Transfer operators for $\Gamma_{0}(n)$ and the Hecke operators for the period functions of $\PSL(2,\mathbb{Z})$

Abstract

In this paper we report on a surprising relation between the transfer operators for the congruence subgroups $\Gamma_{0}(nm),n,m\in \mathbb{N}$, and some kind of Hecke operators on the space of vector valued period functions for the groups $\Gamma_{0}(n)$. We study special eigenfunctions of the transfer operators for the groups $\Gamma_{0}(nm)$ with eigenvalues $\mp$1 which are also solutions of the Lewis equations for these groups and which are determined by eigenfunctions of the transfer operator for the congruence subgroup $\Gamma_{0}(n)$. In the language of the Atkin–Lehner theory of old and new forms one should hence call them old eigenfunctions or old solutions of the Lewis equation for $\Gamma_{0}(n)$. It turns out that certain linear combinations of the components of these old solutions for the group $\Gamma_{0}(nm)$ determine for any $m$ a solution of the Lewis equation for the group $\Gamma_{0}(n)$ and hence also an eigenfunction of the transfer operator for this group.

Our construction gives linear operators $\tilde{T}_{n}$ in the space of vector valued period functions for the group $\Gamma_{0}(n)$ which are rather similar to the Hecke operators. Indeed, in the case of the group $\Gamma_{0}(1)\,{=}\,\SL(2,\mathbb{Z})$ these operators are just the well-known Hecke operators on the space of period functions for the modular group, derived previously using the Eichler–Manin–Shimura correspondence between period polynomials and modular forms for this group, and its extension to Maass wave forms by Lewis and Zagier.