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There is certainly an abundance of advanced books on Galois representations and automorphic forms. What I'm wondering is more simple: What is the basic connection between modular forms and representation theory?

I have a basic grounding in the complex analytic theory of modular forms (their dimension formulas, how they classify isomorphism classes of elliptic curves, some basic examples of level N modular forms and their relation to torsion points on elliptic curves, series expansions, theta functions, Hecke operators). This is all with an undergraduate background in complex analysis and algebra (Galois theory). I also know a little bit about the basics of algebraic number theory and algebraic geometry, if that helps. More importantly, I have a basic background in the representation theory of finite groups. My question is, then, could one example how modular forms and/or theta functions relate to representations of groups?

I'm asking this in part because I imagine a number of students with similar background as I have would have learned about modular forms and thus might be interested to understand how they relate to representation theory, despite not having an extensive background in more advanced results in algebraic geometry and commutative algebra needed for advanced study in the field.

Here are some ideas which might bear fruit: In analytic number theory, one often sees sums over characters - but characters are also very relevant in representation theory. In particular, Jacobi sums come up in both number theory and representation theory (and quadratic forms then relate to theta functions). Is there a connection here?

In addition, Hecke operators are symmetric-like sums over elements of groups, which would suggest a strong connection to representation theory.

Or is the connection to representations of $\mathrm{SL}_2(\mathbb{Z})$? Quotients of this group appear as Galois groups of extensions of spaces of modular forms, so they might be given representations by acting on these spaces?

The point of listing ideas is to show the kind of intuition I might be looking for. One of my ideas might be fruitful, or they all might have nothing to do with why representation theory connects to modular functions. The point is that I'm looking for basic ideas that someone with an elementary background might be able to understand.

I also added "reference request" because I imagine there might be a text which is at my level and discusses these ideas.

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Dear David: A hint of the interaction with rep'n theory is already seen in the fact that the classical upper half-plane is a coset space for the Lie group ${\rm{GL}}_2(\mathbf{R})$, and relation of C-R eqns with Casimir in Lie alg., but need a more adelic formulation to see how the Hecke theory comes out from the action of a group also. (Toy version: adelic formulation of Dirichlet chars, as in CFT.) So ultimately this is part of the richness and power of the Langlands Program. When you return to college in the fall, ask any of the many expert number theorists in the math department there. –  BCnrd Aug 15 '10 at 4:27
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@David: to help you make some sense of the answers below (I was going to answer but clearly I don't need to because you already have lots) let me just comment that there are several distinct links between modular forms and rep theory: first given a modular form you can build a rep of GL_2(adeles) [this is relatively formal], and the theory of auto forms generalises this to GL_n(adeles), but secondly Langlands' philosophy predicts links between modular forms and reps of Galois groups (or Weil groups, which are closely related) and this is relatively deep. Moonshine is a third link and also deep –  Kevin Buzzard Aug 15 '10 at 8:40
    
@Brian: "relation of C-R eqns with Casimir in Lie alg." Could you be more precise about this? Such as, giving a source or writing a small exposition? It is a pity that things like these don't occur in complex analysis courses. –  darij grinberg Aug 15 '10 at 10:03
    
Thanks! I do have basic familiarity with adeles, though I would imagine that other students interested in this question might not have such familiarity, which is why I did not add this. In fact, my main familiarity with them is in showing the equivalence of ideal-theoretic and idelic class field theory, which is precisely your "toy version," so I'm quite curious to eventually understand what you talked about. –  David Corwin Aug 15 '10 at 12:30
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@darij grinberg: briefly, what Brian is saying comes down to the fact that the Casimir operator is the analogue of the Laplacian. As you probably know, the real and imaginary parts of a holomorphic function are harmonic, i.e. killed by the usual Laplace operator. This gives an inkling of the connection. See Bump's book section 2.2 for example (as well as pages 129–130). –  Rob Harron Aug 15 '10 at 15:39

6 Answers 6

Caveat: in order to give you an overview, I've been vague/sloppy in several places.

