Fermat’s Last Theorem

Darmon, Henri; Diamond, Fred; Taylor, Richard

doi:10.4310/cdm.1995.v1995.n1.a1

Cited by 126 publications

(117 citation statements)

References 67 publications

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“…The existence of R in the present context was deduced first by Mazur [8] with Schlessinger's criteria for pro-representability [12]. An alternative construction was given recently by Faltings (see [5] and Section 7 below).The main result of this chapter, formulated below as Theorem (2.3), is actually a little more general than Mazur's. Following Schlessinger, Mazur works only with noetherian rings, and this forces him to assume at the outset that a certain cohomology group is finite.…”

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confidence: 83%

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Explicit Construction of Universal Deformation Rings

Smit¹,

Lenstra²

1997

Modular Forms and Fermat’s Last Theorem

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Abstract. 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 O-algebras A with residue field k the representations of G over A that reduce to V over k are given by O-algebra 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, End k (V )) has finite dimension over k. IntroductionLet G be a profinite group and let k be a field. By a k-representation of G we mean a finite dimensional vector space over k with the discrete topology, equipped with a continuous k-linear action of G. If V is a k-representation of G and A is a complete local ring with residue field k, then a deformation of V in A is an isomorphism class of continuous representations of G over A that reduce to V modulo the maximal ideal of A; precise definitions are given in Section 2. We denote by Def(V, A) the set of such deformations.Let V be an absolutely irreducible k-representation of G. The object of this chapter is to give a straight-forward construction of a ring R, the universal deformation ring, which represents the functor Def(V, −). In a purely algebraic setting, without considerations of continuity, a similar construction was already given by Procesi in the seventies [9, Chap. IV, Lemma 1.7; 10]. The existence of R in the present context was deduced first by Mazur [8] with Schlessinger's criteria for pro-representability [12]. An alternative construction was given recently by Faltings (see [5] and Section 7 below).The main result of this chapter, formulated below as Theorem (2.3), is actually a little more general than Mazur's. Following Schlessinger, Mazur works only with noetherian rings, and this forces him to assume at the outset that a certain cohomology group is finite. For our argument, the noetherian condition is a hindrance, and we find it more convenient to follow Grothendieck [6] and work with not necessarily noetherian rings that are projective limits of artinian rings. This allows us to drop Mazur's cohomological condition; it reappears only at the end, as a necessary and sufficient condition for R to be noetherian.

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confidence: 83%

Section: Introductionmentioning

confidence: 90%

Explicit Construction of Universal Deformation Rings

Smit¹,

Lenstra²

1997

Modular Forms and Fermat’s Last Theorem

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“…v to the annihilator of L v under the perfect pairing given by local Tate duality (see Theorem 2.17 of [16] for instance)…”

Section: 1mentioning

confidence: 99%

Serre’s modularity conjecture (II)

2009

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“…we will denote by ρ f : G Q → GL 2 (E) the Galois representation attached to f by Deligne (cf. e.g., [11], section 3.1). We will write ρ f for the reduction of ρ f modulo a uniformizer of E with respect to some lattice Λ in E 2 .…”

Section: This Implies Thatmentioning

confidence: 99%

Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

Klosin¹

2009

Ann. inst. Fourier

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Let k be a positive integer divisible by 4, ℓ > k a prime, and f an elliptic cuspidal eigenform of weight k−1, level 4, and non-trivial character. Let ρ f be the ℓ-adic Galois representation attached to f . In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρ f in the following way. Let L(Symm 2 f, s) denote the symmetric square L-function of f . We prove that (under certain conditions) ord ℓ (L alg (Symm 2 f, k)) ≤ ord ℓ (#S), where S is the (Pontryagin dual of the) Selmer group attached to the Galois module ad 0 ρ f | G K (−1), and K = Q( √ −1). Our method uses an idea of Ribet [33] in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U(2, 2).

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Fermat’s Last Theorem

Cited by 126 publications

References 67 publications

Explicit Construction of Universal Deformation Rings

Explicit Construction of Universal Deformation Rings

Serre’s modularity conjecture (II)

Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

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