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| Langlands functoriality | |
|---|---|
| Name | Langlands functoriality |
| Conjectured by | Robert Langlands |
| Field | Number theory; Representation theory |
| Date | 1960s |
| Status | conjectural (partial results) |
Langlands functoriality is a central conjecture in modern Number theory and Representation theory proposed by Robert Langlands. It predicts deep correspondences between automorphic representations of different reductive groups linked by morphisms of their L-groups, unifying phenomena observed in the work of Erich Hecke, Atle Selberg, and Harish-Chandra. Functoriality connects objects studied by André Weil, John Tate, and Pierre Deligne to the program advanced by Alexandre Grothendieck and later developed in collaboration with institutions such as Institute for Advanced Study and Princeton University.
The conjecture arose in correspondence between Robert Langlands and André Weil and in lectures at Institute for Advanced Study, inspired by the reciprocity of Richard Dedekind and the work of Emil Artin on L-series. It proposes that a homomorphism between L-groups attached to connected reductive groups induces transfers of automorphic representations, mirroring the transfer of Artin representations in the Artin reciprocity law and generalizing the Taniyama–Shimura–Weil conjecture proved in cases by Andrew Wiles and Richard Taylor. Functoriality thus aims to relate cases handled by Jacquet–Langlands correspondence and the Langlands classification to broader contexts involving Galois group representations studied by Sergey Langlands's predecessors.
Informally, given a morphism φ: {}^L H → {}^L G of L-groups for connected reductive groups H and G over a global field (a number field or a function field), functoriality predicts a transfer from automorphic representations π of H(A) to automorphic representations Π of G(A) preserving L- and ε-factors attached by Godement–Jacquet and Jacquet–Langlands. The conjecture refines the Local Langlands correspondence articulated for p-adic fields by Pierre Deligne and Kazhdan and complements global reciprocity envisioned by Emil Artin. Precise formulations use the language of automorphic L-functions, Arthur parameters developed by James Arthur, and compatibility with the Satake isomorphism for unramified representations, shaped by work at Harvard University and IHÉS.
Classical instances include the base change for GL(2) from Hiroshi Saito's and Langlands–Tunnell theorem approaches, symmetric power functoriality for GL(2) established in low degrees by Kim–Shahidi and Henry Kim, and the local and global Jacquet–Langlands correspondence between GL(2), D^× (a quaternion algebra) and modular forms studied by Yves Jacquet and Ramanathan. The proof of the Taniyama–Shimura–Weil conjecture by Andrew Wiles and Richard Taylor realized a functorial transfer from elliptic curves to modular forms, while the Harris–Taylor proof of the local Langlands correspondence for GL(n) used techniques from Étale cohomology and the geometry of Shimura varietys studied by Michael Harris and Richard Taylor. Results by Laurent Lafforgue resolved global Langlands correspondences for function fields and built on ideas from Drinfeld.
Approaches combine analytic, algebraic, and geometric tools: the trace formula of James Arthur and the stable trace formula refined by Robert Kottwitz; converse theorems of Hecke and Piateski-Shapiro; the Langlands–Shahidi method developed by Frederick Shahidi and Freydoon Shahidi; and methods from the theory of perverse sheafs and the geometric Langlands program advanced by Alexander Beilinson and Vladimir Drinfeld. Cohomological techniques from Étale cohomology and modularity lifting theorems due to Taylor–Wiles use deformation theory developed by Barry Mazur and Mazur–Wiles-style patching. Endoscopic transfer analyzed by Robert Langlands and D. Shelstad and stabilization strategies at Institute for Advanced Study remain pivotal.
Functoriality intertwines with the Local Langlands correspondence, Global Langlands correspondence, and Ramanujan–Petersson conjecture for automorphic forms. It is consistent with predictions of Arthur's conjectures on discrete automorphic spectrum and with the geometric Langlands program articulated by Edward Frenkel. Compatibility with the Sato–Tate conjecture shown in works by Richard Taylor and collaborators reflects functorial lifts preserving statistical properties of L-values, while links to Bloch–Kato conjecture connect to special value conjectures formulated by Kazuya Kato.
Consequences include proofs of modularity results for elliptic curves over Q via the Taniyama–Shimura–Weil conjecture, progress on the Sato–Tate conjecture for families of motives addressed by Taylor and Harris, and explicit constructions of automorphic L-functions used in analytic number theory by Henryk Iwaniec and Peter Sarnak. Functoriality informs classification results in Representation theory such as the Langlands classification and influences arithmetic geometry problems considered by Pierre Deligne and Jean-Pierre Serre.
Major open issues include establishing functoriality for general morphisms of L-groups, completing stabilization of the trace formula as pursued by Robert Kottwitz and James Arthur, and proving symmetric power lifts in full generality as sought by Henry Kim. Active research spans the geometric Langlands program at institutions like University of Chicago and Princeton University, modularity lifting innovations by groups led by Richard Taylor and Mark Kisin, and novel analytic approaches from researchers including Frederick Diamond and David Ben-Zvi. Progress frequently appears in collaborations among researchers at Institute for Advanced Study, IHÉS, Université Paris-Saclay, and Perimeter Institute.