Rigid local systems, Hilbert modular forms, and Fermat's last theorem
type: Item
instance of: scholarly article
title: Rigid local systems, Hilbert modular forms, and Fermat's last theorem
zbMATH Open document ID: 1008.11023
DOI: 10.1215/S0012-7094-00-10233-5
author: Henri Darmon
published in: Duke Mathematical Journal
publication date: 2002-07-07T00:00:00Z
review text: This paper gives a very nice overview of a strategy for proving the non-existence of nontrivial primitive integral solutions to the generalized Fermat equation \(x^p + y^q = z^r\) for certain exponent triples \((p,q,r)\), along the lines of the proof of Fermat's last theorem (FLT). The overall strategy is explained in the introduction. 1. To a putative solution \((a,b,c)\), associate a 2-dimensional mod~\(p\) Galois representation \(\rho\) over a suitable totally real number field \(K\) with ramification bounded independently of \((a,b,c)\). In the FLT case, this is provided by the \(p\)-torsion of the associated Frey elliptic curve. 2. Show that \(\rho\) is modular, i.e., it comes from a Hilbert modular form over \(K\). This was done by Wiles in the FLT case. 3. Deduce that \(\rho\) comes from a modular form of small level, and thence that its image is small (in favorable cases). This is Ribet's level-lowering result. 4. Derive a contradiction by showing that the image of \(\rho\) has to be large. In the FLT case, this is provided by Mazur's results on rational isogenies of elliptic curves. These steps are then discussed in detail in the following sections. The first of these shows which Galois representations one can choose and identifies them as coming from torsion points on certain abelian varieties. The next section discusses modularity. Modularity of the relevant Galois representations would follow from the general modularity conjecture for abelian varieties of GL\(_2\)-type. In certain cases, the abelian varieties are elliptic curves, and so their modularity is known, see \textit{C. Breuil, B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 14, 843-939 (2001; Zbl 0982.11033)]. This leads to a proof that no nontrivial primitive solutions exist for the equations \(x^p + y^p = z^2\) (\(p \geq 4\)) and \(x^p + y^p = z^3\) (\(p \geq 3\)), see \textit{H. Darmon} [Int. Math. Res. Not. 1993, 263-274 (1993; Zbl 0805.11028)] and \textit{H. Darmon} and \te…
reviewed by: Michael Stoll
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1455496
zbMATH Keywords: generalized Fermat equation
zbMATH Keywords: Galois representations
zbMATH Keywords: modular forms
zbMATH Keywords: modular abelian varieties
MaRDI profile type: Publication
cites work: On extensions of number fields obtained by specializing branched coverings.
cites work: On extensions of the maximal cyclotomic field having a given classical Galois group.
cites work: Motives for Hilbert modular forms
cites work: Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert
cites work: Modular embeddings for some non-arithmetic Fuchsian groups
cites work: Modularity of certain potentially Barsotti-Tate Galois representations
cites work: Q3684278
cites work: Q4023809
cites work: Q4289016
cites work: Q4369731
cites work: Q4335074
cites work: Q4366731
cites work: Courbes hyperelliptiques à multiplications réelles et une construction de Shih
cites work: On deformation rings and Hecke rings
cites work: The Taylor-Wiles construction and multiplicity one
cites work: Q3997225
cites work: Q4206859
cites work: Die eindeutige Bestimmung des Modulfunktionen \(q\)-ter Stufe durch algebraische Eigenschaften
cites work: Endliche Gruppen I
cites work: On Galois representations associated to Hilbert modular forms.
cites work: Exponential Sums and Differential Equations. (AM-124)
cites work: Modular curves and the Eisenstein ideal
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Q3972632
cites work: Q3693621
cites work: Some diophantine equations of the form $x^n + y^n = z^m$
cites work: On the levels of mod \(l\) Hilbert modular forms
cites work: A modular construction of unramified p-extensions of \(\mathbb{Q}(\mu_p)\)
cites work: On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
cites work: Galois properties of points of finite order of elliptic curves
cites work: On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\)
cites work: Q4003835
cites work: Ordinary representations and modular forms
cites work: Nearly ordinary deformations of irreducible residual representations
cites work: Residually reducible representations and modular forms
cites work: Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
cites work: On Galois representations associated to Hilbert modular forms
cites work: Ring-theoretic properties of certain Hecke algebras
cites work: Arithmétique des algèbres de quaternions
cites work: The arithmetic genus of the Hilbert modular variety and the elliptic fixed points of the Hilbert modular group
cites work: On \(p\)-adic representations for totally real fields
cites work: On ordinary \(\lambda\)-adic representations associated to modular forms
cites work: Modular elliptic curves and Fermat's Last Theorem
description: scientific article; zbMATH DE number 1455496