Lefschetz formulae for arithmetic varieties
type: Item
instance of: scholarly article
title: Lefschetz formulae for arithmetic varieties
zbMATH Open document ID: 0847.58071
DOI: 10.1007/BF01231760
author: Mark Stern
published in: Inventiones Mathematicae
publication date: 1996-02-21T00:00:00Z
full work available at URL: https://eudml.org/doc/144169
review text: The paper under review is a continuation of the author's work on \(L^2\)-cohomology (with \textit{L. Saper} [Ann. Math., II. Ser. 132, No. 1, 1-69 (1990; Zbl 0722.14009)]) and \(L^2\)-index theory [Invent. Math. 96, No. 2, 231-282 (1989; Zbl 0694.58039), J. Differ. Geom. 31, No. 3, 771-789 (1990; Zbl 0715.53039)] for arithmetic varieties. Its purpose is to prove a Lefschetz formula for Hecke operators on, in general, noncompact arithmetic varieties. More precisely, consider an arithmetic variety \(X_\Gamma = \Gamma \setminus G/K\), where the \(\mathbb{Q}\)-irreducible Hermitean symmetric space \(G/K\) is a quotient of the principal component \(G\) of the group of real points of a semisimple algebraic group over \(\mathbb{Q}\) by a maximal compact subgroup \(K\), and \(\Gamma\) is an arithmetic subgroup of \(G\). Given a finite dimensional complex representation \((\rho, E)\) of \(G\), one obtains a flat vector bundle \(\mathcal E\) over \(X_\Gamma\), and a signature operator \(D_\rho = d + d^*\) acting on smooth square integrable \(\mathcal E\)-valued forms. With respect to an involution \(\tau\) on the space of square integrable \(\mathcal E\)-valued forms anticommuting with \(D_\rho\) and associated to the signature complex, the Lefschetz number of a Hecke operator \(T_G (\alpha)\) is defined by \[ L(\alpha, \tau, E) = \text{tr } \tau T_G (\alpha) |_{\text{ker}} D_\rho. \] For \(\alpha\) an element of the commensurator of \(\Gamma\) the Hecke operator \(T_G (\alpha)\) is defined on \(L^2 (\Gamma \setminus G)\), and extended trivially to commute with \(\tau.\) The problem to find a formula for \(L(\alpha, \tau, E)\) has been approached in different ways by \textit{H. Moscovici} [Lect. Notes Math. 1077, 321-258 (1984; Zbl 0561.58046)] for \(X_\Gamma\) compact, and by \textit{J. Arthur} [Invent. Math. 97, No. 2, 257-290 (1989; Zbl 0692.22004)] for \(X_\Gamma\) non-compact and \(\tau\) associated to the Gauss-Bonnet-complex using the trace formula, and by M. Goresky a…
Mathematics Subject Classification ID: 58J20
Mathematics Subject Classification ID: 11F60
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 549845
zbMATH Keywords: \(L^ 2\)-cohomology
zbMATH Keywords: \(L^ 2\)-index theory
zbMATH Keywords: arithmetic varieties
zbMATH Keywords: Lefschetz formula
zbMATH Keywords: Hecke operators
zbMATH Keywords: heat equation method
zbMATH Keywords: eta terms
zbMATH Keywords: zeta terms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549845