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38
Modular elliptic curves and Fermat’s Last Theorem
 ANNALS OF MATH
, 1995
"... When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n ..."
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Cited by 612 (1 self)
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When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n> 2 such that a n + b n = c n. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
Ringtheoretic properties of certain Hecke algebras
 Ann. of Math
, 1995
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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
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.
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.
Galois representations and modular forms
 Bull. Amer. Math. Soc
, 1995
"... Abstract. In this article, I discuss material which is related to the recent ..."
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Cited by 15 (1 self)
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Abstract. In this article, I discuss material which is related to the recent
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”.
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
Ramified deformation problems
 Duke Math. J
, 1999
"... The proof of the semistable TaniyamaShimura Conjecture by Wiles [22] and TaylorWiles [21] uses as its central tool the deformation theory of Galois representations. In [6], Diamond extends these methods, proving that an elliptic curve E /Q is modular if it is either semistable at 3 and 5 or is jus ..."
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Cited by 11 (4 self)
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The proof of the semistable TaniyamaShimura Conjecture by Wiles [22] and TaylorWiles [21] uses as its central tool the deformation theory of Galois representations. In [6], Diamond extends these methods, proving that an elliptic curve E /Q is modular if it is either semistable at 3 and 5 or is just semistable at 3, provided that the representation