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Well, I'm aware that this question may seem very naive to the several experts on this topic that populate this site: feel free to add the "soft question" tag if you want... So, knowing nothing about modular forms (except they're intrinsically sections of powers of the canonical bundle over some moduli space of elliptic curves, and transcendentally differentials on the upper half plane invariant w.r.t. some specific subgroup of $SL(2,\mathbb{Z})$), I have the curiosity -that many other non experts might have- to understand a bit why that is considered a so vast and important topic in mathematics. The wikipedia page doesn't help: on the contrary, it makes this topic appear as quite narrow and merely technical.

I would roughly divide the question into three (though maybe not neatly distinct) parts:

1) Why are modular forms per se interesting?

That is, do they "generate" some piece of rich self-contained mathematics? To make an analogy: cohomology functors were born as applied tools for studying spaces, but have then evolved to a very rich theory in itself; can the same be said about m.f.'s?

2) How are modular forms deeply related to other, possibly quite distant, mathematical areas?

For example: I've heard about deep relations to some generalized cohomology theories (elliptic cohomology) via formal group laws coming from elliptic curves; and I've heard about the so called moonshine conjecture; there should also be some more classical relations to the theory of integral quadratic forms and diophantine equations, and of course to elliptic curves; and people here always mention Galois representations...

3) Why are modular forms useful as "applied" technical tools?

In this last question I'm ideally expecting indications of cases (or actual theorems) in which some questions that do not involve modular forms are asked about some mathematical objects, and an answer that does not involve m.f.'s is given, but the method used to obtain that answer/proof makes consistent use of m.f.'s.

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The proof of Fermat's last theorem, as an example of (3)? –  David Loeffler May 14 '10 at 11:16
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Certainly more than one. :) –  Qfwfq May 14 '10 at 12:52
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unknown, you chose a particularly unfortunate exponent of 2 in your "illustration" of the Fermat equation (I tend to think z^2 is x^2 + y^2 a fair bit of the time...) –  KConrad May 14 '10 at 12:56
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Ahah! Oh mygod! Maybe Freud, rather than modular forms, could somehow explain that lapsus! :D –  Qfwfq May 14 '10 at 13:09
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I hope the experts don't mind if I add one comment to all these excellent answers. Why are modular forms interesting? We don't quite know...that's why they're so interesting! –  Minhyong Kim May 14 '10 at 17:22

10 Answers 10

Your questions would require an enormous amount of work to answer properly, so let me just suggest a few modest and very partial answers to your 1)2)3).

1) Modular forms are shiny: they satisfy or explain many beautiful and surprising numerical identities (about partitions and sums of square among others). This got them noticed in the first place.

2) Modular forms have Galois representations, and conversely Galois representations often come from modular forms. If you care at all about representations of the absolute Galois group of $\mathbb Q$, then you will first presumably be interested in class field theory, and develop the Kronecker-Weber theorem. But then you will get interested in representations of $G_{\mathbb Q}$ of rank 2. Modular forms provide many examples of such Galois representations, and conversely, only a handful of hypotheses are required for such a Galois representation to come from a modular form. This means concretely that one can identify many Galois representations simply by computing a few traces of Frobenius morphisms and then doing some computations in the complex upper half-plane.

3) If a rational elliptic curve has a non-vanishing $L$-function at 1, it has no non-torsion rational points. The main conjecture of Iwasawa (about class groups in the cyclotomic $\mathbb Z_{p}$-extension of $\mathbb Q$) is true. Fermat's last theorem is true. Here are three extremely famous conjectures solved by an ubiquitous appeal to modular forms. All these conjectures were well-known in the 60s but I don't think it is an exaggeration to say that no one would have suspected that modular forms would come into play.

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This answer addresses one striking example for the second question: how modular forms relate to other mathematical topics.

Values of zeta-functions can arise as the constant terms of nice modular forms, suitably normalized, and this lets knowledge of the higher degree Fourier coefficients impart knowledge to the user about the arithmetic nature of the constant terms that are of more direct interest.

