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Geometric level raising for p-adic automorphic forms

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 August 2010

James Newton
James Newton*
Affiliation:
Department of Mathematics, South Kensington Campus, Imperial College London, SW7 2AZ, London, UK (email: [email protected])
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Abstract

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We present a level-raising result for families of p-adic automorphic forms for a definite quaternion algebra D over ℚ. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor’s theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such a pairing on the infinite dimensional space of p-adic automorphic forms, so instead we introduce a space defined with respect to a dual coefficient system and work with a pairing between the usual forms and the dual space. A key ingredient is an analogue of Ihara’s lemma which shows an interesting asymmetry between the usual and the dual spaces.

Keywords

p-adic modular formslevel raising

MSC classification

Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 335 - 354
Copyright
Copyright © Foundation Compositio Mathematica 2010

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