Lattice points for products of upper half planes

2009/04/20 by Roelof W. Bruggeman, Fritz Grunewald, Bruggeman, Roelof +3
Mathematics · #11F41 (Primary) #11F72 (Secondary) #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0904.3020

openalex publication_date 2009/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be an irreducible lattice in \PSL₂(\RR)^d (d∈\NN) and z a point in the d-fold direct product of the upper half plane. We study the discrete set of componentwise distances \bf D(\Gm,z)⊂ \RR^d defined in (1). We prove asymptotic results on the number of \gm∈\Gm such that d(z,γz is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes % improve the error terms improve the existing error terms (by Gorodnick and Nevo) and generalize the Selberg error term for d=1. We give an asymptotic formula for the number of lattice points γz such that the hyperbolic distance in each of the factors satisfies d((γz)_j, z_j)≤ T. The error term, as T → ∞ generalizes the error term given by Selberg for d=1, also we describe how the counting function depends on z. We also prove asymptotic results when the distance satisfies A_j ≤ d((γz)_j, z_j) < B_j, with fixed A_j < B_j in some factors, while in the remaining factors 0 ≤ d((γz)_j, z_j) ≤ T is satisfied.

Citations

• Cuspidality of symmetric powers with applications
• The spectral function of an elliptic operator

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