Modular Elliptic Curves and Fermat′s Last Theorem(抜粋) (フェルマ-予想がついに解けた!?)

By A. Wiles. 1995

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In Pure Mathematics

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Adaptive Computation Time for Recurrent Neural Networks (2016, background): “Pierre de Fermat was able to write in a margin the conjecture (if not the proof) of a theorem that took three and a half centuries and reams of mathematics to solve [35].”
Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation (2004, methodology): “by the work of Wiles and others [60], [58], [11], and finally apply Ribet’s levellowering Theorem [44].”
Higher-Dimensional Modular Calabi–Yau Manifolds (2007, background): “As a consequence of Wiles’ proof of the Taniyama–Shimura–Weil conjecture [20] there has been considerable interest in the modularity of Calabi–Yau manifolds in recent years.”
A torsion Jacquet--Langlands correspondence (2012, background): “Associated to ρ, we can, in the usual way (see [29, 55, 86]) define a deformation problem that associates to any local Artinian ring (A,m, k) deformations ρ of ρ such that: (1) det(ρ) = · φ̃, where φ is the Teichmuller lift of φ.”
Explicit Chabauty over number fields (2010, background): “The Generalized Fermat Conjecture has been established for many signatures (p, q, r), including for several infinite families of signatures: Fermat’s Last Theorem (p, p, p) by Wiles and Taylor [60], [58]; (p, p, 2) and (p, p, 3) by Darmon and Merel [25]; (2, 4, p) by Ellenberg [27] and Bennett, Elle”
Generalized Fermat equations: A miscellany (2015, methodology): “(p, q, r) reference(s) (n, n, n), n ≥ 3 Wiles [68], Taylor-Wiles [66] (n, n, 2), n ≥ 4 Darmon-Merel [33], Poonen [57] (n, n, 3), n ≥ 3 Darmon-Merel [33], Poonen [57] (2n, 2n, 5), n ≥ 2 Bennett [1] (2, 4, n), n ≥ 4 Ellenberg [39], Bennett-Ellenberg-Ng [5], Bruin [15] (2, 6, n), n ≥ 3 Bennett-Chen [2]”
Faltings heights of CM cycles and derivatives of L-functions (2008, methodology): “By means of the works of Wiles [43] and Breuil et al.”
On the p-adic L-function of a modular form at a supersingular prime (2003, background): “Since E is modular (see [29], [4]), we define the p-adic L-function of E to be the p-adic L-function of the corresponding modular form.”