Well the basic link to representation theory is that modular forms (and automorphic forms) can be viewed as functions in representation spaces of reductive groups. What I mean is the following: take for example a modular form, i.e. a function $f$ on the upper-half plane satisfying certain conditions. Since the upper-half plane is a quotient of $G=\mathrm{GL}(2,\mathbf{R})$, you can pull $f$ back to a function on $G$ (technically you massage it a bit, but this is the main idea) which will be invariant under a discrete subgroup $\Gamma$. Functions that look like this are called automorphic forms on $G$. The space all automorphic forms on $G$ is a representation of $G$ (via the right regular represenation, i.e. $(gf)(x)=f(xg)$). Basically, any irreducible subrepresentation of the space of automorphic forms is what is called an automorphic representation of $G$. So, modular forms can be viewed as certain vectors in certain (generally infinite-dimensional) representations of $G$. In this context, one can define the Hecke algebra of $G$ as the complex-valued $C^\infty$ functions on $G$ with compact support viewed as a ring under convolution. This is a substitute for the group ring that occurs in the representation theory of finite groups, i.e. the (possibly infinite-dimensional) group representations of $G$ should correspond to the (possibly infinite-dimensional) algebra representations of its Hecke algebra. This type of stuff is the basic connection of modular forms to representation theory and it goes back at least to Gelfand–Graev–Piatestkii-Shapiro's Representation theory and automorphic functions. You can replace $G$ with a general reductive group.

To get to more advanced stuff, you need to start viewing modular forms not just as functions on $\mathrm{GL}(2,\mathbf{R})$ but rather on $\mathrm{GL}(2,\mathbf{A})$, where $\mathbf{A}$ are the adeles of $\mathbf{Q}$. This is a "restricted direct product" of $\mathrm{GL}(2,\mathbf{R})$ and $\mathrm{GL}(2,\mathbf{Q}_p)$ for all primes $p$. Again you can define a Hecke algebra. It will break up into a "restricted tensor product" of the local Hecke algebras as $H=\otimes_v^\prime H_v$ where $v$ runs over all primes $p$ and $\infty$ ($\infty$ is the infinite prime and corresponds to $\mathbf{R}$). For a prime $p$, $H_p$ is the space of locally constant compact support complex-valued functions on the double-coset space $K\backslash\mathrm{GL}(2,\mathbf{Q}_p)/K$ where $K$ is the maximal compact subgroup $\mathrm{GL}(2,\mathbf{Z}_p)$. If you take something like the characteristic function of the double coset $KA_pK$ where $A_p$ is the matrix with $p$ and $1$ down the diagonal, and look at how to acts on a modular form you'll see that this is the Hecke operator $T_p$.

Then there's the connection with number theory. This is mostly encompassed under the phrase "Langlands program" and is a significantly more complicated beast than the above stuff. At least part of this started with Langlands classification of the admissible representation of real reductive groups. He noticed that he could phrase the parametrization of the admissible representations say of $\mathrm{GL}(n,\mathbf{R})$ in a way that made sense for $\mathrm{GL}(n,\mathbf{Q}_p)$. This sets up a (conjectural, though known now for $\mathrm{GL}(n)$) correspondence between admissible representations of $\mathrm{GL}(n,\mathbf{Q}_p)$ and certain $n$-dimensional representations of a group that's related to the absolute Galois group of $\mathbf{Q}_p$ (the Weil–Deligne group). This is called the Local Langlands Correspondence. The Global Langlands Correspondence is that a similar kind of relation holds between automorphic representations of $\mathrm{GL}(n,\mathbf{A})$ and $n$-dimensional representations of some group related to Galois group (the conjectural Langlands group). These correspondences should be nice in that things that happen on one side should correspond to things happening on the other. This fits into another part of the Langlands program which is the functoriality conjectures (really the correspondences are special cases). Basically, if you have two reductive groups $G$ and $H$ and a certain type of map from one to the other, then you should be able to transfer automorphic representations from one to the other. From this view point, the algebraic geometry side of the picture enters simply as the source for proving instances of the Langlands conjectures. Pretty much the only way to take an automorphic representation and prove that it has an associated Galois representation is to construct a geometric object whose cohomology has both an action of the Hecke algebra and the Galois group and decompose it into pieces and pick out the one you want.

As for suggestions on what to read, I found Gelbart's book Automorphic forms on adele groups pretty readable. This will get you through some of what I've written in the first two paragraphs for the group $\mathrm{GL}(2)$. The most comprehensive reference is the Corvallis proceedings available freely at ams.org. To get into the Langlands program there's the book an introduction to the Langlands program (google books) you could look at. It's really a vast subject and I didn't learn from any one or few sources. But hopefully what I've written has helped you out a bit. I think I need to go to bed now. G'night.

