2023/09/05 by Biró, András, Tóth, Dávid
#11F72 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2309.02163
A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of K(z,z) (where K(z,w) is an automorphic kernel function) over the fundamental domain, one considers the integral of K(z,z)u(z), where u(z) is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of PSL(2,ℝ), and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways ("geometrically" and "spectrally") the integral of K(z,z)u(z). In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of PSL(2,ℝ)n where n>1. Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for n>1 than in the case n=1.