The Theta Correspondence

July 1, 2022July 10, 2022 / Anton Hilado / Leave a comment

In Siegel modular forms, we mentioned that one could construct Siegel modular forms from elliptic modular forms (see Modular Forms) via a process called “lifting”. In this post, we discuss a more general method that produces new automorphic forms (which generalize modular forms, and are also more properly a part of representation theory, see also Automorphic Forms) out of old ones. There is also a local version that deals with representations of p-adic Lie groups. Both of these form the theory of the (global and local) theta correspondence.

We begin with the local theory. Let $F$ be a nonarchimedean local field of characteristic zero (e.g. $\mathbb{Q}_{p}$ or a finite extension of $\mathbb{Q}_{p}$ ). Let $E$ be a quadratic etale $F$ -algebra. Let $V$ be a vector space over $E$ equipped with a Hermitian form $\langle-,-\rangle_{V}$ , and let $W$ be a vector space over $E$ equipped with a skew-Hermitian form $\langle -,-\rangle_{W}$ . Their respective groups of isometries are the unitary groups $\mathrm{U}(V)$ and $\mathrm{U}(W)$ . These two groups form an example of a reductive dual pair. The theory of the local theta correspondence relates representations of one of these groups to representations of the other.

Now the tensor product $V\otimes_{E} W$ can be viewed as a vector space over $F$ and we can equip it with a symplectic form $(-,-)=\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle -,-\rangle_{W})$ . We have a map

$\displaystyle \mathrm{U}(V)\times\mathrm{U}(W)\to\mathrm{Sp}(V\otimes_{E} W)$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ has a cover called the metaplectic group; we will describe it in more detail shortly, but we say now that the importance of it for our purposes is that it has a special representation called the Weil representation, which as we shall shortly see will be useful in relating representations of $\mathrm{U}(V)$ to $\mathrm{U}(W)$ , and vice-versa.

We first need to construct the Heisenberg group $H(V\otimes_{E} W)$ . Its elements are given by $(V\otimes_{E} W)\oplus F$ , and we give it the group structure

$\displaystyle (x_{1},t_{1})\cdot (x_{2},t_{2})=\left(x_{1}+x_{2},t_{1}+t_{2}+\frac{1}{2}(x_{1},x_{2})\right)$

The Stone-von Neumann theorem tells us that, for every nontrivial character $\psi:F\to\mathbb{C}^{\times}$ the Heisenberg group has a unique irreducible representation $\omega_{\psi}$ with central character $\psi$ . Furthermore, the representation $\omega_{\psi}$ is unitary.

If $V\otimes_{E} W=X\oplus Y$ is a Lagrangian decomposition, we can realize the representation $\omega_{\psi}$ on the vector space of Schwarz functions on either $X$ or $Y$ . Let us take it to be $Y$ . In particular, we can express $\omega_{\psi}$ as follows. We first extend the character $\psi$ to $H(X)$ (defined to be the subgroup $X\oplus F$ of $H(V\otimes_{E}W)$ ) and then define $\omega_{\psi}$ as the induced representation (see also The Local Langlands Correspondence for General Linear Groups for another example of an induced representation)

$\displaystyle \omega_{\psi}=\mathrm{c-Ind}_{H(X)}^{H(V\otimes_{E}W)}\psi$

The symplectic group $\mathrm{Sp}(V\otimes_{E} W)$ acts on the Heisenberg group $H(V\otimes_{E}W)$ by $g\cdot (x,t)=(g\cdot x,t)$ for $g\in \mathrm{Sp}(V\otimes_{E} W)$ and $(x,t)\in H(V\otimes_{E}W)$ . We can compose this action with the representation $\omega_{\psi}$ to get another representation ${}^{g}\omega_{\psi}=\omega_{\psi}\circ g^{-1}$ of $H(W)$ . Now since the action of $\mathrm{Sp}(V\otimes_{E} W)$ on $H(V\otimes_{E}W)$ has trivial center, the central characters of ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ are the same. By the Stone-von Neumann theorem, ${}^{g}\omega_{\psi}$ and $\omega_{\psi}$ have to be isomorphic.

