Group schemes and local densities.
type: Item
instance of: scholarly article
title: Group schemes and local densities.
zbMATH Open document ID: 1048.11028
DOI: 10.1215/S0012-7094-00-10535-2
author: Jiu-Kang Yu
author: Wee Teck Gan
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: Let \(L\) be a lattice over \(\mathbb{Z}\) equipped with an integral quadratic form \(Q\). The celebrated Smith-Minkowski-Siegel mass formula expresses the total mass of \((L,Q)\), which is a weighted class number of the genus of \((L,Q)\), as a product of local factors. These local factors are known as the local densities of \((L,Q)\). Subsequent work of Kneser, Tamagawa and Weil resulted in an elegant formulation of the subject in terms of Tamagawa measures. In particular, the local density at a non-Archimedean place \(p\) can be expressed as the integral of a certain volume form \(w^{\text{Id}}\) over \(\Aut_{\mathbb{Z}_p}(L,Q)\), which is an open compact subgroup of \(\Aut_{\mathbb{Q}_p} (L,Q)\). An explicit formula for the local density for an arbitrary lattice over \(\mathbb{Z}_p\) is known through the work of Pall (for \(p\neq 2)\) and Watson (for \(p=2)\). Unfortunately, the known proof involves complicated recursions. On the other hand, Conway and Sloane have given a heuristic explanation of the formula. In this paper, a simple and conceptual proof of the local density fomula for \(p\neq 2\) is given. The proof is based on the observation that there exists a smooth affine group scheme \({\mathcal G}\) over \(\mathbb{Z}_p\) with generic fibre \(\Aut_{\mathbb{Q}_p} (L,Q)\) which satisfies \({\mathcal G}(\mathbb{Z}_p)= \Aut_{\mathbb{Z}_p}(L,Q)\). The main contribution of this paper is to give an explicit construction of \({\mathcal G}\) and to determine its special fibre. Finally, by comparing \(w^{\text{Id}}\) and the canonical volume form \(w^{\text{can}}\) of \({\mathcal G}\), the explicit formula for local density is obtained. As a consequence of the mass formula and the result on local densities, an explicit formula for the mass of an arbitrary lattice in a quaternionic Hermitian space is obtained which extends the result of Shimura for the mass of maximal lattice.
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 1820769
zbMATH Keywords: local densities
zbMATH Keywords: mass formula
zbMATH Keywords: Tamagawa measure
zbMATH Keywords: lattice
zbMATH Keywords: integral quadratic form
zbMATH Keywords: quaternionic Hermitian space
MaRDI profile type: Publication
cites work: Seminar of algebraic geometry du Bois-Marie 1963--1964. Topos theory and étale cohomology of schemes (SGA 4). Vol. 1: Topos theory. Exp. I--IV
cites work: Néron Models
cites work: Schémas en groupes et immeubles des groupes classiques sur un corps local. II : groupes unitaires
cites work: Low-dimensional lattices. IV. The mass formula
cites work: On an exact mass formula of Shimura
cites work: On the motive of a reductive group
cites work: Haar Measure and the Artin Conductor
cites work: Q4143433
cites work: Local zeta functions on Hermitian forms and its application to local densities
cites work: Spherical Functions and Local Densities of Alternating Forms
cites work: Local densities of alternating forms
cites work: Q4271318
cites work: Q5509050
cites work: Some exact formulas on quaternion unitary groups
cites work: Linear algebraic groups.
cites work: Q3050630
cites work: Q3884236
cites work: The 2‐adic density of a quadratic form
full work available at URL: s0012-7094-00-10535-2
OpenAlex ID: W2076752446
description: scientific article; zbMATH DE number 1820769