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One question: What kind of math do you need to know to do these books? It seems like some kind of group theory. Is it Lie theory (and if so what aspects of it)? Or is it Fourier analysis on groups? Would a course on measure theory teach me what I need to know (or at least be a necessary prerequisite)? Or is it just that I need to learn some more algebraic geometry? –  David Corwin Aug 15 '10 at 12:32
    
Also, when you say "not just as functions on GL2(R) but rather on GL2(A)," does that mean that given a modular form (in the standard sense), there is a corresponding function on GL2(A)? Or does it just mean that modular forms for adeles and modular forms for the complex numbers are slightly different beasts? –  David Corwin Aug 15 '10 at 12:51
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To answer the question in your second comment I'll say this: if Gamma is a congruence subgroup of SL(2,Z), then there is an open compact subgroup K of $GL(2,A_f)$ (where $A_f$ are the finite adeles, i.e. Zhat tensor Q) such that $GL(2,Q)\backslash GL(2,A)/(R^\times O(2,R)K)$ is basically the upper-half plane modulo Gamma. In this way, you can take a modular form and construct an associated automorphic form as a function on $GL(2,A)$. For details, I'd refer you to any one of the books I suggested. –  Rob Harron Aug 15 '10 at 15:15
    
As for your comment: there's a lot of background and while I am all for learning background first, in this situation there's just so much that you might want to learn background as you go along. I suggest you take a look at Borel's article Introduction to automorphic forms in the Boulder conference (available freely at ams.org/publications/online-books/pspum9-index). It'll give you a bit of an idea of the type of thing that goes into simply the definition of an automorphic form. If you were to pick up Bump's book Automorphic forms and representations he'll go over some background. –  Rob Harron Aug 15 '10 at 15:23
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But yes, to get a good understanding of the basics Lie theory is required (e.g. knowing why the universal enveloping algebra of the Lie algebra acts as differential operators on functions on the group). But also the representation theory of reductive Lie groups (both the finite-dimensional case and the infinite-dimensional case, at least the "admissible" case). And if you want to get into the whole automorphic representations on adeles groups then some knowledge of algebraic groups and representations of reductive algebraic groups. Somehow one manages to understand a bit without knowing all. –  Rob Harron Aug 15 '10 at 15:30

A one line answer:
A modular form is a highest weight vector of a discrete series summand of L2(SL2(Z)\SL2(R)).

There are numerous variations of this: one can replace the reals by the adeles, or SL2 by another group, or replace discrete series representations by principal series to get Maass wave forms, and so on. This is explained in detail in Automorphic forms on adele groups by Gelbart. An introduction to the Langlands program by Bernstein and others is also good.

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In your one line answer "modular form" should be "cusp form of weight $\geq 2$", right? –  Rob Harron Aug 15 '10 at 21:20
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Yes, plus other conditions such as level 1, but this would have taken up more than 1 line. –  Richard Borcherds Aug 15 '10 at 23:15

A short answer is that given a modular form $f$ for a congruence subgroup of $SL_2(\mathbb{Z}),$ one can lift $f$ to a function $F$ on the adelic group $G=GL_2(\mathbb{A})$ with certain properties that generates a representation $\pi$ of $G$ under the left regular action. Various properties of $f$ translate into properties of $\pi;$ conversely, representation theoretic techniques applied to $\pi$ yield information about $f.$ For example,

  $f$ is a Hecke eigenform $\iff \pi$ is irreducible.

In the case of theta functions in $g$ variables, they can be lifted to the appropriate symplectic group $Sp_{2g}(\mathbb{A})$ and become matrix coefficients of the Weil representation. Here, too, the results from representation theory can be translated back into information about theta functions.

Here are two fairly old books that explain and exploit representation theory behind the theory of theta functions and automorphic forms neither assuming nor using algebraic geometry and commutative algebra in a serious way:

  1. Gelfand, Graev, Piatetskii-Shapiro, Representation theory and automorphic functions. Generalized Functions, 6. Academic Press, 1990 (original edition published in Russian in 1966)

  2. Lion, Vergne, The Weil representation, Maslov index and theta series. Progress in Mathematics, 6. Birkhäuser, 1980

In fact, while recently the role of Galois representations has been highlighted (Langlands program, modularity theorem), this is an entirely separate and higher level issue compared with the basic dictionary between modular forms and automorphic representations. Thus most books on automorphic forms (e.g. Bump or Goldfeld) will explain the latter, without necessarily touching on the former.