What this means now, is that for every $g\in \mathrm{Sp}(V\otimes_{E}W)$ , we have a linear transformation $A_{\psi}(g)$ of the underlying vector space $\mathcal{S}$ of the representation $\omega_{\psi}$ , so that

$\displaystyle A_{\psi}(g)\circ {}^{g}\omega_{\psi}=\omega_{\psi}\circ A_{\psi}(g)$

This action however is only defined up to a factor of $\mathbb{C}^{\times}$ . Since $\omega_{\psi}$ is unitary, we can also require $A_{\psi}$ to be unitary, and so the action becomes well-defined up to $S^{1}$ . All in all, this means that we have a representation

$\displaystyle A_{\psi}:\mathrm{Sp}(V\otimes W)\to \mathrm{GL}(\mathcal{S})/S^{1}$

Now if we pull back the map $\mathrm{GL}(\mathcal{S})\to\mathrm{GL}(\mathcal{S})/S^{1}$ by the map $A_{\psi}:\mathrm{Sp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})/S^{1}$ , we get a map $\widetilde{A}_{\psi}:\mathrm{Mp}(V\otimes_{E} W)\to \mathrm{GL}(\mathcal{S})$ , where the group $\mathrm{Mp}(V\otimes_{E} W)$ is an $S^{1}$ -cover of $\mathrm{Sp}(V\otimes_{E} W)$ . This group $\mathrm{Mp}(V\otimes_{E} W)$ is the metaplectic group mentioned earlier.

Our construction allows us to extend the representation $\omega_{\psi}$ of $H(V\otimes_{E }W)$ to the semidirect product $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ . This representation of $\mathrm{Mp}(V\otimes_{E} W)\ltimes H(V\otimes_{E} W)$ is called the Heisenberg-Weil representation. The representation of $\mathrm{Mp}(V\otimes_{E} W)$ obtained by restriction is called the Weil representation.

Recall that we have a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Sp}(V\otimes_{E}W)$ . If we could lift this to a map $\mathrm{U}(V)\times \mathrm{U}(W)\to \mathrm{Mp}(V\otimes_{E}W)$ , then we could obtain a representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by restricting the Weil representation $\omega_{\psi}$ from $\mathrm{Mp}(V\otimes_{E}W)$ to $\mathrm{U}(V)\times \mathrm{U}(W)$ . It turns out such a lifting can be defined and is determined by a pair $(\chi_{V},\chi_{W})$ of characters of $E^{\times}$ satisfying certain conditions. Once we have this lifting, we denote the resulting representation of $\mathrm{U}(V)\times \mathrm{U}(W)$ by $\Omega$ .

Now let $\pi$ be an irreducible representation of $V$ . We consider the maximal $\pi$ -isotypic quotient of $\Omega$ , which is its quotient by the intersection of all the kernels of morphisms of representations of $U(V)$ from $\Omega$ to $\pi$ . This quotient is of the form $\pi\otimes\theta(\pi)$ , where $\Theta(\pi)$ is a representation of $U(W)$ called the big theta lift of $\pi$ . The maximal semisimple quotient of $\Theta(\pi)$ is denoted $\theta(\pi)$ , and is called the small theta lift of $\pi$ .

Let us now look at the global picture. Let $k$ be a number field and let $k_{v}$ be the completion of $k$ at one of its places $v$ . Let $E$ be a quadratic extension of $k$ . Now we let $V$ and $W$ be vector spaces over $E$ equipped with Hermitian and skew-Hermitian forms $\langle-,-\rangle_{B}$ and $\langle--\rangle_{W}$ , as in the local case, and consider the tensor product $V\otimes_{E} W$ as a vector space over $k$ , and equip it with the symplectic form $\mathrm{Tr}(\langle-,-\rangle_{V}\otimes\langle-,-\rangle_{W})$ . We have localizations $(V\otimes_{E} W)_{v}$ for every $v$ , and we have already seen that in this case we can construct the metaplectic group $\mathrm{Mp}((V\otimes_{E} W)_{v})$ . We want to put each of these together for every $v$ to construct an “adelic” metaplectic group.

First we take the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ . “Restricted” means that all but finitely many of the factors in this product belong to the hyperspecial maximal compact subgroup $K_{v}$ of $\mathrm{Sp}((V\otimes_{E} W)_{v})$ , which is also a compact open subgroup of $\mathrm{Mp}((V\otimes_{E} W)_{v})$ . This restricted product contains $\bigoplus_{v}S^{1}$ as a central subgroup. Now if we quotient out the restricted product $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ by the central subgroup $Z$ given by the set of all $(z_{v})\in\bigoplus_{v}S^{1}$ such that $\prod_{v}z_{v}=1$ , the resulting quotient is the “adelic” metaplectic group $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ that we are looking for.

