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On Ihara’s lemma for Hilbert modular varieties

Part of: Arithmetic algebraic geometry Algebraic number theory: global fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  09 September 2009

Mladen Dimitrov
Mladen Dimitrov*
Affiliation:
Université Paris 7, UFR Mathématiques, Site Chevaleret, Case 7012, 75205 Paris cedex 13, France (email: [email protected])
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Abstract

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Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

Keywords

Hilbert modular varietiescohomologyHecke algebrasGalois representationsadjoint L-functions

MSC classification

Secondary: 11F80: Galois representations 11G18: Arithmetic aspects of modular and Shimura varieties 11F41: Automorphic forms on $(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11R34: Galois cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1114 - 1146
Copyright
Copyright © Foundation Compositio Mathematica 2009

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