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a) Do these books require knowledge of several complex variables? b) I think what I'm more interested in is precisely "the basic dictionary between modular forms and automorphic representations." –  David Corwin Aug 15 '10 at 4:22
    
And if these books require knowledge of several complex variables and/or other areas an undergraduate like myself would not be expected to know, are there any books which don't? –  David Corwin Aug 15 '10 at 4:22
    
G-G-PS book is about automorphic forms on $GL_2,$ so there is only one complex variable to worry about (coming from identifying the upper half plane with a homogeneous space for $PGL_2$), but in fact, complex analysis plays almost no role. It's impossible to determine ahead of time whether you know enough to fully understand these books (certainly, functional analysis would be helpful to know), but the good news is that you can start right away and pick up some pieces as you go. You can read the dictionary in many places, but I suggest working through the proofs, at least for $PGL_2.$ –  Victor Protsak Aug 15 '10 at 5:40

Since you mentioned Galois representations, I can briefly discuss the simplest version of the connection there and point you to Diamond and Shurman's excellent book which discusses modular forms with an aim towards this perspective.

The connection here is to representations of the absolute Galois group $G = \text{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. By the Kronecker-Weber theorem, one-dimensional (continuous, complex) representations of $G$ are classified by Dirichlet characters, so it is natural to ask about the next hardest case, the two-dimensional representations. A large class of them can be constructed as follows. Given an elliptic curve $E$ defined over $\mathbb{Q}$, the elements of order $n$ (hereby designated by $E[n]$) form a group isomorphic to $(\mathbb{Z}/n\mathbb{Z})^2$, and since their coordinates are algebraic numbers, $G$ acts on them. This gives a representation

$$G \to \text{GL}_2(\mathbb{Z}/n\mathbb{Z}).$$

As is, this representation causes problems because $\mathbb{Z}/n\mathbb{Z}$ isn't an integral domain. So what we do is we take $n$ to be all the powers of $\ell$ for a fixed prime $\ell$ and take the inverse limit over all the corresponding $E[\ell^n]$. The result is a gadget called a Tate module, which is a $G$-module isomorphic (as an abstract group) to $\mathbb{Z}_{\ell}^2$, and which therefore defines a representation

$$G \to \text{GL}_2(\mathbb{Z}_{\ell}).$$

So how does one identify the representation corresponding to $E$? The standard answer is to look at certain ("conjugacy classes" of) elements of $G$ called Frobenius elements, which come from lifts of Frobenius morphisms. Although Frobenius elements aren't always well-defined, it turns out that the trace $a_{p,E}$ of the Frobenius element corresponding to $p$ in a representation is, and so we can identify a representation by giving the numbers $a_p$ for all $p$. (I am not really familiar with the details here, but I believe this works because Frobenius elements are dense in $G$.) It turns out that if $p$ is a prime of good reduction, $a_{p,E} = p + 1 - |E(\mathbb{F}_p)|$, so these numbers can actually be obtained in a fairly concrete manner (where $E(\mathbb{F}_p)$ is the set of points of $E$ over $\mathbb{F}_p$). (Again, I am not really familiar with the details here, including what happens when $p$ doesn't have good reduction.)

Now: one statement of the modularity theorem, formerly the Taniyama-Shimura conjecture, is that there exists a cusp eigenform $f$ of weight $2$ for $\Gamma_0(N)$ for some $N$ (called the conductor of $E$) such that, whenever $p$ is a prime of good reduction,

$$a_{p, f} = a_{p, E}$$

where $a_{p, f}$ is the $p^{th}$ Fourier coefficient of $f$. In other words, cusp eigenforms of weight $2$ "are the same thing as" a large class of two-dimensional representations of $G$. The Langlands program is at least in part about generalizations of this statement to higher-dimensional representations of $G$, but there are many qualified number theorists here who can tell you what this is all about.

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You might also be interested in this survey article by Darmon, Diamond, and Taylor: math.mcgill.ca/darmon/pub/Articles/Expository/05.DDT/paper.pdf –  Qiaochu Yuan Aug 15 '10 at 9:32

Probably the most notable example is monstrous moonshine. See Terry Gannon's book.

