What is the difference between an automorphic form and a modular form?

The term "automorphic form" is clearly a generalization of the term "modular form." What is not clear is exactly which generalization it is. Many sources use the term in different ways.

Any classical holomorphic modular form for $\mathrm{SL}_2(\mathbb{Z})$ is called a modular form, and usually (but not always) so are modular forms for congruence subgroups. Often, "automorphic form" is used when one considers either other Fuchsian groups, forms on groups other than $\mathrm{GL}_2$, or non-holomorphic forms (such as Maass forms and real analytic Eisenstein series). Alternatively, Diamond and Shurman define an "automorphic form" in Section 3.2 as being like a modular form but possibly meromorphic instead of holomorphic.

As another example, Miyake's book Modular Forms writes on p.114, "Automorphic functions and automorphic forms for modular groups are called modular functions and modular forms, respectively," and his definition of "modular group" seems to coincide with that of congruence subgroup.

The Princeton Companion to Mathematics writes in section III.21, "automorphic forms, which are generalized versions of the classical analytic functions called modular forms [III.61]," but it does not specify what the generalization is. The book contains other, similar statements (in III.61, "And indeed, automorphic forms, which are generalizations of modular forms").

So what exactly is the* definition of modular form as opposed to automorphic form? Since there is likely no "right" answer, what I really want to know is what is the history and what are the different conventions and the relations between them.

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I think you are making a mistake by using "the" in "the definition of modular form". I think it is standard practice to explicitly describe a definition that is convenient for one's work, and one usually selects from a reasonably small list. One way that modular forms are not subsumed by automorphic forms is that one has a notion of modular form for a noncongruence subgroup of $SL_2(\mathbb{Z})$, and these are generally not considered automorphic. –  S. Carnahan Mar 17 '13 at 7:10
I think that Shimura might use the term "automorphic form" to mean a modular form with possible poles, and perhaps Diamond (who was educated in Princeton so maybe learnt from Shimura or his book) and Shurman followed his lead. I've always thought it was terrible notation though, given that "automorphic form" means a different thing to almost everyone else :-/ I'm not sure I know any other place where "automorphic form" means "modular form with possible poles" other than Shimura's book and D-S. –  user30035 Mar 17 '13 at 9:11
Holomorphic automorphic forms are the sections of automorphic vector bundles on Shimura varieties --- this definition is essentially due to Deligne. When the Shimura variety is a moduli variety, they are often called modular forms. There are also nonholomorphic automorphic forms. –  anon Mar 17 '13 at 12:58
@S. Carnahan: Yes, and that's part of my reason for trying to ask for a historical answer and an overview of different definitions instead of one definitive answer. Also, see Miyake's book, which seems to precisely define modular forms as referring only to congruence groups, with automorphic being the more general notion. –  David Corwin Mar 17 '13 at 14:48

Very briefly: until work of Hans Maass c. 1949, "modular" or "automorphic" both referred to holomorphic functions invariant-up-to-cocycle (that is, invariant holomorphic sections of a bundle) on a quotient $\Gamma\backslash X$. For $X$ the upper half-plane, these were ellipic modular forms, visible since the 19th century. For $X$ a product of upper half-planes, these were Hilbert-Blumenthal modular forms, also studied by Hecke and Siegel. For $X$ a Siegel upper half-space, Siegel modular forms, studied by Siegel and Hel Braun in 1939. The "elliptic" case arose partly from moduli problems concerning elliptic curves, indeed, and had its roots in the work of Abel and Jacobi in the early 19th century. The Hilbert-Blumenthal story seems to have been a conscious generalization, once the number-theoretic content of the elliptic modular case was illustrated in the late 19th century, e.g., as in in Fricke-Klein. The "Siegel modular" case has physical setting the moduli space for principally polarized abelian varieties of a fixed size.

The "general" case of discrete subgroups (Fuchsian, etc.) of $PSL(2,\mathbb R)$ acting on the upper half-plane (or, equivalently, $PSU(1,1)$ on the disk) was investigated by Poincare and others in the very late 19th century, and called "automorphic".

