automorphic representations

automorphic representations
Name	Automorphic representations
Field	Representation theory, Number theory, Harmonic analysis
Notable people	Robert Langlands, Atle Selberg, Harish-Chandra, André Weil, David Kazhdan
Related concepts	L-function, Modular form, Adèle, Reductive group, Local field

automorphic representations

Automorphic representations provide a unifying language connecting Representation theory, Number theory, Harmonic analysis, Algebraic geometry, and Lie groups through the study of symmetries of spaces associated to adèles, Reductive group actions, and spectral decompositions on quotients like modular curve and Shimura variety. They formalize objects such as Modular form, Maass form, and Hecke character as irreducible constituents of representation spaces for groups like GL_n over Local fields and global Adèle groups, enabling deep links between automorphic spectra, L-functions, and arithmetic invariants arising in conjectures of Robert Langlands and results by Atle Selberg, Harish-Chandra, and André Weil.

Introduction

Automorphic representations arise from the action of an Adèle group of a Reductive group on spaces of functions on quotient spaces such as Gamma\G for arithmetic subgroups like SL_2(Z), admitting decompositions into irreducible unitary representations studied by Harish-Chandra, Atle Selberg, I. M. Gelfand, and H. P. Garland. They generalize classical notions including Modular form, Theta function, and Hecke eigenform and serve as the input for constructions of L-functions central to conjectures by Robert Langlands and proofs by teams led by Andrew Wiles, Richard Taylor, and Michael Harris.

Basic Definitions and Examples

An automorphic representation of a Reductive group G over a global field is an irreducible constituent of the right-regular representation on L^2 of G(A)/G(K) with respect to an arithmetic subgroup like GL_2(Z), often factorizable as a restricted tensor product of local components at places associated to Local fields such as p-adic number fields and Real numbers; classical instances include representations attached to Modular forms for SL_2(Z), Eisenstein series studied by Atle Selberg and Hecke operator eigenfunctions appearing in the work of Hecke, Maass, and Harish-Chandra. Important examples are cuspidal automorphic representations linked to Cusp forms, principal series representations arising from parabolic induction studied by J. A. Shalika and I. N. Bernstein, and representations attached to Hecke characters in the work of André Weil and Emil Artin.

Local and Global Theory

Local components live in categories of smooth representations of groups over Local fields, with classification results for GL_n by Jacquet–Langlands and the local Langlands conjecture proved for GL_n by Michael Harris, Richard Taylor, and Michael Harris–Taylor methods; these interact with global automorphic representations via the restricted tensor product and local factors of L-functions studied by Godement–Jacquet and Tate. The trace formula, developed by Atle Selberg and extended by James Arthur, links spectral decomposition of automorphic representations to orbital integrals and has been pivotal in comparing distributions for Endoscopy and stabilizing comparisons used by Jean-Loup Waldspurger and Robert Langlands in proving cases of functoriality.

Langlands Correspondence and Functoriality

The Langlands program posits correspondences between automorphic representations of Reductive groups and n-dimensional representations of global Galois groups or hypothetical Langlands dual group-valued parameters; instances include the modularity theorem for elliptic curves proved by Andrew Wiles, Richard Taylor, and Breuil–Conrad–Diamond–Taylor and reciprocity statements by Langlands–Tunnell theorem and Taylor–Wiles method. Functoriality predicts transfer of automorphic representations along homomorphisms of Langlands dual groups, with foundational cases established by Jacquet, Shalika, and results on base change by Langlands and Arthur–Clozel; recent progress involves techniques from Trace formula comparisons, endoscopic classification by Arthur, and potential automorphy strategies used by Clozel, Harris, and Taylor.

Applications in Number Theory and Arithmetic Geometry

Automorphic representations underpin proofs and conjectures connecting Elliptic curve arithmetic, special values of L-functions, and rationality results for motives appearing in works by Andrew Wiles, Richard Taylor, Freeman Dyson (contextual influence), Shinichi Mochizuki (related conjectures), and Pierre Deligne; the Bloch–Kato conjecture and Birch and Swinnerton-Dyer conjecture link automorphic L-values to arithmetic invariants of Elliptic curves and Motivic cohomology while modularity lifting techniques inform proofs about Fermat's Last Theorem and modularity of Galois representations by Wiles and Taylor. Automorphic methods also yield equidistribution results like the Sato–Tate conjecture proved in many cases by Barnet-Lamb, Harris, Taylor, and collaborators, and explicate arithmetic of Shimura varietys via the cohomology of locally symmetric spaces studied by Kottwitz and Harris–Taylor.

Key Constructions and Techniques

Central constructions include Eisenstein series developed by Atle Selberg and Langlands, the Arthur–Selberg trace formula, Langlands–Shahidi method for L-functions by Frederick Shahidi, converse theorems by Jacquet–Piatetski-Shapiro–Shalika, and theta correspondence (Howe duality) by Roger Howe connecting automorphic representations of dual reductive pairs; cohomological methods involve the study of automorphic forms in the cohomology of Shimura varietys and arithmetic applications via Kisin and Faltings-style comparisons. Newer techniques employ p-adic Langlands programs and patching methods by Calegari, Geraghty, and Taylor to construct and relate refinements of automorphic representations to p-adic Galois representations.

Historical Development and Major Results

The subject evolved from classical studies of modular forms by Hecke, spectral theory of automorphic forms by Atle Selberg and harmonic analysis by Harish-Chandra, through formulation of the Langlands conjectures by Robert Langlands, to deep arithmetic applications including the proof of the Taniyama–Shimura–Weil conjecture (modularity theorem) by Andrew Wiles and Richard Taylor and classification results by James Arthur. Landmark achievements include local and global Langlands correspondences for GL_n by Harris–Taylor and Henniart, endoscopic classification by Arthur, and modularity lifting and potential automorphy results by Clozel, Harris, Taylor, Kisin, and collaborators, which together transformed modern Number theory and Arithmetic geometry.

Category:Representation theory