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Thanks, though I also imagine there are other, more direct examples. –  David Corwin Aug 15 '10 at 4:04
    
Actually, that book does seem particularly good as an answer to this question. Section 2.4 seems to begin to get to the heart of my question. –  David Corwin Aug 15 '10 at 4:19
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Dear David: A nice concrete example arises when considering the action of ${\rm{SL}}_2(\mathbf{Z}/(p))$ on the space of full level-$p$ modular forms of some fixed weight, and its interaction with Hecke operators at primes away from $p$ as well as viewing various newforms as generating vectors of irreducible subrepresentations. This is a finite-level "shadow" of the rich interaction of Hecke theory and the representation theory of adelic groups, a perspective which allows one to "localize" problems with modular forms in a manner that can be difficult to express in purely classical terms. –  BCnrd Aug 15 '10 at 4:31
    
You might also be interested in chapter 7 of An Introduction to the Langlands Program: books.google.com/books?id=x3XR0ljIV6YC&pg=PA133 –  Steve Huntsman Aug 15 '10 at 4:34
  1. Observation: The modular forms/automorphic representations should be seen as generalization of Hecke quasi characters, which are simply characters $\chi: K^{\times }\backslash \mathbb{A}_K^{\times} \rightarrow \mathbb{C}^{\times}$. Tate's thesis gives a proof of the functional equation for the associated L-functions purely in terms of representation theory. He studies the right regular representation on the space of some functions $f:K^{\times }\backslash \mathbb{A} \rightarrow \mathbb{C}$. This is the so called $\mathrm{GL}_1$ case, since $\mathrm{GL}_1(K)=K^{\times}$. The next obvious choice is then to consider other reductive groups instead of $\mathrm{GL}_1$ e.g. $\mathrm{GL}_2$. One remark how I think about Tate's thesis: The right regular representation on an (locally compact) abelian groups is in direct connection with its Fourier transform. The functional equation can be seen as an adelic version of the Poisson summation formula.

  2. Observation: The special linear group $\mathrm{SL}_2(\mathbb{R})$ acts on upper half plane $\mathbb{H}$ by Moebius transformations. Moreover this group is actually the group of all biholomorphic mappings $\mathbb{H} \rightarrow \mathbb{H}$. The isotropy group of $\mathrm{i}$ is $\mathrm{SO}_2$, i.e. the group of elements, which fix $\mathrm{i}$. Since the action is associative, i.e. for any $x, y \in \mathbb{H}$ there exists $g \in \mathrm{SL}_2(\mathbb{R})$ such that $gx = y$, we get an isomorphism of the orbit space with the space we act on. Hence, we actually have $\mathbb{H} \cong \mathrm{SL}_2(\mathbb{R}) /\mathrm{SO}_2$.

  3. Observation You can lift through weak approximation Dirichlet character $\chi : (\mathbb{Z}/n \mathbb{Z})^{\times} \rightarrow \mathbb{C}^{\times}$ to the adele space $K^{\times }\backslash \mathbb{A}_K^{\times}$. You can proceed similiar with strong approximation and obtain that the double coset space $ SL_2(\mathbb{Q}) \backslash SL_2( \mathbb{A} ) / K $ is actually isomorphic to the orbit space $\mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}.$ Here $K$ means the product of all $\mathrm{SL}_2(\mathbb{Z}_p)$ for the finite places and $\mathrm{SO}_2$ for the archimedian place.

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Wait, how does a Hecke character give a modular form? I understand that Hecke characters relate to adeles, but you seem to be implying that Hecke characters lifting to characters on adeles in the first example of a classical modular form becoming a function on adeles. But I think your comment #3 is implying that it's in analogy with L-functions of Hecke characters, not that the latter are actually automorphic forms of some kind. Is that true? –  David Corwin Nov 12 '10 at 5:15
    
At David: I said Hecke quasi characters are the modular forms of $\mathrm{GL}_1$ and the classical modular forms are associated to $\mathrm{SL}_2$. You should not see modular forms or Hecke characters as functions on these spaces. The right point of view is that the right regular representation decomposes into irreducible representations. In the case of $\mathrm{GL}_1$, these irreducible representations are in one-to-one correspondance with Hecke characters (or better holomorphic one parameter families of Hecke quasi characters). –  plusepsilon.de Nov 12 '10 at 6:46
    
urggh, replace associative by transitive –  plusepsilon.de Dec 7 '12 at 11:58

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