For example, there are modular forms whose constant term is the value of the zeta-function of a totally real number field at a negative integer. It's a mysterious number but the higher-degree coefficients of the modular form are more down-to-earth things (something like sums of powers of divisors of integers) and in particular are visibly rational numbers. Spaces of modular forms are finite-dimensional while at the same time each modular form has infinitely many Fourier coefficients, which introduces a lot of redundancy, so knowing enough coefficients in a sense determines what all the others must be. In particular, if the higher-degree Fourier coefficients are rational then the constant term is also rational. Applying this fact to the specially constructed modular forms with zeta-values as constant terms proves that the zeta-values are rational. This is how Klingen and Siegel proved zeta-functions of totally real number fields are rational at negative integers.

Of course this leaves out the issue of how you find modular forms with zeta-values as their constant term in a natural way. It requires experience and cleverness. Classical examples of modular forms (Eisenstein series on ${\rm SL}_2({\mathbf Z})$) have values of the Riemann zeta-function as the constant term, even though people were not originally looking at them for that reason; it just happened to show up. Then you try to generalize that construction to get values of other zeta-functions as the constant terms.

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I loved this post! Thanks. –  Unknown May 15 '10 at 13:02

As an answer to (2): I think the easiest to state application of modular forms to a domain not obviously connected to them is in the theory of lattices.

A lattice is a discrete subgroup of $\mathbb{R}^n$ that is generated by a basis of $\mathbb{R}^n$. The unimodular lattices are the ones with fundamental domains of volume 1. The even lattices are the ones where the norms of the elements are all even integers (with the usual Euclidean norm on $\mathbb{R}^n$, except in this field it's conventional to use the word norm to refer to the length squared).

Given a lattice L, write $a_n$ for the number of vectors in the lattice of norm $n$. We can form a generating function $\Theta_L(q)=\sum_n a_nq^n$.

Now comes the surprise: if $L$ is an even unimodular lattice, then $\Theta_L(\exp(\pi i \tau))$, as a function of $\tau$, is a modular form of weight $n/2$. As mentioned in another post, spaces of modular forms of given weight are finite dimensional. So we now have a vast amount of information at our disposal on the possible norms of lattice vectors.

This gives some beautiful relationships between well known modular forms and lattices like $E_8$ and the 24-dimensional Leech lattice. It's neat how the geometrical problem of finding pockets of space into which you can squeeze lattice elements can translate into discovering modular forms. (And of course the appearance of the Leech lattice hints at Monstrous Moonshine.)

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One can generalize this relationship to lattices which are not necessarily even, but one has to take more general kinds of modular forms. Probably the most classical application of this idea is the formula for the number of ways to represent a positive integer as the sum of four squares (one writes down the theta function of Z^4 and then relates it to other modular forms of the same weight and congruence subgroup). –  Qiaochu Yuan May 14 '10 at 17:59

A super short answer (I haven't had enough coffee yet this morning to expand): The space of modular forms of a given weight and level form a finite dimensional vector space. This allows one to easilty derive interesting (e.g. combinatorial) relations between their Fourier coefficients, which often are formed from a generating series or encode otherwise meaningful information.

A super short example of why modular forms are cool: using elliptic modular forms and Heegner points gives the (only) method for computing non-torsion rational points on elliptic curves $E/\mathbb{Q}$.

A super great chapter on modular forms: Chapter 1-Elliptic modular forms and their applications- by Zagier in `The 1-2-3 of Modular forms'. It can be read in an evening and answers more of your questions very nicely.

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What do you mean by "the (only) method"? The most naive approach is to try all rationals up to the height bound in Lang's conjecture (on heights of generators). –  Dror Speiser May 14 '10 at 13:47
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Dror, I don't know if I count that as "computing" - I think Kim's point, no pun intended, is that Heegner point constructions yield a priori rational points with no fiddling or search. –  David Hansen May 14 '10 at 14:02
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The comment on Zagier's chapter is seconded! –  Daniel Larsson May 14 '10 at 20:36
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@David: in that case there's the even more naive method of choosing the coefficients of the curve to contain a predetermined point... –  Dror Speiser May 14 '10 at 22:18
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Heegner points only work for (analytic) rank 1 to find points, largely over Q, and still take time linear in the conductor. For the "visible" range of Cremona's database (as an initial segment of the infinite), the simple searches and descents are now more practical, even after one applies the many tricks (Atkin-Lehner and more) to the Heegner protocol. I would guess that for about 5% of the curves of Cremona (with points) it were found by Heegner methods. What I think you mean is that the Heegner construction is the method that is //proven// to work, thanks to the formula of Gross and Zagier. –  Junkie May 15 '10 at 4:22

As for the second question, there is a conjectural relation between automorphic objects and number theory - the right keyword to search for is "Langlands philosophy".

