I know some theory of "classical" modular forms, that is functions in the complex upper-half plane satisfying

$f(\frac {az+b} {cz+d})=(cz+d)^kf(z)$

I know one can study modular forms on finite-index subgroups of $SL(2,\mathbb Z)$. But I have not seen much theory of modular forms on arbitrary Fuchsian groups. Which are the most interesting cases of such groups? Can somebody recommend a good reference?
I have also come across Hilbert and Siegel modular forms, but I don't have these in mind as an answer to this question. I wonder whether one can use arbitrary Lie group instead of $SL(2,\mathbb R)$, which is what the Wikipedia page about automorphic forms suggests, but I am not on a level to tackle the theory.

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    $\begingroup$ The problem is not lack of references but their abundance. To begin with: Are you interested in $SL(2,R)$ or more general groups? What aspects of the theory are you mostly interested in? $\endgroup$
    – Misha
    Nov 5, 2017 at 2:51
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    $\begingroup$ Because I am just starting with groups other than $SL(2,\mathbb Z)$, I would prefer $SL(2,\mathbb R)$. But if the references are much better on the "more general groups", it's not such a big problem. $\endgroup$
    – FusRoDah
    Nov 5, 2017 at 9:09

2 Answers 2


Yes, as you surmise, since about 1950, there has developed a general theory of automorphic forms on (semi-simple or reductive, mostly, "Jacobi forms" are a sort of exception) real Lie groups... and also on the corresponding adele groups when the group is defined over (some localization of) $\mathbb Z$. The general development is due to Harish-Chandra, Borel, Gelfand-PiatetskiShapiro, Godement, Langlands, and many others subsequently.

One of the earliest overviews of various aspects is the 1965 Boulder Conference, which appeared in 1966 as AMS Proc. Symp. Pure Math 9. The next iconic source is the 1977 Corvallis conference, which occurred in two volumes as AMS Proc Symp Pure Math 33.

There were and are many more sources...

I note that, apart from the relatively isolated studies on Siegel and Hilbert modular forms, and Maass' waveforms, by Maass, Siegel, Shimura, Klingen, and a few others, until 1960 in the U.S. "automorphic forms" exactly meant "with respect to (suitable) Fuchsian subgroups of $SL(2,\mathbb R)$, and/or intense examination of ratios of products of the weight $1/2$ modular form $\eta(z)$, for purposes of examining partition functions...

The renaissance of the general theory was perhaps due to Selberg-Roelcke's study of spectral theory in the 1950's, and Shimura's study of arithmetic consequences throughout the 1960's (and later), and then Langlands' generalizations and abstractions.


You can start with Lester Ford's Automorphic Functions - an oldie but a goodie (republished by AMS Chelsea, surely available for free as an ebook from somewhere [ok, I know where, but I can't say].

  • $\begingroup$ You don't mean this, do you: "An introduction to the theory of automorphic functions", archive.org/details/introductiontoth00forduoft ? (that's a 1915 book, there's also a 1929 book called simply "Automorphic functions") $\endgroup$
    – David Roberts
    Nov 3, 2017 at 23:19
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    $\begingroup$ FYI: the 1915 book is out of copyright, so it's not like it's "borrowing it from Library Genesis" $\endgroup$
    – David Roberts
    Nov 4, 2017 at 1:36
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    $\begingroup$ @reuns The OP asks for a reference. $\endgroup$
    – Igor Rivin
    Dec 4, 2017 at 15:36
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    $\begingroup$ You know what I mean : if you are capable to explain the mathematical idea, do it, if you are not capable, post a comment. This is what MO is useful for. $\endgroup$
    – reuns
    Dec 4, 2017 at 15:39
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    $\begingroup$ @reuns No, I don't know what you mean. You are entitled to your opinion, of course, but please don't try to force it onto the rest of us. $\endgroup$
    – Igor Rivin
    Dec 4, 2017 at 18:56

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