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27
The SatoTate Conjecture for Hilbert Modular Forms
"... We prove the SatoTate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the SatoTate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF), F a totally real field, which are not of CM type. The argument is based on the potentia ..."
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We prove the SatoTate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the SatoTate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential
Explicit determination of images of Galois representations attached to Hilbert modular forms
 J. Number Theory, 117, Issue
"... Abstract. In a previous article [6], the second author proved that the images of the Galois representations mod λ attached to a Hilbert modular form without Complex Multiplication are “large ” for all but finitely many primes λ. In this brief note, we give an explicit bound for this exceptional fini ..."
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Abstract. In a previous article [6], the second author proved that the images of the Galois representations mod λ attached to a Hilbert modular form without Complex Multiplication are “large ” for all but finitely many primes λ. In this brief note, we give an explicit bound for this exceptional finite set of primes and determine the images in three different examples. Our examples are of Hilbert newforms on real quadratic fields, of parallel or nonparallel weight and of different levels. 1.
On Ihara’s lemma for Hilbert Modular Varieties
, 2005
"... Let ρ be a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has large image and admits a minimal modular deformation we show that every low weight crystalline deformation of ρ unramified outside a finite set of primes is again modular. ..."
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Cited by 7 (1 self)
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Let ρ be a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has large image and admits a minimal modular deformation we show that every low weight crystalline deformation of ρ unramified outside a finite set of primes is again modular. We use the approach of Wiles and 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 ppart of the value at 1 of the adjoint Lfunction associated to a Hilbert modular newform to the cardinality of the corresponding Selmer group.
On a generalization of the conjecture of Mazur–Tate–Teitelbaum
 Article ID rnm102
"... We propose a generalization of the conjecture of Mazur–Tate–Teitelbaum predicting an exact shape of the padic Linvariant of rational Tate curves (which is now a theorem of GreenbergStevens) to the symmetric powers of motivic two dimensional odd Galois representations over totally real fields. At ..."
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We propose a generalization of the conjecture of Mazur–Tate–Teitelbaum predicting an exact shape of the padic Linvariant of rational Tate curves (which is now a theorem of GreenbergStevens) to the symmetric powers of motivic two dimensional odd Galois representations over totally real fields. At padic places where the motive is multiplicative, the Linvariant is conjectured to have the same shape as predicted by them. Then we prove our conjecture assuming a precise ring theoretic structure of the universal infinitesimal Galois deformation ring of the symmetric power. In References [17, 19], we made explicit a conjectural formula of the Linvariant of symmetric powers of a Tate curve over a totally real field. In this paper, we generalize the conjecture for more general two dimensional nearly ordinary odd Galois representations and prove the formula for Greenberg’s Linvariant when the symmetric power is of adjoint type, assuming a conjecture (Conjecture 0.1) on the ring structure of a Galois deformation ring of the symmetric power. Let p be an odd prime and F be a totally real field of degree d < ∞ with integer ring O. WriteSpforthe set of primes factors of p in F, we use the symbol p for a member Downloaded from
On reductions of families of crystalline Galois representations
 Doc. Math
"... Let K be any finite unramified extension of Qp. We construct analytic families of étale (ϕ,ΓK)modules which correspond to all the effective crystalline characters and some families of ndimensional crystalline Galois representations of GK =Gal ( ¯ Qp/K). As an application, we compute semisimplifie ..."
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Cited by 5 (1 self)
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Let K be any finite unramified extension of Qp. We construct analytic families of étale (ϕ,ΓK)modules which correspond to all the effective crystalline characters and some families of ndimensional crystalline Galois representations of GK =Gal ( ¯ Qp/K). As an application, we compute semisimplified modulo p reductions for some of these families. Contents
Density of crystalline points on unitary Shimura varieties
 Int. J. Number Theory
, 2013
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The SatoTate Conjecture for Modular Forms of Weight 3
 DOCUMENTA MATH.
, 2009
"... We prove a natural analogue of the SatoTate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place. ..."
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We prove a natural analogue of the SatoTate conjecture for modular forms of weight 2 or 3 whose associated automorphic representations are a twist of the Steinberg representation at some finite place.
On the freeness of the integral cohomology groups of HilbertBlumenthal varieties as Hecke modules
"... Let F be a totally real field of degree d ≥ 1. Let f be a Hilbert modular cusp form defined over F of level N ⊂ OF and parallel weight (k, k,..., k) with k ≥ 2. Assume that f is a normalized newform and a common eigenform of all the Hecke operators. Such a Hilbert modular form will be called a primi ..."
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Let F be a totally real field of degree d ≥ 1. Let f be a Hilbert modular cusp form defined over F of level N ⊂ OF and parallel weight (k, k,..., k) with k ≥ 2. Assume that f is a normalized newform and a common eigenform of all the Hecke operators. Such a Hilbert modular form will be called a primitive