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

Published online by Cambridge University Press:  04 December 2007

Payman L Kassaei
Payman L Kassaei
Affiliation:
Department of Mathematics, Brandeis University, MS 050, P.O. Box 9110, Waltham, MA 02454-9110, [email protected] Department of Mathematics, McGill University, Montreal, Quebec H3A 2K6, Canada (e-mail: [email protected])
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Abstract

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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.

Keywords

p-adic modular formsquaternionic and Hilbert modular formsShimura curvesthe U-operator11F85 (primary)11F5511F3314G35 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 140 , Issue 2 , March 2004 , pp. 359 - 395
Copyright
Foundation Compositio Mathematica 2004