Density of integer points on affine homogeneous varieties
type: Item
instance of: scholarly article
title: Density of integer points on affine homogeneous varieties
zbMATH Open document ID: 0798.11024
DOI: 10.1215/S0012-7094-93-07107-4
published in: Duke Mathematical Journal
publication date: 1994-11-01T00:00:00Z
review text: This paper is an important contribution to the fundamental problem in diophantine analysis to investigate the asymptotics of \(T \to \infty\) of \(N(T,V) = \{m \in V (\mathbb{Z}) : \| m \| \leq T\}\) of integral points of norm \(\leq T\) in an affine variety. The only general method available is the Hardy-Littlewood circle method which has certain limitations (e.g. on the codimension, on the degree of the defining polynomials and the size of the singular sets of the related varieties). This paper seems to be inspired by \textit{J. Franke}, \textit{Yu. Manin} and \textit{Yu. Tschinkel} [Invent. Math. 95, 421-435 (1989; Zbl 0674.14012)] who considered the problem of counting rational points of height \(\leq T\) on certain flag varieties \(G/P\), \(G\) reductive, \(P\) parabolic, involving Eisenstein series. The authors consider varieties \(V = G \cdot w_ 0 \subseteq W\) coming from rational representations of a \(\mathbb{Q}\)-semisimple (usually simple) connected group \(G\) on a \(\mathbb{Q}\)-vector space \(W\). Since \(V(\mathbb{Z})\) breaks up into finitely many \(G(\mathbb{Z})\)-orbits (by a classical theorem of Borel and Harish-Chandra) the problem can be reduced to study a single orbit \({\mathcal O} \cong G (\mathbb{Z})/H(\mathbb{Z})\) \((H\), the stabilizer of \(w_ 0\) is reductive), i.e. one has to study \[ N(T,{\mathcal O}) = \Bigl | \biggl\{ \gamma \in G (\mathbb{Z})/H(\mathbb{Z}) : \bigl \| \gamma (w_ 0) \bigr \| \leq T \biggr\} \Bigr |. \] This can be done, making full use of harmonic analysis in \(L^ 2 (G(\mathbb{R})/G(\mathbb{Z}))\) if \(V(\mathbb{R}) = G (\mathbb{R})/H(\mathbb{R})\) is symmetric, i.e. if \(H(\mathbb{R})\) is the fixed point set of some (not necessarily Cartan-) involution \(\tau\) of \(G(\mathbb{R})\) \((V(\mathbb{R})\) need to be Riemannian symmetric). In this paper it is also assumed that \(H(\mathbb{Z})\) is a lattice in \(H\), i.e. \(H(\mathbb{R})/H(\mathbb{Z})\) has finite volume (in the general case the asymptotics change by a f…
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 495440
zbMATH Keywords: density of integer points
zbMATH Keywords: symmetric space
zbMATH Keywords: affine variety
zbMATH Keywords: volume function
zbMATH Keywords: regularizing Eisenstein periods
author: W. Duke
author: Zeév Rudnick
author: Peter Sarnak
reviewed by: Harald Rindler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 495440