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| Arthur trace formula | |
|---|---|
| Name | Arthur trace formula |
| Field | Number theory; Representation theory; Automorphic forms |
| Introduced | 1970s–1980s |
| Named after | James Arthur |
Arthur trace formula
The Arthur trace formula is a deep analytic and algebraic identity linking harmonic analysis on adelic groups and arithmetic invariants of reductive groups, developed to study automorphic representations and L-functions. It refines earlier trace formulas by Selberg trace formula techniques and provides a framework for comparing spectral data with orbital integrals across adelic and local field settings. The formula underpins major progress in the Langlands program, automorphic representations, and the classification of discrete spectra for reductive groups.
Arthur developed the trace formula as an extension of methods used by Atle Selberg and tools from the theory of Harish-Chandra characters, synthesizing concepts from Harmonic analysis on Lie groups, adelic groups, p-adic groups, and the theory of Eisenstein series. The approach builds on the structure theory of reductive algebraic groups such as GL(n), SL(2), SO(n), Sp(2n), and their Levi subgroups, exploiting the geometry of Bruhat–Tits buildings and conjugacy classes in local fields. The trace formula has both a global form over number fields and a local form over local fields, and it interacts with the study of automorphic L-functions, Arthur parameters, and the Langlands dual group.
The genesis of the trace formula lies in work of Atle Selberg on the Selberg trace formula for Riemann surfaces and later extensions by Harish-Chandra for noncompact groups. James Arthur synthesized these strands while addressing problems posed by the Langlands conjectures and the classification of automorphic discrete spectra. Influential contributors and contexts include I. M. Gelfand's representation theory, Robert Langlands's conjectural reciprocity, and advances in the theory of Eisenstein series by Jean-Pierre Serre and Harish-Chandra. The formula responded to needs arising from the study of functoriality, endoscopy, and the stabilization problems formulated in workshops and research programs at institutions such as Institute for Advanced Study, Princeton University, and University of Cambridge.
Arthur's invariant trace formula equates two distributions: a spectral distribution assembled from traces of operators on spaces of automorphic forms and a geometric distribution built from weighted orbital integrals over conjugacy classes in a reductive group G over a number field. The spectral side involves sums over discrete automorphic representations, continuous spectra via Eisenstein series, and contributions indexed by Arthur packets and parameters associated to the Langlands group. The geometric side organizes contributions by conjugacy classes, involving weighted orbital integrals for semisimple, unipotent, and mixed classes, and sums over Levi subgroupes and parabolic subgroups. Precise formulations require choices of test functions in Hecke algebras, truncation operators in the style of Arthur truncation, and measures on adele ring quotients.
The spectral side decomposes according to automorphic discrete spectrum components attached to cuspidal representations, residual spectra arising from poles of Eisenstein series, and continuous integrals parameterized by unitary dual data. It features Plancherel measure terms and multiplicities for representations of groups like GL(n), GSp(2n), and SO(n,n). The geometric side is a sum of weighted orbital integrals, with contributions from elliptic, hyperbolic, and unipotent conjugacy classes; it invokes the theory of Shalika germs, Kottwitz's work on rational conjugacy, and explicit matching of orbital integrals across endoscopic groups. The comparison requires delicate convergence arguments and the use of invariant distributions developed by Arthur.
Stabilization transforms the trace formula into a form amenable to comparison between different groups by isolating stable distributions and transferring orbital integrals via endoscopic transfer. This program was motivated by Langlands' functoriality conjectures and realized through contributions by Robert Kottwitz, R. P. Langlands (Robert Langlands), and Arthur, leading to the notion of stable trace formula and classification results linking automorphic representations to Arthur parameters and endoscopic data. Stabilization underlies the proof of endoscopic classification for classical groups, enabling comparison between spectra of groups such as SO(2n+1), Sp(2n), and U(n) and those of GL(n), via packet constructions and trace identities.
Arthur's trace formula has yielded major consequences: the endoscopic classification of automorphic representations for classical groups, proofs of instances of functoriality between classical groups and GL(n), and results on the analytic properties of automorphic L-functions including nonvanishing and pole structure. It has been instrumental in progress on the Ramanujan conjecture for various families, the study of cohomology of locally symmetric spaces and Shimura varietys, and arithmetic applications such as counting rational points and trace computations for Hecke operators on modular forms and Siegel modular forms. Work building on the trace formula interacts with results of Wiles, Taylor, Harris, Clozel, and Blasius in automorphy lifting and reciprocity contexts.
Classical instances include Selberg's formula for SL(2,R) and trace computations for congruence subgroups of SL(2,Z), trace formulas for GL(2) and the Jacquet–Langlands correspondence between GL(2), quaternionic forms, and applications to the spectrum of the Laplacian on arithmetic surfaces. For higher-rank groups, Arthur's formula has been applied to GL(n), yielding insights into discrete series and residual spectra, and to classical groups such as Sp(2n), SO(n), and unitary groups U(n), where endoscopic classification produces explicit packet descriptions. Local versions contribute to the study of representations of p-adic groups and the proof of local Langlands correspondences in many settings.