Density of integer points on affine homogeneous varieties
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 factor \(\log T)\) and that \(G(\mathbb{R})\) is noncompact (in order that certain matrix-coefficients decay at infinity). The main result is that \(N (T,{\mathcal O}) \sim \mu(T)\) as \(T \to \infty\), where \(\mu (T)\) is the volume of the set \(\{\dot g : \dot g \in G(\mathbb{R})/H (\mathbb{R})\), \(\| g(w_ 0) \| \leq T\}\). This theorem also gives a ``mass formula'' à la Siegel [see \textit{A. Eskin}, \textit{Z. Rudnick} and \textit{P. Sarnak} [Int. Math. Res. Not. 65-69 (1991; Zbl 0743.11023)] if \(V\) is in addition a Hardy-Littlewood system, i.e. if \(N(T,V)\) is a product of local densities [see \textit{W. Schmidt}, Acta Math. 154, 243-296 (1985; Zbl 0561.10010)]; many homogeneous affine varieties are not far from being such. The proof is not given in full generality, since \textit{A. Eskin} and \textit{C. McMullen} [Duke Math. J. 71, 181-209 (1993)] have given a technically much simpler proof (see the review below). If \(W_ n = \{f(x,y)\), \(a_ ox^ n + a_ 1x^{n-1}y + \cdots + a_ ny^ n\}\), \(n \geq 3\), \(\| (a_ 0, \dots,a_ n) \|^ 2 = \sum^ n_{i=0} {h \choose i}^{- 1} a^ 1_ i\), then \({\mathcal O} \cong G (\mathbb{R})/M(\mathbb{R})\) is not (affine) symmetric. Nevertheless it is proved that \(N(T,{\mathcal O}) \sim C_{\mathcal O} T^{2/n}\), \(T \to \infty\). In general the symmetry condition can not be dropped (for an example of Eskin see the review below). The authors consider also the affine symmetric case \(V_{n,k} = \{x \in \text{Mat}_ n,\text{det} x = k\}\). \(V_{n,k} \cong G/H\), \(G = Sl_ n \times Sl_ n\), \(H = \{(g,g),g \in Sl_ n\}\), \(\tau (g_ 1, g_ 2) = (g_ 2,g_ 1)\) and prove even an error-estimate \[ N(T,V_{n,k}) = \mu(T) + O \Bigl( T^{{n^ 2-n-1 \over n+1+ \eta}} \Bigr), \] for all \(\eta>0\) (for \(n=2\) one obtains \(O(T^{5/3})\); the best known remainder is \(O(T^{4/3})\) due to Selberg [see \textit{P. Lax} and \textit{R. Phillips}, J. Funct. Anal. 46, 280-350 (1982; Zbl 0497.30036)]). The paper contains also interesting related results and appendices on the volume function \(\mu(T)\) and on regularizing Eisenstein periods on \(Sl_ 2 (\mathbb{C})/Sl_ 2 (\mathbb{R})\).
scientific article; zbMATH DE number 495440
scientific article