Integral formula and asymptotic behaviour of the number of lattice points in complex hyperbolic space. (Formule intégrale et développement asymptotique du nombre de points d’un réseau dans l’espace hyperbolique complexe.) (French) Zbl 0857.11024

Let \(\Gamma\) be a torsion-free discrete group of isometries of the complex hyperbolic space of dimension \(n \geq 1\). The author sketches a general integral representation for the counting function of the orbit of one point in a ball centered on another. He does not discuss the related convergence problems; these are to appear in a more extensive paper. The method used is based on spectral theory and is a development of one used by Levitan. It is essentially equivalent to Selberg’s method of point-pair invariants. He applies his result to the case where the number of eigenvalues corresponding to the “complementary series” is finite to obtain an explicit asymptotic development.


11F72 Spectral theory; trace formulas (e.g., that of Selberg)
11P21 Lattice points in specified regions
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)