2003/03/20 by Joachim Hilgert, J. Hilgert, Dieter Mayer +6 · 2 citations
Mathematics · #11F25 #11F67 #37C40 #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Spectral Theory in Mathematical Physics #math.DS #math.NT #msc:11F25 #msc:11F67 #msc:37C40
paper · pdf · doi:10.48550/arxiv.math/0303251
33 pages
arxiv created 2003/03/20 · openalex publication_date 2003/03/20 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
In this article we report on a surprising relation between the transfer operators for the congruence subgroups Γ0(n) and the Hecke operators on the space of period functions for the modular group \PSL (2,ℤ). For this we study special eigenfunctions of the transfer operators with eigenvalues ∓ 1, which are also solutions of the Lewis equations for the groups Γ0(n) and which are determined by eigenfunctions of the transfer operator for the modular group \PSL (2,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 Γ0(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.