We have a representation $\bigotimes_{v}'\omega_{\psi_{v}}$ of $\prod_{v}'\mathrm{Mp}((V\otimes_{E} W)_{v})$ which acts trivially on the central subgroup $Z$ defined above and therefore gives us a representation $\omega_{\psi}$ of $\mathrm{Mp}(V\otimes_{E} W)(\mathbb{A})$ .

What is the underlying vector space of the representation $\omega_{\psi}$ ? If $V\otimes_{E}W=X\oplus Y$ is a Lagrangian decomposition, we have seen that we can realize the local Weil representation $\omega_{\psi_{v}}$ on $\mathcal{S}(Y_{v})$ , the vector space of Schwarz functions of $Y_{v}$ (the corresponding localization of $Y$ ). Likewise we can also realize the global Weil representation $\omega_{\psi}$ as functions on the vector space $\mathcal{S}(Y_{\mathbb{A}})$ , defined to be the restricted product $\bigotimes'\mathcal{S}(Y_{v})$ .

So now we have the global Weil representation $\omega_{\psi}$ , which is a representation of the group $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ on the vector space $\mathcal{S}(Y_{\mathbb{A}})$ . But suppose we want an automorphic representation, i.e. one realized on the vector space of automorphic forms for $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ (recall that one of our motivations in this post is to “lift” automorphic forms from one group to another). This is accomplished by the formation of theta functions $\theta(f)(g)$ , so-called because it is a generalization of the Jacobi theta function discussed in Sums of squares and the Jacobi theta function. Let $f$ be a vector in the underlying vector space of the Weil representation. Then the theta function $\theta(f)(g)$ is obtained by summing the evaluations of the output of the action of Weil representation on $f$ over all rational points $y\in Y(k)$ :

$\displaystyle \theta(f)(g)=\sum_{y\in Y(k)}(\omega_{\psi}(g)\cdot f)(y)$

Now suppose we have a pair of characters $\chi_{1},\chi_{2}$ of $E^{\times}$ , so that we have a lifting of $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ to $\mathrm{Mp}(V\otimes_{E}W)(\mathbb{A})$ . This lifting sends $U(V)(k)\times U(W)(k)$ to $\mathrm{Mp}(V\otimes_{E}W)(k)$ , which means that we can consider $\theta(f)(g)$ as an automorphic form for $U(V)(\mathbb{A})\times U(W)(\mathbb{A})$ .

Now we can perform our lifting. Let $f$ be a cuspidal automorphic form for $U(V)$ , and let $\varphi$ be a vector in the underlying vector space of the Weil representation. We can now obtain an automorphic form $\theta(\varphi,f)(g)$ on $U(W)$ as follows:

$\displaystyle \theta(\varphi,f)(g)=\int_{[\mathrm{U}(V)]}\theta(\varphi)(g,h)\cdot \overline{f(h)}dh$

The space generated in this way, for all vectors $f$ in a cuspidal automorphic representation $\pi$ of $U(V)$ , and all vectors $\varphi$ in the in the underlying vector space of the Weil representation, is called the global theta lift of $\pi$ , denoted $\Theta(\pi)$ . It is an automorphic representation of $U(W)$ .

There is also an analogue of all that we discussed for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ when $V$ and $W$ are vector space over some field, equipped with a quadratic form and symplectic form respectively.

Many cases of “lifting”, for instance the Saito-Kurokawa lift from elliptic modular forms to Siegel modular forms, can be considered special cases of the global theta lift (in particular for the reductive dual pair $\mathrm{SO}(V)\times \mathrm{Sp}(W)$ ). The theory of theta lifting is itself part of the theory of Langlands functoriality (see also Trace Formulas). More aspects and examples of the theta correspondence will be discussed in future posts on this blog.

References:

Theta correspondence on Wikipedia

Heisenberg group on Wikipedia

Metaplectic group on Wikipedia

Saito-Kurokawa lift on Wikipedia

Automorphic forms and the theta correspondence by Wee Teck Gan

A brief survey of the theta correspondence by Dipendra Prasad

Non-tempered Arthur packets of G2 by Wee Teck Gan and Nadia Gurevich

A quaternionic Saito-Kurokawa lift and cusp forms on G2 by Aaron Pollack

Completed Cohomology

November 1, 2021February 22, 2022 / Anton Hilado / 1 Comment

Let $F$ be a number field, and let $G_{F,S}$ be the Galois group over $F$ of the maximal extension of $F$ unramified outside a finite set of primes $S$ . It should follow from the Langlands correspondence that $n$ -dimensional continuous (we shall only be talking about continuous Galois representations in this post, so we omit the word “continuous” from here on) representations $\rho:G_{F,S}\to \mathrm{GL_{n}}(\overline{\mathbb{Q}}_{p})$ should correspond to certain automorphic representations $\pi$ of $\mathrm{GL}_{n}$ unramified outside $S$ (see also Automorphic Forms).

