Stabilization of the twisted trace formula. Vol. 1
type: Item
instance of: scholarly article
title: Stabilization of the twisted trace formula. Vol. 1
zbMATH Open document ID: 1361.11004
DOI: 10.1007/978-3-319-30049-8
author: Colette Moeglin
author: Jean-Loup Waldspurger
published in: Progress in Mathematics
publication date: 2016-09-29T00:00:00Z
review text: The Arthur-Selberg trace formula is an important identity in the spectral geometry of the space \(G(F)\backslash G({\mathbb A})\) for a number field \(F\), where \(\mathbb A\) is the adele ring and \(G\) a reductive linear group over \(F\). It comes about by comparing two ways of computing the trace of an integral operator, spectrally on the one side and geometrically on the other. The trace formula has profound applications in the Langlands theory, where it can be used to show instances of functoriality. The strategy of these proofs is to compare the trace formulae of two different groups, applied to suitable test functions, whose geometric sides of the formula match, so that the ensuing identity of the spectral sides yields the wanted functoriality. In order to match the geometric sides, which are organized by conjugacy classes in the groups, one needs a suitable correspondence between these conjugacy classes. Such a correspondence is much easier obtained if one uses conjugacy over the respective algebraic closures instead of the ground field, i.e., the so called \textit{stable conjugacy classes}. To exploit such a correspondence, one needs a different version of the trace formula, the \textit{stable trace formula}. The stabilization of the trace formula was carried out by \textit{J. Arthur} in three groundbreaking papers in 2001--2003 [Invent. Math. 143, No. 1, 157--220 (2001; Zbl 0978.11025); J. Inst. Math. Jussieu 1, No. 2, 175--277 (2002; Zbl 1040.11038); Ann. Math. (2) 158, No. 3, 769--873 (2003; Zbl 1051.11027)], at the time conditionally based on the fundamental lemma, which in the meantime has been proven \textit{G. Laumon} and \textit{B. C. Ngô} [Ann. Math. (2) 168, No. 2, 477--573 (2008; Zbl 1179.22019)] in the unitary case and in general by \textit{B. C. Ngô} [Publ. Math., Inst. Hautes Étud. Sci. 111, 1--271 (2010; Zbl 1200.22011)]. Langlands functoriality in its full version also incorporates more general assertions than those provable with the stable …
reviewed by: Anton Deitmar
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6630983
zbMATH Keywords: trace formula
zbMATH Keywords: twisted trace formula
zbMATH Keywords: stable trace formula
zbMATH Keywords: Langlands functoriality
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-30049-8
OpenAlex ID: W2556709788
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description: scientific article; zbMATH DE number 6630983