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79
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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1. BreuilMézard conjecture and the padic local Langlands 644 (1.1) The BreuilMézard conjecture 644 (1.2) Review of Colmez’s functor 647
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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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.
NEARLY ORDINARY GALOIS DEFORMATIONS OVER ARBITRARY NUMBER FIELDS
, 2009
"... Let K be an arbitrary number field, and let ρ: Gal ( ¯ K/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation ..."
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Let K be an arbitrary number field, and let ρ: Gal ( ¯ K/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic ’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM ’ forms, and (3) ‘even ’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many ’ positivedimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘padic functoriality’, as well as some remarks on how our methods should apply to ndimensional representations of Gal ( ¯ Q/Q) when n>2.
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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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”.
Adjoint motives of modular forms and the Tamagawa number conjecture
, 2001
"... This paper concerns the Tamagawa number conjecture of Bloch and Kato [BK] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated Lfunction to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” Th ..."
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This paper concerns the Tamagawa number conjecture of Bloch and Kato [BK] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated Lfunction to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” The strategy for achieving
Φmodules and coefficient spaces
 Moscow Math. J
"... This paper is inspired by Kisin’s article [Ki1], in which he studies deformations of Galois representations of a local padic field which are defined by finite flat group schemes. The result of Kisin most relevant to our paper is his construction of a kind of resolution of the formal deformation spa ..."
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This paper is inspired by Kisin’s article [Ki1], in which he studies deformations of Galois representations of a local padic field which are defined by finite flat group schemes. The result of Kisin most relevant to our paper is his construction of a kind of resolution of the formal deformation space of the given Galois representation, by constructing a scheme which
Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image
, 2013
"... This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation sa ..."
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This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem. MSC (2010): 11F80 (Galois representations); 20C25 (Projective representations and multipliers), 12F12 (Inverse Galois theory). 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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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 ↦→