Maass Wave Forms of Real Weight

In this chapter, we introduce Maass wave/cusp forms of real weight on subgroups of finite index in the full modular group with compatible multiplier systems. After introducing multiplier systems and the unitary automorphic factor and discussing the hyperbolic Laplace operator and Maass operators, we define Maass waveforms. We then use Hecke operators in the special case where the Maass waveform is of weight 0 and trivial multiplier system on the full group. We conclude the chapter by mentioning some spectral theory for the Laplace operator and its Friedrichs extension and discuss briefly Selberg’s conjecture.