Conjugacy classes and rational period functions for the Hecke groups

TL;DR

This paper explores the relationship between conjugacy classes of Hecke groups and the poles of rational period functions, providing new constructions and characterizations for automorphic integrals.

Contribution

It establishes a one-to-one correspondence between conjugacy classes and poles, and introduces new rational period functions for Hecke groups.

Findings

Established a correspondence between conjugacy classes and poles

Constructed new rational period functions

Counted poles and characterized symmetries

Abstract

We establish a one-to-one correspondence between conjugacy classes of any Hecke group and irreducible systems of poles of rational period functions for automorphic integrals on the same group. We use this correspondence to construct irreducible systems of poles and to count poles. We characterize Hecke-conjugation and Hecke-symmetry for poles of rational period functions in terms of the transpose of matrices in conjugacy classes. We construct new rational period functions and families of rational period functions.