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Automorphic forms are ubiquitous in modern number theory and stands as a mysterious Graal lying at the intersection of many fields, if not building valuable bridges between them. However, since this aim has been erected as one of the topmost paradigm of research in number theory, reasons for formulating motivations for studying automorphic forms may have faded. I am seeking here for such motivations and facts leading to making so many efforts toward the theory of automorphic forms.

Why are we interested in automorphic forms/representations?

There are many different levels of reading of this question, and it might be of use to both have in mind the possibility of some of them -- for we often stay at one particular of them, depending on our interests and culture --and the different treatment they require.

Type of audience

  • (A) General Public (at most high school background, with no particular interest in science)

  • (B) Scientists (some undergraduate years or curiosity and interest toward science)

  • (C) Mathematicians (think of the question as an analog to Sarnak's book "Some applications of modular forms", but for automorphic forms/representations)

  • (D) Number Theorists (inner motivations, e.g. Langlands program, allowed)

Type of argument

  • (0) Meta (e.g. unification of different notions; "representations are a way to makes a group a group of motions, so endow it with a more geometrical structure, change the insight, or note that we know better many objects by their actions than by their elements", etc.)

  • (1) Outside motivations (e.g. PDE model many dynamics coming from physics or biology; high-dimensional spaces as efficient setting for robotics; formal logic leading to specification and verification, etc.)

  • (2) Math Facts (e.g. historical reasons of why automorphic forms are powerful; "such fact reduces the study of a certain kind of objects to a simpler one", how it intervenes in a classification as a relevant part of the field, etc.)

  • (3) Math Hopes (e.g. what a given result on automorphic forms would bring)

This is a proposition and every comment or suggestion towards those classification are welcome.

Some comments on the classification

  • the "Audience" classification can be superfluous because it can be an issue concerning language and culture (replacing "PDE" and "representation" by "evolution law" or "way to move" can allow to make an argument work for both C and A as well) more than contents, but not always
  • there is no necessary relation between the "Audience" tag and the "Argument" one. Even if some of those categories seems to be mutually exclusive (as A2), this is also part of the challenge to be able to present the whole diversity of motivations for automorphic forms to every audience.

The purpose of this thread if both to gather ideas to answer the natural questions of mathematicians as well as friends about the reason of spending so much time working on those topics, and also to reinvent our field in facing different aspects and motivations for the automorphic world as well as having to put words on motivations that are sometimes far away from our mind (and I do not claim they are necessary). Those could also lead to natural introductions and motivations for students.

Ideas, precise references, detailed answers or specific examples are all welcome.

Note: Even if many examples motivating modular forms or Maass forms arise in the literature, I fail for months to find something for automorphic forms without specific distinction.

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    $\begingroup$ Please change to a less provocative title (e.g., "What is the motivation for..."), since one could ask the same thing about plenty of advanced but important math topics that look opaque to non-specialists (e.g., microlocal analysis, schemes, $\infty$-categories, various models of set theory, etc.). The Corvallis volumes are a good answer for (D). But if you intend something about automorphic methods beyond classical-style topics as in Sarnak's book that you mention then your cases (A) and (B) are hopeless: see 13:10-14:10 of vimeo.com/24751596 for an entertaining version. $\endgroup$ – nfdc23 Jan 15 '18 at 16:19
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    $\begingroup$ @nfdc23 I change the title, even if provocation was part of the com :) I am perfectly fine with answers to (D), in particular more encapsulated than referring to the whole Corvallis volumes (and also of other fashions). For Sarnak's book, I'd like to say that I do not necessarily want something beyond, yet something more focused on automorphic forms at large (without considering only modular forms as in Sarnak's book, or only Maass forms). Every motivation I have in mind without brandishing very abstract theory (what I do not call efficient motivation...) unfortunately fail to do so... $\endgroup$ – Desiderius Severus Jan 15 '18 at 17:52
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    $\begingroup$ Speaking of provocation, would you mind changing the label of Audience (A)? In English, "profane" carries a pejorative connotation. $\endgroup$ – Neal Jan 15 '18 at 18:55
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    $\begingroup$ Maybe "general public" is better, in English... $\endgroup$ – paul garrett Jan 15 '18 at 19:15
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    $\begingroup$ With the solution of the quintic via elliptic modular forms in hand, in 1858, Kronecker conjectured that his solution might be a special case of a more general result that yields solutions of an arbitrary-degree polynomial. In 1984 Umemura [1] showed that Kronecker was right, and that Siegel modular forms were the relevant generalisation of elliptic modular forms.$$.$$ [1] Umemura, H. "Resolution of algebraic equations by theta constants", Tata Lectures on Theta II (Mumford). $\endgroup$ – Oliver Nash Jan 17 '18 at 13:17
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The specific issue of what automorphic forms on bigger groups than $GL(2)$ over $\mathbb Q$ (for example) may tell us about automorphic forms (and L-functions) for $GL(2,\mathbb Q)$ or $GL(1,k)$ for number fields $k$ does have at least a few good answers. First, about 1960 and a little before, Klingen's proof that zeta functions of totally real number fields $k$ (and L-functions of totally even characters on such fields) have good special values at positive even integers used the idea of pulling back holomorphic Hilbert modular Eisenstein series to elliptic modular forms. (I heard G. Shimura lecture on this c. 1975, and it was quite striking.)

