modularity lifting theorems

modularity lifting theorems
Name	Modularity lifting theorems
Area	Number theory
Introduced	1986
Contributors	Andrew Wiles, Richard Taylor, Jean-Pierre Serre, Jean-Marc Fontaine

modularity lifting theorems

Modularity lifting theorems are results in Number theory that connect Galois representations arising from arithmetic geometry with automorphic forms such as Modular forms and Hilbert modular forms. They provide criteria under which a p-adic or l-adic representation of the absolute Galois group of a number field is shown to come from a modular or automorphic object, bridging ideas from Andrew Wiles, Richard Taylor, Jean-Pierre Serre, Ken Ribet, and Barry Mazur.

Introduction

Modularity lifting theorems assert that a Galois representation which is congruent mod p to a representation known to be modular (or automorphic) is itself modular, given local and global hypotheses involving deformation theory and Selmer groups; this framework builds on the work of Andrew Wiles, Richard Taylor, Florian Herzig, Tom Weston, and Mark Kisin. The results unify methods from Iwasawa theory, Hida theory, Fontaine–Mazur conjecture, Taniyama–Shimura conjecture, and the theory of Hecke algebras and have been central to major achievements such as the proof of Fermat's Last Theorem and progress toward the Langlands program.

Statement of Main Theorems

Typical modularity lifting formulations start with a continuous irreducible representation rho: Gal(Fbar/F) -> GL_n(E) for a number field F and p-adic field E, together with a residual representation rhobar: Gal(Fbar/F) -> GL_n(k). Under hypotheses such as: - rhobar is modular coming from a cuspidal eigenform as in results of Ken Ribet and Jean-Pierre Serre, - local conditions at primes dividing p modeled on the classifications of Jean-Marc Fontaine and Gerd Faltings, - minimal ramification or prescribed level lowering conditions inspired by Edixhoven and Serre's conjectures, one obtains that rho is modular (or automorphic) in the sense of arising from a Newform, a Hilbert modular form, or a Cuspidal representation of GL_n(A_F). Major variants include the Wiles–Taylor theorem for weight two forms and later generalizations by Fred Diamond, Brian Conrad, Mark Kisin, and Toby Gee to higher weight, higher dimension, and potential automorphy statements related to Clozel and Harris.

Historical Development and Key Contributors

The subject traces from conjectures of Yutaka Taniyama and Goro Shimura formalized by Taniyama–Shimura conjecture and the influential modularity criterion proven by Andrew Wiles with Richard Taylor to settle Fermat's Last Theorem. Foundational algebraic deformation techniques were developed by Barry Mazur and refined by Ken Ribet, Ribet's level lowering theorem contributors such as Ken Ribet and Serre; subsequent expansion to p-adic Hodge theoretic local conditions used innovations by Jean-Marc Fontaine, Michael Harris, Richard Taylor, Mark Kisin, and Clozel–Harris–Taylor. Ongoing extensions to GL_n and Potential automorphy owe much to Taylor, Clozel, Harris, Shepherd-Barron, and Barnet-Lamb progeny including Toby Gee and Christophe Breuil.

Techniques and Proof Strategies

Proofs combine deformation theory of Galois representations as in Barry Mazur with comparison of deformation rings and Hecke algebras (R = T) developed in the work of Andrew Wiles and Richard Taylor. Key inputs use p-adic Hodge theory results of Jean-Marc Fontaine and Gerd Faltings to control local deformation rings, while control of global Selmer groups uses ideas from Iwasawa theory and Mazur control theorem analogues. Patchings and Taylor–Wiles systems introduced by Andrew Wiles and Richard Taylor are supplemented by the patching method of Calegari–Geraghty and integral p-adic automorphic lifting techniques by Mark Kisin, with modularity lifting often relying on properties of Hecke algebra actions on completed cohomology developed by Matthew Emerton.

Applications and Consequences

Modularity lifting theorems yielded the proof of Fermat's Last Theorem via the modularity of semistable elliptic curves, and they underpin progress on the Langlands reciprocity conjectures for GL_2 and higher-rank groups, impacting results in Arithmetic geometry, Diophantine equations, and the classification of torsion in elliptic curves over number fields studied by Mazur and Merel. They enable potential automorphy results used to prove cases of the Sato–Tate conjecture for elliptic curves by Richard Taylor and collaborators, and they contribute to proofs of modularity for Galois representations arising from Shimura variety cohomology in work of Harris, Clozel, and Taylor.

Examples and Notable Cases

Notable instances include the original Wiles–Taylor application to semistable elliptic curves over Q culminating in the proof of the semistable case of the Taniyama–Shimura conjecture, level-lowering applications due to Ken Ribet that linked Fermat to modular forms, and later modularity for elliptic curves over real quadratic fields proven by Freitas–Le Hung–Siksek. Further milestones are potential automorphy for compatible systems of Galois representations proved by Clozel–Harris–Taylor and modularity lifting for GL_n systems advanced by Barnet-Lamb, Geraghty, Harris, and Taylor.

Category:Number theory