Since I not know understand enough to explain the general setting, let me shortly summarize the connection between elliptic curves and modular forms. In this special case the conjectures are no conjectures anymore but this is what was proved by Wiles, Taylor...

So let $E: Y^2=X^3+aX+b$ be an elliptic curve with integral coefficients. Then for all but finitely primes, the curve is also non-singular over the field $\mathbb{F}_p$ and hence also an elliptic curve over $\mathbb{F}_p$. By work of Hasse it is known that the number of points on $E$ over $\mathbb{F}_p$ is $p+1-a_p(E)$ and $|a_p(E)|\leq 2\sqrt{p}$.

Now the big step which was realised by Wiles et al was to connect the numbers $a_p(E)$ to modular forms:

For any elliptic curve $E$ there is a modular form $f$ with the following Fourier expansion $$ f(\tau)=\sum_{n=1}^\infty a_n e^{2 \pi i n \tau}$$ where $a_p$ is just the number $a_p(E)$ defined above for the elliptic curve and the coefficients $a_n$ are multiplicative, i.e. we know them all if we know $a_p$ for primes $p$.

Why is this important? For example consider the $L$-function associated to the elliptic curve. This is essentially (omitting the bad primes) $$ L_E(s)=\prod_{p \ \text{prime}}(1-a_p(E)p^{-s}+p^{1-2s})^{-1} $$.

Now, for example the Birch and Swinnerton-Dyer conjecture makes a statement about this function at $s=1$. But the product representation is convergent only for $\Re s > \frac{3}{2}$, so we first need to prove analytic continuation for the BSD conjecture to make sense.

As far as I know, the only known method to prove this, is to use the theorem above. The fact that coefficients in the $L$-function correspond to a modular form, make it possible to prove analytic continuation as was first proved by Hecke.

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That big step wasn't exactly "realized" by Wiles but was a known conjecture to everyone working in the area. That ell. curves and modular forms could be related was suggested earlier essentially by the similar functional equations of their L-functions. –  KConrad May 14 '10 at 14:40
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It may sound slightly puzzling to some people when you say this (recent) result is "the only known method" of proof of analytic continuation while at the same time you say the analytic continuation was "first proved" by Hecke (long ago). Hecke worked on the special case of elliptic curves with complex multiplication, which can be approached by special methods. Also, the BSD conj. was first discovered numerically on CM ell. curves, where Hecke's work gave the basis for doing experiments when non-CM cases were out of reach. The conj., once found, was naturally extended to the non-CM case. –  KConrad May 14 '10 at 14:44
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"Fearless Symmetry" by Ash and Gross is a great popular account of some of these ideas. –  Qiaochu Yuan May 14 '10 at 17:56
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Let me also briefly mention that a_p(E) is the trace of the corresponding Frobenius element(s) acting on a 2-dimensional Galois representation attached to E coming from its points of finite order, so one can think of the modular forms appearing in the above theorem as analytic objects which encode number-theoretic information about the absolute Galois group of Q. (This should be contrasted to the situation with 1-dimensional Galois representations, where by Kronecker-Weber all you get are Dirichlet characters and the behavior of Frobenius elements is really simple by comparison.) –  Qiaochu Yuan May 14 '10 at 18:10
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@KConrad: I admit that the conjecture was known to the everyone working in the area and that the functional equation for the L-function of the elliptic curve was shown to hold, only if it came from a modular from: However, it was Wiles,Breuil,Conrad,Diamond and Taylor in the general case,or am I mistaken. Analytic continuation for modular L functions was proved a long time ago. But isn't it correct that there is only one method known at the moment for proving the functional equation for L functions of elliptic curves in general (not only for CM elliptic curves) namely the modularity theorem –  wood May 15 '10 at 2:52

Lots of good answers so far, and I hope that someone talks about Moonshine at some point. Let me just briefly mention an application which hasn't been mentioned so far to class field theory. Special values of the $j$-invariant generate Hilbert class fields of imaginary quadratic fields; the connection is that the $j$-invariant parameterizes elliptic curves over $\mathbb{C}$, which look like $\mathbb{C}/\Lambda$ for some lattice $\Lambda$, so one looks for lattices $\Lambda$ which are acted on by the ring of integers in an imaginary quadratic field and magic happens.