Another example: already in the 1960s, J.-P. Serre and others saw that holomorphic-ness of symmetric-power L-functions for $GL(2)$ holomorphic modular forms would prove Ramanujan-conjecture-type results. How to prove that holomorphy? By finding an integral representation of such L-functions, and using that. This has met with varying degrees of limited success, e.g., in papers of H. Kim and F. Shahidi.

The previous example was grounded in the general pattern of Langlands-Shahidi treatment of L-functions in terms of constant terms of cuspidal-data Eisenstein series on (necessarily larger) reductive groups. The specifica cases where various Levi-Malcev components of parabolics were products of $GL2$'s or $SL2$'s produced several "higher" L-functions for $GL2$.

As variation on that, already in the Budapest conference in 1971, Piatetski-Shapiro observed that (what we often nowadays call) Gelfand pairs could produce Euler produces via integral representations (usually involving Eisenstein series). Various success-examples of this idea included work of PiatetskiShapiro-Rallis, Shimura, myself, M. Harris, S. Kudla, various collaborations among these people, and several others, beginning in the late 1970s. E.g., I was fortunate enough to stumble upon an integral representation for triple tensor product L-functions for $GL2$ in terms of an integral representation against Siegel Eisenstein series on $Sp(3)$ (or $Sp(6)$, if one prefers). M. Harris and S. Kudla found another such integral representation that covered special value results in the "other range" (in terms of P. Deligne's conjectures).

In yet other terms, Jacquet-Lapid-Rogawski (and several others) have demonstrated that a variety of L-functions appear as periods of Eisenstein series on "larger" reductive groups. (One novelty is using relative trace formulas to exhibit Euler products when the simpler "Gelfand pair" idea is not quite sufficient.)

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To give a brief answer, which I think applies to all audiences, and I hope is not too "elementary" for you (I'm not attempting to give details, which of course need to be specialized for the intended audience, and I'm not entirely sure what you're looking for):

Automorphic forms provide analytic ways to study solutions to equations in integers or rational numbers.

Ex 1 How many ways can one represent $n$ as a sum of $k$ squares, i.e., how many integer solutions does $x_1^2 + \cdots + x_k^2 = n$ have?

If we call this number $r_{k}(n)$, then the Fourier series $$ f_{k}(z) = \sum_{n=0}^\infty r_{2k}(n) e^{2\pi i nz}$$ is a modular form of weight $k/2$ and level $4$. Studying the analytic behaviour of this function allows one to get asymptotics for $r_{2k}(n)$, and more precise studies of Fourier coefficients of modular forms allows one to get actual formulas for $r_{2k}(n)$, at least for certain values of $k$.

The study of quadratic forms motivated much of the arithmetic study of modular forms, of which automorphic forms generalize. Asking more complicated questions leads to other kinds of automorphic forms (e.g., what quadratic forms represent what forms leads to Siegel modular forms).

Ex 2 What integers are sums of two rational cubes, i.e., when does $x^3+y^3=n$ have a solution over $\mathbb Q$?

This is a special case of asking when elliptic curves have infinitely many rational points, which is related to modular forms via $L$-functions and the Birch and Swinnerton-Dyer conjecture. Namely one wants to study zeroes of $L$-functions, but one needs to pass to modular forms/automorphic representations to get $L$-value formulas. Fermat's last theorem is of course more famous, but I like this problem a lot.

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    $\begingroup$ Perhaps it is my own ignorance but this answer could be improved a lot by providing some details on how exactly the connection between the topics it raises and automorphic forms is established. I think this is the part that is the least known among novices from outside the field (like me). $\endgroup$ – Saal Hardali Jan 16 '18 at 13:49
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    $\begingroup$ @SaalHardali Well I read the question as asking for examples of motivations, rather than a detailed presentation to each possible audience, and I didn't feel like writing a long essay last night (e.g., like math.ou.edu/~kmartin/papers/quatcubforms.pdf or math.ou.edu/~kmartin/papers/mfs.pdf which I wrote for general math grad students), but since this is CW others should feel free to improve this answer by providing more details. $\endgroup$ – Kimball Jan 16 '18 at 15:24
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    $\begingroup$ @SaalHardali Okay, I decided to add a few details for Ex 1. Maybe when I have more time, I will do the same for Ex 2 (but again, others should feel free to add this in). $\endgroup$ – Kimball Jan 16 '18 at 15:39
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    $\begingroup$ Great examples, though I think this is really more of an answer to why number theorists care about modular forms, rather than why someone would bother going to more general automorphic forms and representations for groups other than GL(2). $\endgroup$ – Jason Polak Jan 17 '18 at 8:59
  • $\begingroup$ @JasonPolak And this lack of capacity of talking about automorphic forms or representations per se is the very seed of my question! $\endgroup$ – Desiderius Severus Jan 17 '18 at 9:26
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Here's an answer for audiences (C), (D) about class field theory:

Class field theory provides a way classify abelian extensions of number fields $F$. In the case of $F=\mathbb Q$, this is answered by the Kronecker-Weber theorem, which says any abelian extension is contained in $\mathbb Q(e^{2\pi i/n})$ for some $n$. The fields $\mathbb Q(e^{2\pi i/n})$ are the ring class fields of $\mathbb Q$. To generalize this to other number fields $F$, one would like an analogue of the extensions $\mathbb Q(e^{2\pi i/n})$. If $F$ is imaginary quadratic, these are extensions obtained by adjoint special values of a certain transcendental function called the $j$-invariant, which is a quotient of classical modular forms.

