Hybrid trace formula for a non-uniform irreducible lattice in $\PSL_2(\bbR)^n$

TL;DR

This paper extends Selberg's trace formula to non-uniform lattices in (bR)^n, enabling analysis of conjugacy classes and connections to number theory, especially class numbers and units in quadratic forms.

Contribution

It derives a new trace formula for non-uniform lattices in (bR)^n, generalizing previous uniform lattice results and linking to algebraic number theory.

Findings

Distribution of elliptic-hyperbolic conjugacy classes analyzed

Connections established between trace formula and class numbers

Results interpreted in terms of quadratic forms over totally real fields

Abstract

In a series of lectures Selberg introduced a trace formula on the space of hybrid Maass-modular forms of an irreducible uniform lattice in $\PSL_2(\bbR)^n$. In this paper we derive the analogous formula for a non-uniform lattice and use it to study the distribution of elliptic-hyperbolic conjugacy classes. In particular, for Hilbert modular groups there is a nice interpretation of these results in terms of class numbers and fundamental units of quadratic forms over totally real number fields.