After Hecke saw that binary theta series give Dedekind zetas of complex quadratic extensions of $\mathbb Q$, he gave Maass the thesis problem of finding an analogue for real quadratic. This led Maass to discover "waveforms", solutions of $(\Delta-\lambda)f=0$ where $\Delta=y^2({\partial^2\over \partial x^2}+{\partial^2\over \partial y^2})$ is the $SL(2,\mathbb R)$-invariant Laplacian on the upper half-plane. The Eisenstein series $E_s(z)=\sum'_{c,d} y^s/|cz+d|^{2s}$ are the most explicit examples, and also the "special waveforms" Maass found (that Mellin transform essentially to Dedekind zetas of real fields and Hecke L-functions thereupon) are explicit. These are "automorphic forms/functions".

Selberg, Roelcke, Gelfand-et-al, and a few others continued looking at the "analytic" side of these things in the 1950s. Siegel and Braun continued to work on holomorphic modular/automorphic forms on higher-dimensional spaces, with Braun initiating the investigation of "Hermitian" modular forms, that is, attached to the group $U(n,n)$ rather than $Sp(n,\mathbb R)$ for Siegel modular forms.

Starting in the late 1950s and 1960s, Shimura considered the algebraic geometry and arithmetic (that is, Hasse-Weil zeta functions, generation of classfields, and such) on many higher-dimensional "modular" varieties, subsumed mostly under the "PEL-type" label: polarization, endomorphism, level. By this point, it seems that "automorphic" was used to refer to these "general" situations, even though they had connections with moduli problems.

Also in the late 1950s and 1960s, Gelfand and his collaborators (Pieatetski-Shapiro, especially) emphasized the representation-theoretic possibilities in studying not only holomorphic modular/automorphic forms/function, but also Maass waveforms and other "generalizations". In particular, by about 1960 it was clear that from a repn theoretic viewpoint "holomorphic modular forms" and "real-analytic waveforms" had the commonality that both generated irreducible repns of the Lie group acting. Further, for congruence subgroups, being an eigenfunction for Hecke operators essentially meant generating irreducibles for the p-adic groups acting, as well. (In fact, even the "bad prime" behaviors are included nicely, if less formulaically, under this umbrella.)

Langlands' work on the spectral theory of automorphic forms and Eisenstein series in general, in the 1960s, and conjectures relating Artin L-functions to general cuspforms on $GL(n)$, etc., gave a big impetus to the "general" theory starting in the late 1960s. In part, this was made feasible by progress in the repn theory of semi-simple real Lie groups, especially by Harish-Chandra. The repn theory of p-adic groups, initiated mostly by MacDonald and Gelfand-et-al, started off a little more slowly, but also proved to be sufficiently robust as to be a "help" rather than "hindrance" in this aspect of the theory of afms.

The general study of moduli problems similarly needed additional inputs to continue to make progress, and Grothendieck-et-al's newer algebraic geometry, in the hands of Deligne and others, turned out to be a good language/viewpoint for this.

There is a lot more to be said, naturally. By this year, "modular" suggests "holomorphic", as well as "related to moduli problem". "Automorphic" suggests "something more general", but also can be used as an umbrella term. Holomorphic-except-for-singularities, that is, meromorphic forms, have also arisen in higher-dimensional settings, as in Borcherds products. On another hand, the "weak" automorphic/modular forms of Zwegers-et-al allow a controlled extension of the moderate-growth condition on waveforms.

Edit... : and it may be worth noting that various more-formal "definitions" are highly non-trivial to compare to each other, often depending upon appreciation of big theorems from repn theory, algebraic geometry, and number theory, that are usually _not_named_ during a formal discussion of "definitions". Further, there are more-elementary incompatibilities that are often harmless in a given context, but are not overtly acknowledged. For example, the "moderate growth" condition is not met by $L^2$ automorphic forms, for the same reason that functions in $L^2(\mathbb R)$ are typically not of moderate growth. Another is that requiring $\mathfrak z$-finiteness precludes taking an $L^2$ closure. But these awkwardnesses of the formal language are not genuine obstacles.