To study extensions to more general number fields $F$, Hilbert was led to consider a generalization of classical modular forms to what are now known as Hilbert modular forms, again special cases of automorphic forms. Then, at least in some cases, one can use automorphic forms to construct certain special functions whose special values can be used to construct abelian extensions of $F$.

Much more generally, but also much more vague (at least at present), there is the Langlands program. (Here I will be quite vague on details---feel free to edit more in or make another answer about this.) This has many aspects, but the one related to class field theory is the point of view that class field theory can be regarded as a correspondence between 1-dimensional Galois representations (representations of Gal($\bar F/F$)) and automorphic representations of GL(1) (Hecke characters, like Dirichlet characters). At the heart of the Langlands program is a conjectural correspondence between $n$-dimensional Galois representations and "automorphic representations of GL($n$)". This is often called nonabelian class field theory, because it should provide a way to describe the non-abelian extensions of $F$.

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The following answer will be an attempt for the 'general mathematical audience', and so contains a little imprecision.

Here are two reasons (somewhat?) related to number theory why one might consider automorphic representations other than in the typical incarnation of modular forms:

(1) One of the main focuses of number theory is Galois representations. That is, representations of the absolute Galois group $\Gamma$ of the rational numbers. It is hard in itself to motivate this object in an elementary way but let's just say knowing something about this profinite group should tell us something about number theory. Given a complex n-dimensional representation of $\Gamma$, one can attach a complex function to it, Artin's L-function. Artin's conjecture is that this L-function is entire.

Where automorphic forms come in is that sometimes (and I don't know the current state of the art) you can get a Galois representation from an automorphic one. The automorphic representation also has an L-function attached, and it is easier to prove that this is entire. The L-function of the automorphic representation and the Galois representation are the same, so this is one way of proving certain cases of Artin's conjecture.

Since you want to know about arbitrary n-dimensional Galois representations, you need arbitrary automorphic representations, not just for GL(2), where it reduces to the classical modular form case.

(2) Maybe you're interested in geometry and cycles on Shimura varieties? Then automorphic representations give you a way to study these cycles. In fact, you can use the techniques in [1] to produce nontrivial cycles on Shimura varieties. So then you might be able to say something about their cohomology. This needs automorphic representations for general unitary groups, not just classical ones. This is somewhat related to the first point as the Galois representations in this case in some sense are 'easier' than the general case. However, you might also be interested in Shimura varieties for more geometric reasons, and there is a lot of literature you can search out for on them in this regard.

[1] Getz, J. R. & Wambach, E. Twisted relative trace formulae with a view towards unitary groups Amer. J. Math., 2013, 135, 1-57; arXiv:1701.01762

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  • $\begingroup$ In (1), I guess you mean you can sometimes attach an automorphic representation to a Galois representation. Also, I think motivation can be stated in a fairly elementary way, e.g., Artin $L$-functions tell you about about how primes split ($\approx$ factor) in finite extensions. $\endgroup$ – Kimball Jan 28 '18 at 14:02
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Two other applications where automorphic forms (again: only some of them, so it is more a motivation for modular forms or for Maass forms than for automorphic forms) arise naturally are provided in Kowalski's article, Classical Automorphic Forms, in Bernstein-Gelbart volume.

Kloosterman Sums. Estimating cancellations arising in Kloosterman sums, for which best results are given by automorphic methods. For instance, it is used to address the problem of estimating the size of $L$-functions on the critical line, and more precisely associated moments $$\int_{-T}^T \left|L\left(\frac12+it, \pi\right)\right|^k dt$$

The strong implications of good bounds on those moments (e.g. Lindelöf) are a motivation for addressing the problem, and since automorphic forms are for now among the best way to address it, it is a motivation to automorphic forms.

Elliptic curves. Modular functions can be seen as meromorphic functions homogeneous on lattices, and for instance de Weierstrass function $\mathcal{P}$ give invariants determining the elliptic curve.

Again, for the reason stated above, these fail to totally answer my question (even for audience (D) I believe, for it is only a partial answer, or better: an answer to part of the question).

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For (possibly) "Outside motivations" (B): automorphic forms can be used to get ample divisors on moduli spaces. This was put to spectacular use by Davesh Maulik in his proof of the Tate conjecture for supersingular K3's for big primes.

See Maulik's article or a Bourbaki article of Benoist.

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