On the cuspidal spectrum of the arithmetic Hecke groups

Abstract

Let Γ ( m ) ( m = 1 , 2 , 3 ) \Gamma (\sqrt m )\;(m = 1,2,3) be the three arithmetic Hecke groups, generated by the translation z ↦ z + m z \mapsto z + \sqrt m and the inversion z ↦ − 1 z z \mapsto - \frac {1}{z} . ( Γ ( 1 ) \Gamma (1) is the modular group P S L ( 2 , Z ) PSL(2,Z) .) The purpose of this article is to present several theoretical results on Σ ( m ) \Sigma (\sqrt m ) , the weight-zero cuspidal spectrum of Γ ( m ) \Gamma (\sqrt m ) , which are of interest from a computational perspective, since they have application to the numerical study both of the spectra themselves and the Fourier coefficients of the associated Maass wave forms. The first of these is the theorem that Σ ( 1 ) ⊂ Σ ( m ) \Sigma (1) \subset \Sigma (\sqrt m ) for m = 2 m = 2 and m = 3 m = 3 . Additional results explicate the action of the Hecke operators, for the three groups in question, upon Maass wave forms, in particular upon their Fourier coefficients. These results are motivated in part by work of Stark—and more recent work of Hejhal and Hejhal and Arno—which demonstrates the importance of the Hecke operators for the numerical study of the Fourier coefficients of Maass wave forms on Γ ( 1 ) \Gamma (1) .