On an exact mass formula of Shimura
type: Item
instance of: scholarly article
title: On an exact mass formula of Shimura
zbMATH Open document ID: 1023.11019
DOI: 10.1215/S0012-7094-01-10716-3
author: Jonathan Hanke
author: Jiu-Kang Yu
author: Wee Teck Gan
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: In a recent series of papers, G. Shimura obtained an exact formula for the mass of a maximal lattice in a quadratic or Hermitian space over a totally real number field \(k\). The mass formula was obtained by Shimura as a consequence of his theory of Euler products and Eisenstein series for the corresponding orthogonal or unitary group. On the other hand, if \(G\) is a connected reductive group over \(k\) which is anisotropic at all Archimedean places and quasi split at all finite places, an explicit mass formula was obtained by \textit{B. H. Gross} and \textit{W. T. Gan} [Trans. Am. Math. Soc. 351, 1691-1704 (1999; Zbl 0991.20033)]. This is an extension of a fundamental result of \textit{G. Prasad} [Publ. Math., Inst. Hautes Étud. Sci. 69, 91-117 (1989; Zbl 0695.22005)]. The aim of this paper is to rederive Shimura's formula from the general mass formula using Bruhat-Tits theory at least in the cases where the quadratic or Hermitian form is totally definite. This derivation is based on the observation that the stabilizer of a maximal lattice at a finite place is always a maximal parahoric subgroup of the orthogonal or unitary group.
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11E57
Mathematics Subject Classification ID: 11E41
Mathematics Subject Classification ID: 20G35
zbMATH DE Number: 1820796
zbMATH Keywords: maximal lattice
zbMATH Keywords: Hermitian space
zbMATH Keywords: mass formula
zbMATH Keywords: reductive group
zbMATH Keywords: Shimura's formula
zbMATH Keywords: Bruhat-Tits theory
MaRDI profile type: Publication
cites work: Reductive groups on a local field. II. Groups schemes. Existence of valuated root datum
cites work: Schémas en groupes et immeubles des groupes classiques sur un corps local. II : groupes unitaires
cites work: On the motive of a reductive group
cites work: Haar Measure and the Artin Conductor
cites work: Tamagawa numbers
cites work: Unrefined minimal \(K\)-types for \(p\)-adic groups
cites work: Jacquet functors and unrefined minimal \(K\)-types
cites work: Volumes of S-arithmetic quotients of semi-simple groups. With an appendix by Moshe Jarden and Gopal Prasad
cites work: Q4346717
cites work: An exact mass formula for orthogonal groups.
cites work: Some exact formulas on quaternion unitary groups
cites work: Q3050630
cites work: Construction of tame supercuspidal representations
full work available at URL: s0012-7094-01-10716-3
OpenAlex ID: W2006403544
description: scientific article; zbMATH DE number 1820796