On the Euler characteristic of the discrete spectrum
type: Item
instance of: scholarly article
title: On the Euler characteristic of the discrete spectrum
zbMATH Open document ID: 1080.11041
author: David Pollack
author: Benedict H. Gross
published in: Journal of Number Theory
publication date: 2005-03-08T00:00:00Z
review text: Let \(G\) be a simply connected, semi-simple algebraic group defined over \(\mathbb Q\). Let \(\mathbb A\) be the ring of adeles of \(\mathbb Q\). Let \(dg\) be a fixed Haar measure on \(G(\mathbb A)\). The group \(G(\mathbb A)\) acts unitarily, by right translation, on the Hilbert space \[ L^2=L^2(G(\mathbb Q)G(\mathbb A),dg). \] Let \[ L=L^2_{\text{disc}}\subset L^2 \] be the sum of all irreducible \(G(\mathbb A)\)-subspaces of \(L^2\). \(L\) is called the discrete spectrum of \(G(\mathbb A)\). If \(S\) is a finite set of places of \(\mathbb Q\) which contains the real place and all finite primes \(p\) where \(G\) is ramified, there exists an integral model \(\underline G\) for \(G\) over the ring \({\mathbb Z}_S\) of \(S\)-integers, with \({\underline G}\) having good reduction at all primes \(p\) outside of \(S\). Fix such a finite set \(S\). The product \[ G_S(\mathbb A)=\prod_{v\in S}G({\mathbb Q}_v)\times\prod_{p\not\in S}{\underline G}({\mathbb Z}_p) \] is locally compact, open in \(G(\mathbb A)\). Let \(V\) be an irreducible, finite-dimensional representation of the real Lie group \(G(\mathbb R)\), such that \(V\) has trivial central character. The tensor product \(L\otimes V\) is a continuous, complex representation of the group \(G_S(\mathbb A)\). The continuous cohomology groups \[ H^i(G_S(\mathbb A), L\otimes V) \] are finite dimensional complex vector spaces, zero for \(i\gg 0\). Let \[ \chi=\chi(G, S, V)=\sum_{i\geq 0}\dim H^i(G_S(\mathbb A), L\otimes V) \] be the Euler characteristic of \(L\otimes V\). The goal of the authors is to give an explicit formula for \(\chi\) under the hypotheses that \(S\) contains a finite prime and \(G(\mathbb R)\) contains a maximal compact torus. This Euler characteristic is first expressed as a trace of a certain test function on the space of automorphic forms, and then, by the stable trace formula, is converted into a sum of orbital integrals, which are calculated in terms of the values of certain Artin \(L\)-functio…
reviewed by: Florin Nicolae
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E41
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 2142241
zbMATH Keywords: automorphic form
zbMATH Keywords: trace formula
zbMATH Keywords: orbital integrals
zbMATH Keywords: Artin L-function
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.03.008
OpenAlex ID: W2005444229
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DOI: 10.1016/J.JNT.2004.03.008
description: scientific article; zbMATH DE number 2142241