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p-adic modular forms of non-integral weight over Shimura curves

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 November 2012

Riccardo Brasca
Riccardo Brasca*
Affiliation:
Dipartimento di Matematica, Università degli studi di Milano, Milan, Italy (email: riccardo.brasca@gmail.com)
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Abstract

In this work, we set up a theory of p-adic modular forms over Shimura curves over totally real fields which allows us to consider also non-integral weights. In particular, we define an analogue of the sheaves of kth invariant differentials over the Shimura curves we are interested in, for any p-adic character. In this way, we are able to introduce the notion of overconvergent modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic families over a suitable rigid analytic space, that parametrizes the weights. Finally, we define Hecke operators, including the U operator, that acts compactly on the space of overconvergent modular forms. We also construct the eigencurve.

Keywords

p-adic modular formsquaternionic modular formsmodular forms of non-integral weight

MSC classification

Secondary: 11F85: $p$-adic theory, local fields 14G35: Modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 1 , January 2013 , pp. 32 - 62
Copyright
Copyright © The Author(s) 2012

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