How is representation theory used in modular/automorphic forms?

Question

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.

EDIT: The answer of paul garett here actually gives a nice history of how modular forms came to be viewed in terms of representations theory: What is the difference between an automorphic form and a modular form?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

Answer 6

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.

Answer 7

As often seems to be the case, I'm late to this party. :)

But, perhaps, a few more items are worth noting, that put the question into a more general context, so that it's less a special thing dependent on definitions or style.

First, there are points to be made about "the adeles" not merely being a thing that we "define", and then insist that everyone use if they want to be conformist. Rather, $\mathbb A/\mathbb Q$ (and similarly for any global field) arises naturally as a (proj) lim of natural "more classical" objects, namely, $\mathbb R/n\mathbb Z$ where $n$ runs over positive integers partially ordered by divisibility. (Perhaps see my notes on "solenoids" for the obvious/natural details...) One big operational point is that, while, sure, $\mathbb R$ acts on each of those circles, by translation, the $p$-adic completions $\mathbb Q_p$ only act on the whole family, but/and then do act on that limit.

Similarly, although "lifting" to the group quotient $\Gamma\backslash SL_2(\mathbb R)$ shows that $G=SL_2(\mathbb R)$ acts on all of these modular curves, it's only in the projective limit over congruence subgroups $\Gamma_N$, to get $SL_2(\mathbb Q)\backslash SL_2(\mathbb A)$ (in effect giving modular/automorphic forms of all levels) that we get/see the right-translation action of all $SL_2(\mathbb Q_p)$. This is lots more structure! ... especially since, as it turns out, we can understand quite a lot about the repn theory of these groups.