Transfer operators for $\Gamma_0(n)$ and the Hecke operators for the period functions for $PSL(2,Z)$

TL;DR

This paper uncovers a surprising connection between transfer operators for congruence subgroups and Hecke operators on period functions for the modular group, revealing how special eigenfunctions relate across these structures.

Contribution

It demonstrates that eigenfunctions of transfer operators for $\Gamma_0(n)$ are linked to solutions of Lewis equations for the modular group, bridging transfer operators and Hecke operators in a novel way.

Findings

Old eigenfunctions for $\Gamma_0(n)$ determine solutions for the modular group.

Sum of components of old solutions yields eigenfunctions for the modular group.

The relation clarifies the structure of transfer operators and Hecke operators in this context.

Abstract

In this article we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(n)$ and the Hecke operators on the space of period functions for the modular group $\PSL (2,\mathbb{Z})$. For this we study special eigenfunctions of the transfer operators with eigenvalues $\mp 1$, which are also solutions of the Lewis equations for the groups $\Gamma_{0}(n)$ and which are determined by eigenfunctions of the transfer operator for the modular group $\PSL (2,\mathbb{Z})$. In the language of the Atkin-Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of Lewis equation. It turns out that the sum of the components of these old solutions for the group $\Gamma_{0}(n)$ determine for any $n$ a solution of the Lewis equation for the modular group and hence also an eigenfunction of the transfer operator for this group.