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66
Ringtheoretic properties of certain Hecke algebras
 Ann. of Math
, 1995
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Fermat’s Last Theorem
 Current Developments in Mathematics
, 1995
"... The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this ..."
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Cited by 78 (12 self)
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The authors would like to give special thanks to N. Boston, K. Buzzard, and B. Conrad for providing so much valuable feedback on earlier versions of this
Localglobal compatibility in the padic Langlands programme for GL2/Q
, 2010
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On the Meromorphic Continuation of Degree Two LFunctions
 DOCUMENTA MATH.
, 2006
"... We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation. ..."
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Cited by 37 (2 self)
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We prove that the Lfunction of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.
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
Explicit construction of universal deformation rings
 MODULAR FORMS AND FERMAT’S LAST THEOREM
, 1997
"... Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete Oalgebras A with residue field k the representations of G over A that reduce to V over k are given by Oalgebra homomorphisms R → A, where R is the uni ..."
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Cited by 32 (0 self)
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Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete Oalgebras A with residue field k the representations of G over A that reduce to V over k are given by Oalgebra homomorphisms R → A, where R is the universal deformation ring of V. We show this with an explicit construction of R. The ring R is noetherian if and only if H 1 (G, Endk(V)) has finite dimension over k.
Variation of Iwasawa invariants in Hida families
 Invent. Math
"... Let ¯ρ: GQ → GL2(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p. Assume further that ¯ρ is pordinary and pdistinguished in the sense that the restriction of ¯ρ to a decomposition group at p is reducible and nonscalar. The Hida family H(¯ρ) ..."
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Cited by 30 (5 self)
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Let ¯ρ: GQ → GL2(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p. Assume further that ¯ρ is pordinary and pdistinguished in the sense that the restriction of ¯ρ to a decomposition group at p is reducible and nonscalar. The Hida family H(¯ρ) of ¯ρ is the set of all pordinary
On Serre’s conjecture for 2dimensional mod p representations of Gal(Q̄/Q)
"... We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weig ..."
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Cited by 27 (1 self)
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We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that a ¯ρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Deltafunction. In the last part of the paper we present arguments that reduce Serre’s conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.
Companion forms for unitary and symplectic groups
, 2009
"... Abstract. We prove a companion forms theorem for ordinary ndimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre ..."
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Cited by 15 (11 self)
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Abstract. We prove a companion forms theorem for ordinary ndimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp 4 over totally real fields.