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47
On the modularity of elliptic curves over Q: Wild 3adic exercises
 Journal of the Amer. Math. Soc
"... In this paper, building on work of Wiles [Wi] and of Taylor and Wiles [TW], we will prove the following two theorems (see x2.2). Theorem A. If E=Q is an elliptic curve, then E is modular. Theorem B. If : Gal(Q=Q) ! GL2(F5) is an irreducible continuous represen ..."
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Cited by 346 (0 self)
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In this paper, building on work of Wiles [Wi] and of Taylor and Wiles [TW], we will prove the following two theorems (see x2.2). Theorem A. If E=Q is an elliptic curve, then E is modular. Theorem B. If : Gal(Q=Q) ! GL2(F5) is an irreducible continuous represen
Serre's modularity conjecture (I)
, 2007
"... This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases p ̸ = 2 and odd conductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [13]. We then reduce the general case to a modularity statement ..."
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Cited by 97 (0 self)
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This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases p ̸ = 2 and odd conductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [13]. We then reduce the general case to a modularity statement for 2adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin [19].
Modularity of certain potentially BarsottiTate Galois representations
 J. Amer. Math. Soc
, 1999
"... Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate ..."
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Cited by 81 (7 self)
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Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate
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 modularity of elliptic curves over Q
 J. of the AMS
"... In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ: Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic ..."
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Cited by 30 (4 self)
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In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E /Q is an elliptic curve, then E is modular. Theorem B. If ρ: Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic
Report on mod representations of Gal(Q/Q
 In Motives
, 1991
"... Let N ≥ 1 and k ≥ 2 be integers. Let Γ1(N) be the group{( a b ..."
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Cited by 19 (0 self)
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Let N ≥ 1 and k ≥ 2 be integers. Let Γ1(N) be the group{( a b
Constructing elements in Shafarevich–Tate groups of modular motives
 in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303
, 2003
"... We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of ..."
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Cited by 17 (3 self)
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We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1
Mazur’s Principle for totally real fields of odd degree
, 1998
"... Abstract. In this paper, we prove an analogue of the result known as Mazur’s Principle concerning optimal levels of mod ` Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of ..."
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Cited by 16 (2 self)
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Abstract. In this paper, we prove an analogue of the result known as Mazur’s Principle concerning optimal levels of mod ` Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of the special fibre. It is this study which allows us to generalise, in the second part of this paper, Mazur’s result to totally real fields of odd degree.
lowering for modular mod ` representations over totally real
, 1999
"... In this paper, we continue the study of part of the analogue of Serre’s conjecture for mod ` Galois representations for totally real fields. More precisely, one knows, through results of Carayol and Taylor, that to any Hilbert cuspidal eigenform over a totally real field F, one can attach a compatib ..."
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Cited by 14 (0 self)
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In this paper, we continue the study of part of the analogue of Serre’s conjecture for mod ` Galois representations for totally real fields. More precisely, one knows, through results of Carayol and Taylor, that to any Hilbert cuspidal eigenform over a totally real field F, one can attach a compatible system of λadic representations of the corresponding absolute Galois group. One may ask if a given λadic or modulo ` representation is attached by this process to a Hilbert modular form, and, if so, what weights and levels this form can have. We prove some analogues of results known in the case F = Q.