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68
Automorphy for some ladic lifts of automorphic mod l Galois representations. II
, 2006
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The FontaineMazur conjecture for GL2
 Journal of the A.M.S
"... 1. BreuilMézard conjecture and the padic local Langlands 644 (1.1) The BreuilMézard conjecture 644 (1.2) Review of Colmez’s functor 647 ..."
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Cited by 33 (0 self)
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1. BreuilMézard conjecture and the padic local Langlands 644 (1.1) The BreuilMézard conjecture 644 (1.2) Review of Colmez’s functor 647
Construction of some families of 2dimensional crystalline representations
, 2004
"... Abstract. We construct explicitly some analytic families of étale (ϕ, Ɣ)modules, which give rise to analytic families of 2dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and ex ..."
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Cited by 30 (4 self)
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Abstract. We construct explicitly some analytic families of étale (ϕ, Ɣ)modules, which give rise to analytic families of 2dimensional crystalline representations. As an application of our constructions, we verify some conjectures of Breuil on the reduction modulo p of those representations, and extend some results (of Deligne, Edixhoven, Fontaine and Serre) on the representations arising from modular forms. Mathematics Subject Classification (2000): 11F80, 11F33, 11F85, 14F30
First steps towards padic Langlands functoriality, preprint math.NT/0603499
"... Abstract. — By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the WeilDeligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlan ..."
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Cited by 26 (1 self)
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Abstract. — By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the WeilDeligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a WeilDeligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the WeilDeligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and
Automorphic lifts of prescribed types
, 2006
"... Abstract. We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger ..."
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Cited by 24 (8 self)
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Abstract. We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of ndimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for ndimensional automorphic Galois representations. Contents
ON SERRE’S MODULARITY CONJECTURE FOR 2DIMENSIONAL MOD p REPRESENTATIONS OF ... Unramified Outside p
, 2005
"... We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint wor ..."
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Cited by 13 (0 self)
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We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint work with JP. Wintenberger, together with a new method of “weight reduction”.
two filtered (ϕ, N)modules with Galois descent data and coefficients
"... Let K be any finite extension of Qp, F any finite Galois extension of K and E any finite, large enough coefficient field containing F. We classify twodimensional, Fsemistable Erepresentations of GK, by listing the isomorphism classes of rank two weakly admissible filtered (ϕ, N, F/K, E)modules. ..."
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Cited by 11 (3 self)
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Let K be any finite extension of Qp, F any finite Galois extension of K and E any finite, large enough coefficient field containing F. We classify twodimensional, Fsemistable Erepresentations of GK, by listing the isomorphism classes of rank two weakly admissible filtered (ϕ, N, F/K, E)modules. For simplicity, we restrict ourselves to the Fsemisimple nonscalar case but our method works in complete generality. Contents 1
Modularity of some potentially BarsottiTate Galois representations
 Compos. Math
"... We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDi ..."
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Cited by 11 (6 self)
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We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the FontaineMazur conjectures, specifically, the modularity of certain potentially BarsottiTate Galois representations. The proof follows the template of Wiles, TaylorWiles, and BreuilConradDiamondTaylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the ring of integers of a local field. This makes crucial use of the filtered φ1modules of C. Breuil. 1. Notation, terminology, and results Throughout this article, we let l be an odd prime, and we fix an algebraic closure Ql of Ql with residue field Fl. The fields K, L, and E will always be finite extensions of Ql inside Ql. We denote by GK the Galois group Gal(Ql/K), by WK the Weil group of K, and by IK the inertia group of K. The group IQl will be abbreviated Il. The character ωn: GQl → Fln ⊂ Fl is defined via u ωn: u ↦→
Serre’s modularity conjecture (II)
, 2007
"... We provide proofs of Theorems 4.1 and 5.1 of [30]. ..."