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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.

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    $\begingroup$ Search under Maass cusp forms or automorphic forms. There is an extensive literature. For example see the book by Iwaniec called Spectral theory of automorphic forms. $\endgroup$
    – Lucia
    Dec 11, 2013 at 18:01
  • $\begingroup$ Maass form are not real-valued. They are real-analytic as oposed to holomorphic. That distinguishes them from modular forms. $\endgroup$
    – Marc Palm
    Dec 13, 2013 at 8:43

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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.

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  • $\begingroup$ Maass forms are not real-valued in general. $\endgroup$
    – Marc Palm
    Dec 13, 2013 at 8:28
  • $\begingroup$ They're real analytic and as far as I can tell, that is what the question is about, so I don't think I wrote something wrong or out of subject. $\endgroup$ Dec 13, 2013 at 13:27
  • $\begingroup$ Since the invariant Laplacian has real coefficients, its eigenspaces have bases consisting of real-valued functions, since $f+\overline{f}$ is also a $\lambda$-eigenfunction if $f$ is, etc. $\endgroup$ Dec 13, 2013 at 19:09

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