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369
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 617 (2 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.
On the modularity of elliptic curves over Q: Wild 3-adic 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 345 (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 95 (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 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin [19].
Modularity of certain potentially Barsotti-Tate 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 ℓ-adic Tate ..."
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Cited by 82 (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 ℓ-adic Tate
Elliptic and modular curves over finite fields and related computational issues. Computational perspectives on number theory
- AMS/IP Stud. Adv. Math
, 1995
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Another Look at “Provable Security"
, 2004
"... We give an informal analysis and critique of several typical “provable security” results. In some cases there are intuitive but convincing arguments for rejecting the conclusions suggested by the formal terminology and “proofs,” whereas in other cases the formalism seems to be consistent with common ..."
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Cited by 73 (13 self)
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We give an informal analysis and critique of several typical “provable security” results. In some cases there are intuitive but convincing arguments for rejecting the conclusions suggested by the formal terminology and “proofs,” whereas in other cases the formalism seems to be consistent with common sense. We discuss the reasons why the search for mathematically convincing theoretical evidence to support the security of public-key systems has been an important theme of researchers. But we argue that the theorem-proof paradigm of theoretical mathematics is often of limited relevance here and frequently leads to papers that are confusing and misleading. Because our paper is aimed at the general mathematical public, it is self-contained and as jargon-free as possible.
Mahler's Measure and Special Values of L-functions
, 1998
"... this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula, ..."
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Cited by 70 (1 self)
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this paper is to describe an attempt to understand and generalize a recent formula of Deninger [1997] by means of systematic numerical experiment. This conjectural formula,
Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
- Annals of Math
"... Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general ..."
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Cited by 63 (15 self)
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Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue-Nagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 1.
