potential automorphy

potential automorphy
Name	Potential automorphy
Field	Number theory
Introduced	1990s
Notable people	Richard Taylor, Andrew Wiles, Nicholas Katz, Michael Harris, Mark Kisin, Fabrizio Calegari, Taylor–Wiles method
Related concepts	Langlands program, Modularity theorem, Galois representation, Automorphic representation, Shimura variety

potential automorphy Potential automorphy is a concept in modern number theory asserting that a Galois representation becomes automorphic after restriction to the Galois group of a finite extension. It lies at the intersection of the Langlands program, the Modularity theorem, and the study of Galois representations, and has been pivotal in proofs involving Fermat's Last Theorem, the proof of potential modularity for elliptic curves, and advances by researchers like Andrew Wiles, Richard Taylor, and Michael Harris. The idea enables transfer of arithmetic information via passage to suitable extensions related to CM fields, totally real fields, and Shimura varietys.

Introduction

Potential automorphy emerged as a flexible weakening of automorphy that allows arithmetic objects to be shown automorphic only after base change to an auxiliary extension such as a CM field or a solvable extension. It played a central role in the proof strategies of Wiles, Taylor, and collaborators for modularity lifting, became a standard tool in the work of Harris, Taylor, Kisin, Calegari, and Geraghty on symmetric powers and potential modularity, and interfaces with methods developed in the context of the Langlands correspondence and the study of Shimura variety cohomology.

Definitions and Background

A continuous l-adic Galois representation ρ: Gal( \overline{F}/F ) → GL_n( Q_l ) is called potentially automorphic if there exists a finite extension E/F such that the restriction ρ|_{Gal(\overline{F}/E)} corresponds to an automorphic representation of GL_n over E under the expected reciprocity of the Langlands program. Foundational backgrounds include the Fontaine–Mazur conjecture, the Brauer induction-style reductions used in the Modularity theorem, and the development of deformation theory by Mazur and the Taylor–Wiles method by Andrew Wiles and Richard Taylor. Constructions often use CM field base changes, totally real fields, and automorphic forms on unitary groups studied by Arthur and Clozel.

Key Theorems and Results

Major results include the potential modularity of elliptic curves over totally real fields proved by Taylor and Borcherds techniques extended by Breuil and Conrad-style local analysis, and the potential automorphy theorems of Harris–Taylor and Harris–Shepherd-Barron–Taylor for symmetric powers of modular forms. Kisin produced refinements of modularity lifting applicable to potentially Barsotti–Tate representations, and Calegari with Geraghty developed new patching methods yielding potential automorphy for families of Galois representations. Results by Skinner–Wiles and Wiles underpin many lifting arguments, while contributions from Taylor–Wiles method, Diamond, and Böckle shaped the deformation-theoretic framework. The Fontaine–Mazur conjecture and successes toward it provide motivational context for these theorems.

Methods and Techniques

Techniques include deformation theory of Galois representations as formulated by Mazur, the Taylor–Wiles method and its patching variants developed by Taylor, Wiles, and later generalized by Calegari and Geraghty, base change and level-raising via Langlands–Tunnell-type arguments, and construction of congruences using cohomology of Shimura varietys following work of Harris, Taylor–Harris and Scholze. Local-global compatibility uses Fontaine–Laffaille theory, Breuil–Mezard conjectures, and local deformation rings analyzed by Kisin and Emerton. Solvable base change and cyclic base change techniques derive from classical results of Langlands and Clozel, while automorphic descent and lifting use trace formula input associated with Arthur and Labesse.

Examples and Applications

Classic applications include the proof of modularity for semistable elliptic curves over Q which implied Fermat's Last Theorem via work of Wiles and Taylor–Wiles method, potential automorphy proofs for symmetric powers of modular forms by Harris–Taylor that led to analytic continuation of L-functions studied by Gelbart and Jacquet, and modularity lifting for elliptic curves over totally real fields by Taylor and Skinner–Wiles. Potential automorphy has been applied to special value results influenced by the Bloch–Kato conjecture, to instances of the Fontaine–Mazur conjecture via work of Kisin, and to constructing compatible systems of l-adic representations in the spirit of Serre and Deligne. Studies of cohomology on Shimura varietys by Scholze and Caraiani have provided further explicit realizations.

Open Problems and Conjectures

Important open problems include proving automorphy (without the potential) for broader classes of Galois representations predicted by the full Langlands program and the Fontaine–Mazur conjecture, establishing potential automorphy over arbitrary number fields beyond CM field and totally real field cases, and refining local-global compatibility at places dividing l in more generality as conjectured in work of Buzzard and Gee. Further challenges arise in extending patching and derived deformation methods of Calegari, Geraghty, and Scholze to non-GL_n settings, proving automorphy lifting theorems with minimal local hypotheses as envisioned by Diamond and Kisin, and connecting potential automorphy to explicit arithmetic predictions of Bloch–Kato and Beilinson.

Category:Number theory