There's also a nice list of applications in Stein's new book on the computational side of the subject. The application to constructing Ramanujan graphs is one of the wackier ones; see the Ramanujan-Petersson conjecture.

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In this list, for instance, Stein fails to mention equidistribution results. Sarnak's book (1990) "Some applications of modular forms" has these (chapter 2), and includes the "wackier" Ramanujan graphs (chapter 3) too. Also, denoting Stein's book as "new" made me think he had something more recent than Feb 2007. Sigh. –  Junkie May 15 '10 at 10:40

Bryan Birch's view is that they form a bottomless area for research problems. All answers to the question fall into two types: showing examples of why this is true, and asking why it should be true. G. H. Hardy by temperament seems to have been convinced of the first part by the tau-function, while banning reference to "modular forms" in the general theory of Hecke, which was the 1930s answer to the second part. This all came round again in the 1960s, with the BS-D conjecture on the one hand and Langlands on the other. Most answers to why "special functions" are special are question-begging, because intrinsic interest in mathematics can't really be faked, and "symmetry" as an answer doesn't have a definitive formulation. It looks like the Shimura variety concept will have a big explanatory value in future mathematics, but we can't anticipate the real answers.

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I quite like the following specific result, confirming a conjecture of Gross:

Theorem: For every prime $p$, there is a finite non-solvable Galois extension of $\mathbf{Q}$ ramified only at $p$.

For $p\geq 11$ this is not so hard, but for smaller primes the only known examples come via the Galois representations associated with modular forms. See e.g. this paper on the case $p=2$, which uses Hilbert modular forms!

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2),3) A further application of modular forms that relates two quite distant mathematical fields arises in string theory. There the problem of understanding spacetime geometry (more precisely Calabi-Yau varieties) as a derived mathematical object leads to an attempt to relate modular forms that appear in the context of Kac-Moody algebras to motivic modular forms.

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I agree with the leading poster. One can't really answer this in a few paragraphs. To really get a complete answer you have to study modular forms in detail and you will see why they are so enchanting. I will give my partial answers to questions 1,2.

(Answer to Question 1) They are interesting for two reasons. First, a modular form satisfies so many functional identities their existence almost seems unreal. There is so much structure involved with Modular forms that one can prove beautiful results. Second, modular forms are deeply connected with several number theoretic objects.

(Answer to Question 2) The easiest application of modular forms to understand is in classic analytic number theory. Modular forms often act as generating functions for several interesting arithmetic functions. I will give three examples: 1) the eta function almost generates for the number of integer partitions of $n$, 2) Powers of theta functions generate the number of ways a number can be written as a sum of squares, 3)Eisenstein series generate weighted divisor functions.

Knowing these generating functions are modular allows for extremely precise approximation of such generating function, and by Fourier analysis, we can obtain information about the behavior of the arithmetical function you are studying.

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"First, a modular form satisfies so many functional identities their existence almost seems unreal." This is an argument I have already heard several times. But I don't find it convincing. Why should it come as a surprise that a holomorphic form on the poincare half-plane satisfies one functional equation for every $\gamma \in SL_2(\mathbb{Z})$? After all, $f(x)=e^{2 i \pi x}$ satisfies infinitely many functional equations, $f(x+n)=f(x)$, one for each $n \in \Z$. –  Joël Nov 2 '11 at 22:19
    
Jo\"el, of course, exponentials are the simplest automorphic form :) Though maybe saying "so many complicated functional identities" is more appropriate. Depending on how you emphasize it, I think it is a good point, since, after all, having a reductive group generate your function's symmetries turns out to be a wonderfully restrictive condition! –  B R Nov 3 '11 at 0:35

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