The Hecke algebras for the Fricke groups

Abstract

We discuss the Hecke algebra \({\cal H}(\Gamma_0(N),GL_2^+({\rm Q}))\)⋌e for the standard subgroup Г0 (N) of S L2(ℤ). In particular the center of this Hecke algebra is determined. The results are applied to the Fricke group, which is a normal and maximal discrete extension of Г0 (N), and to the attached Hecke algebra, which turns out to be a commutative polynomial ring. In this case the Hecke groups \(G(\sqrt 2)\)⋌e and \(G(\sqrt 3)\)⋌e are included. Finally we give a description of modular forms for the Fricke groups in terms of new-forms in the sense of Atkin-Lehner.