TITLE: What is the relationship between modular forms and Maass forms?
QUESTION [21 upvotes]: Modular forms are defined here:
http://en.wikipedia.org/wiki/Modular_form#General_definitions
Maass forms are defined here:
http://en.wikipedia.org/wiki/Maass_wave_form
I wonder if modular forms can be transfered into Maass forms.
Or they two are different categories of automorphic forms.

REPLY [31 votes]: Automorphic forms correspond to representations that occur in $L^2(G/\Gamma)$. In the case when $G$ is $SL_2$, holomorphic modular forms correspond to (highest weight vectors of) discrete series representations of $G$, while Maass wave forms correspond to (spherical vectors of) continuous series representations of $G$.