On Ihara's lemma for Hilbert Modular Varieties

Abstract

Let ρbe a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρhas large image and admits a low weight crystalline modular deformation we show that any low weight crystalline deformation of ρunramified 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 ρof the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at 1 of the adjoint L-function associated to a Hilbert modular newform to the cardinality of the corresponding Selmer group.

This is a completely revised version (30 pages)

References in corpus (4)

• Deformation rings and Hecke algebras in the totally real case
• On Ihara's lemma for Hilbert Modular Varieties
• Cohomology of Siegel Varieties with p-adic integral coefficients and Applications
• Level optimization in the totally real case

Cited by in corpus (15)

• On Ihara's lemma for Hilbert Modular Varieties
• Euler systems for Hilbert modular surfaces
• Modularity of the Consani-Scholten quintic
• Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form
• Unramifiedness of Galois representations attached to weight one Hilbert modular eigenforms mod p
• Proof of de Smit's conjecture: a freeness criterion
• On the quantitative variation of congruence ideals and integral periods of modular forms
• Iwasawa theory for quadratic Hilbert modular forms
• Motivic cohomology of quaternionic Shimura varieties and level raising
• Unramifiedness of weight one Hilbert Hecke algebras
• Congruences between Hilbert modular forms of weight $2$, and special values of their $L$-functions
• Congruence modules in higher codimension and zeta lines in Galois cohomology
• Galois representations modulo $p$ and cohomology of Hilbert modular varieties
• Control Theorems for Hilbert Modular Varieties
• Explicit determination of images of Galois representations attached to Hilbert modular forms