Most literature on modular functions (invariant or covariant with weight k under the full modular PSL\_2(Z) group) treats holomorphic functions and introduce the notion of cusp forms (modular functions that decay at the cusps).


I am interested in modular invariant, real functions. All literature I could find in the subject introduce the Real or non-holomorphic Eisenstein series:


$G\_s(\tau)=\frac{1}{2} \sum'\_{m,n} \frac{Im(\tau)^s}{|m \tau+n|^{2s}}$


However such functions diverge at the cusp $\tau=i\infty$. Does anyone know families of real, modular invariant functions that decay at the cusp? (for me is is important that they are real and modular invariant). Something like real cusps forms. And are these objects studied in the literature? is there a standard basis for such objects? (I am sorry if this question is too trivial for the forum, being a physicists I tried to characterize as precisely as possible the objects I need) . Thanks so much! f. 



Real cusp forms
Katz' "$p$-adic interpolation of real analytic Eisenstein series" (Annals of mathematics 1976, volume 104(2)) is an article you might want to read -- sure its goal is $p$-adic interpolation, but for that he has to discuss the real objects to some length.


EDIT: Bump's "Automorphic forms and representations" (Cambridge university press, 1997) also discusses them ; they are called Maass forms and are discussed in several places in the book.


