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1 results

  • On Ihara’s lemma for Hilbert modular varieties

  • Part of
  • Mladen Dimitrov
  • Journal:
    Compositio Mathematica / Volume 145 / Issue 5 / September 2009
    Published online by Cambridge University Press:
    09 September 2009, pp. 1114-1146
    Print publication:
    September 2009
    • Article
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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.