham, MA 02454-9110, U.S.A. payman@brandeis.edu

Article author query
kassaei p   [Google Scholar] 
 

Abstract

We set up the basic theory of $\mathcal P$-adic modular forms over certain unitary PEL Shimura curves MK. For any PEL abelian scheme classified by MK, 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.

Maths Classification

11F85 (primary); 11F55; 11F33; 14G35 (secondary).


Dedication:
Dedicated to S. Shahshahani

Correspondence:
p1 Department of Mathematics, McGill University, Montreal, Quebec H3A 2K6, Canada (e-mail: kassaei@math.mcgill.ca)


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