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I thought it was a result that a $K$ finite $z$ finite function $f$ on $G/\Gamma$ which was $L^2$ indeed has moderate growth; $f=f*\phi$ where $\phi$ is a suitable compactly supported function? Is this incorrect? –  Venkataramana Mar 17 '13 at 16:13
@Aakumadula: oh, no, you are entirely correct. But this fundamental result is quite non-trivial to prove, and dropping assumptions leads to failure, illustrating my point. E.g., $\mathfrak z$-finite but not $K$-finite, or vice-versa, with $L^2$, does not yield moderate growth. –  paul garrett Mar 17 '13 at 17:03
@paul garrett, thank you very much. I did not notice that you had dropped the assumption of $K$ finiteness. –  Venkataramana Mar 17 '13 at 23:25

There are many aspects to the relation between modular forms and automorphic forms. First let me recall what a modular form of weight $2k$ for, say $SL(2,{\mathbb Z})$ is. This is a function on the upper half plane, which is $holomorphic$, has an equivariance property $$f(\frac{az+b}{cz+d})=(cz+d)^{2k}f(z)$$ for $\begin{pmatrix} a & b \cr c & d\end{pmatrix}\in SL(2,{\mathbb Z})$ and at infinity (the "cusp" for $SL(2,{\mathbb Z})$ in the fundamental domain) has the Fourier expansion $$f(z)= \sum _{k=0}^{\infty} a(n)q^n,$$ where $q=e^{2\pi i z}$ for $z$ in the upper half plane. For example, the Ramanujan $\Delta$ is such a modular form. But, $1/\Delta (z)$, while it is still holomorphic on the upper halp plane, has its Fourier expansion at infinity starting from $q^{-1}$ and is hence not a modular form.

We can convert a modular form into a function on the group $SL(2,{\mathbb R})$ by taking the related function $F_f(g)= (ci+d)^{-2k}f(g(i))$. This is a function on $SL(2,{\mathbb R})$ on which the centre of the enveloping algebra acts by scalars (translating the holomorphy condition on $f$), is $SL(2,{\mathbb Z})$ invariant (translating the equivariance for the modular form $f$), and has moderate growth on the fundamental domain for $SL(2,{\mathbb Z})$ in $SL(2,{\mathbb R})$.

This can be generalised to any congruence subgroup of the modular group, and any weight etc, and you get $K$-finite , $Z$ finite functions on $G$, which are $\Gamma$ invariant. These are called automorphic forms. If you do this construction for the inverse of $\Delta$ you get most properties, but the moderate growth, and hence you do not get an automorphic form.

On the other hand, any function on $G/\Gamma$ (say, $G$ semi-simple and $\Gamma$ an arithmetic group) which is $K$ finite, $z$ finite and has moderate growth (along with its derivatives ) on a fundamental domain, is defined to be an automorphic form. So even for $SL(2,{\mathbb R})$ there are many more automorphic forms than those arising from modular forms (those from modular forms generally have infinite component of the discrete series (this keeps track of the weight $k$ of the modular form)).

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The most common definition of an automorphic form is that it a K-finite Z-finite vector in an automorphic representation. Often one requires the automorphic rep to be irreducible for being able to get a reasonable L function from it. For the purpose of the Langlands stuff, this is the only reasonable definition, also for Taniyama-Shimura conjecture etc. These are the only functions being analyzable by trace formulas etc.

When we consider Hecke Maass or modular cusp forms , this is a specialization. For this, you can look up strong approximation, e.g. in Bumps book.

Depending on their focus, some authors consider being automorphic not related to a congruence setting, but this is rare. (Note also that division algebras give rise to uniform lattices. At least, these forms must be called automorphic. ) Also one assumes reasonable growth conditions,i.e., no poles.

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@Marc: Take a look at the book "Automorphic forms and Kleinian groups" by I.Kra, where he uses automorphic forms in the context of Kleinian groups (the latter are infinite covolume discrete subgroups of $PSL(2, C)$, typically, not contained in $PSL(2, R)$; such groups are very far from being congruence subgroups of the modular group). Thus, "rare" depends on what field of mathematics you belong to. –  Misha Mar 17 '13 at 18:25
Thanks. Automorphic forms over quadratic imaginary fields are as well of this form. Too general definitions give not rise to L functions, eg, general lattices. So I meant rare in the sense of number theoretic applications. –  Marc Palm Mar 17 '13 at 18:47