Calegari–Geraghty

Calegari–Geraghty
Name	Calegari–Geraghty
Field	Number theory
Authors	Frank Calegari; David Geraghty
Main topics	Modular forms; Galois representations; Automorphy lifting; Taylor–Wiles method; Shimura varieties
First appeared	2018

Calegari–Geraghty

The Calegari–Geraghty result is a framework in number theory and arithmetic geometry that generalizes the Taylor–Wiles method for proving automorphy lifting theorems for Galois representations, particularly in settings with non-minimal level and torsion classes. It was introduced by Frank Calegari and David Geraghty to address obstacles encountered by Richard Taylor and Andrew Wiles in the proofs of modularity and Fermat's Last Theorem, and has influenced work of Peter Scholze, Thomas Barnet-Lamb, Toby Gee, and Mark Kisin on potential automorphy and modularity lifting.

Introduction

The Calegari–Geraghty framework modifies the Taylor–Wiles method by incorporating congruences in torsion cohomology of Shimura varieties and the use of derived deformation rings, enabling new automorphy lifting results for Galois representations of Gal(Qbar/Q). It brings together techniques from modular forms, étale cohomology, Hecke algebras, and the Langlands program to treat cases where the classical minimal patching hypotheses fail, and it has been applied to questions linked to Serre's conjecture, Fontaine–Mazur conjecture, and potential automorphy for odd-dimensional motives.

Statement of the Theorem

Roughly stated, the Calegari–Geraghty theorem asserts that under suitable local-global compatibility hypotheses for a residual Galois representation r̄: Gal(Qbar/Q) → GL_n(F_p), one can deduce automorphy of certain p-adic lifts r: Gal(Qbar/Q) → GL_n(E) by patching completed cohomology groups of arithmetic locally symmetric spaces, provided a suitable "R = T" identification holds in a derived sense. The precise formulation involves: deformation rings representing Mazur-style deformation problems, Hecke algebras acting on torsion in completed cohomology of Shimura varieties or locally symmetric spaces for GL_n, and a "Taylor–Wiles system" of auxiliary primes satisfying level-raising congruences by means of Ihara's lemma analogues and local deformation conditions. Authors require hypotheses such as adequate image (à la Jack Thorne), Fontaine–Laffaille or potentially crystalline local conditions (related to Jean-Marc Fontaine and Gerd Faltings), and existence of congruences with cohomological automorphic forms for groups like GL_n or unitary groups.

Context and Motivation

The need for Calegari–Geraghty arose from limitations in the Taylor–Wiles argument when torsion classes in cohomology obstructed the usual numerical coincidence between deformation rings and Hecke algebras observed in works of Wiles, Diamond, Skinner–Wiles, and Breuil–Conrad–Diamond–Taylor. Research on torsion in the cohomology of arithmetic manifolds by Matt Emerton, Michael Harris, Richard Taylor, and Peter Scholze showed that automorphic forms mod p and completed cohomology carry important information beyond classical cuspidal eigenforms, motivating a framework that treats torsion systematically. The Calegari–Geraghty approach also connects to developments by Kisin on local deformation rings and patching, and to innovations in derived algebraic geometry by Jacob Lurie and Bertrand Toën that inform derived R = T perspectives.

Outline of the Proof

The proof strategy adapts the Taylor–Wiles patching method with critical changes: (1) one constructs spaces of completed cohomology for a tower of levels of a locally symmetric space attached to a reductive group such as a unitary group considered by Michael Harris and Laurent Clozel; (2) one introduces auxiliary Taylor–Wiles primes to kill dual Selmer groups while tracking torsion congruences in cohomology via Hecke operators as in John Coates's perspective; (3) one builds a patched module over a patched deformation ring and a patched Hecke algebra, using homological algebra and derived completions inspired by Pierre Deligne's and Alexander Beilinson's ideas; (4) one proves a derived version of R = T by comparing support and depth, using local-global compatibility results proved by Caraiani, Gee, and Newton, and adequacy hypotheses from Jack Thorne; and (5) one deduces automorphy of lifts by showing the patched module is nonzero on components corresponding to desired deformations. Key technical inputs include potential automorphy theorems of BLGGT (Barnet-Lamb, Gee, Geraghty, Taylor), Ihara-type lemmas, and control of Fontaine–Laffaille or crystalline deformation conditions as in Kisin.

Applications and Consequences

Calegari–Geraghty has been used to prove new instances of modularity and potential automorphy for Galois representations arising from algebraic varieties and motives, impacting results attributable to Scholze, Caraiani–Scholze, and the BLGGT collaboration. It has led to progress on the Fontaine–Mazur conjecture for certain representations, refinements of Serre's conjecture formulations for higher rank groups, and the construction of Galois representations attached to torsion classes in cohomology, which connect to the conjectural categorical Langlands correspondences studied by Edward Frenkel and Dennis Gaitsgory. The framework also motivated advances in the study of eigenvarieties of Buzzard and Coleman–Mazur and informed computational approaches in the work of William Stein.

Related Results and Extensions

Subsequent work has expanded or refined the Calegari–Geraghty ideas: derived R = T results by Frank Calegari with collaborators and by Peter Scholze draw on derived algebraic geometry of Lurie; Jack Thorne and Toby Gee have extended automorphy lifting under adequacy hypotheses; Caraiani and Scholze established powerful local-global compatibility theorems for torsion cohomology; Ana Caraiani's work with Scholze on Kottwitz conjectures interacts with this framework; and recent progress on modularity of abelian varieties and odd-dimensional motives owes to these methods. Ongoing research explores variants for symplectic groups studied by Weissauer and Kret–Shin, and potential interaction with categorical and p-adic Langlands programs pursued by Colmez, Emerton, and Breuil.

Category:Number theory