The Fontaine-Mazur-Langlands conjecture further states that such Galois representations $\rho$ that are irreducible and “geometric” (i.e. its restrictions to the primes above $p$ satisfy some conditions related to p-adic Hodge theory, see also p-adic Hodge Theory: An Overview) should match up with “algebraic” (we shall explain this shortly) cuspidal $\pi$ . Furthermore this conjecture expects that certain “Hodge numbers” associated to the Galois representation $\rho$ via p-adic Hodge theory should match up to “Hodge numbers” defined for the automorphic representation $\pi$ via its “infinitesimal character” at the archimedean primes (note that they are defined differently, since they are associated to different kinds of representations; they only share the same name because they are expected to coincide).

Generally, whether $\rho$ is “geometric” or not, its Hodge numbers going to be $p$ -adic numbers, and whether $\pi$ is “algebraic” or not, its Hodge numbers are complex numbers. However, if $\rho$ is geometric, then its Hodge numbers are integers, and if $\pi$ is algebraic, its Hodge numbers are also integers (in fact the definition of “algebraic” here just means that its Hodge numbers are integers), and this allows us to match them up.

To see things in a little more detail, let us consider the case of a $1$ -dimensional representation $\rho:G_{F,S}\to \overline{\mathbb{Q}}_{p}$ . We have seen in Galois Representations that an example of this is given by the p-adic cyclotomic character which we can also view as follows. Let $S=\lbrace p,\infty\rbrace$ . Let $G_{F,S}^{\mathrm{ab}}$ be the abelianization of $G_{F,S}$ . It follows from the Kronecker-Weber theorem that $G_{F,S}^{\mathrm{ab}}$ is isomorphic to $\mathbb{Z}_{p}^{\times}$ , and it is precisely the p-adic cyclotomic character that gives this isomorphism. Since $\mathbb{Z}_{p}^{\times}$ embeds into $\overline{\mathbb{Q}}_{p}^{\times}$ , which is also $\mathrm{GL}_{1}(\overline{\mathbb{Q}_{p}})$ , we have our $1$ -dimensional Galois representation. We can also take a power of the p-adic cyclotomic character to get another $1$ -dimensional Galois representation.

But the p-adic cyclotomic character and its powers are not the only $1$ -dimensional Galois representations. For instance, we have a map from $\mathbb{Z}_{p}^{\times}\to \mathbb{Q}_{p}^{\times}$ given by reducing $\mathbb{Z}_{p}$ mod $p^{r}$ and then composing it with the map $\chi$ that sends this element of $(\mathbb{Z}/p^{r})^{\times}$ to the corresponding $p^{r}$ -th root of unity in $\overline{\mathbb{Q}}_{p}^{\times}$ . This is a finite-order character. We also have another map from $\mathbb{Z}_{p}^{\times}\to \overline{\mathbb{Q}}_{p}^{\times}$ which sends $x$ to $x^{s}$ , for some $s$ in $\overline{\mathbb{Q}}_{p}$ such that $\vert s\vert<\frac{p}{p-1}$ . If we compose the p-adic cyclotomic character with either of these maps, we get another $1$ -dimensional Galois representation. It turns out the Hodge number of the latter representation is given by $s$ .

The $1$ -dimensional Galois representations form a rigid analytic space (see also Rigid Analytic Spaces), and their Hodge numbers form p-adic analytic functions on this space. The geometric representations are the ones that are from a power of the p-adic cyclotomic character composed with a finite-order character, and these form a countable dense subset of this rigid analytic space.

Some form of this phenomena happens more generally for higher dimensional Galois representations – they form a rigid analytic space and the geometric ones are a subset of these.

It is convenient that our Galois representations form a rigid analytic space, and suppose we want to do something similar for our automorphic representations. The problem is that the automorphic representations aren’t really “p-adic”, as we may see from the fact that their Hodge numbers are complex instead of p-adic. This is the problem that p-adically completed cohomology, also simply known as completed cohomology, aims to solve.

Let us look at how we want to find automorphic representations in cohomology. Let $G_{\infty}=\mathrm{GL_{n}}(F\otimes_{\mathbb{Q}}\mathbb{R})$ . If $F$ has $r_{1}$ real embeddings and $r_{2}$ complex embeddings, then $G_{\infty}$ will be isomorphic to $\mathrm{GL}_{n}(\mathbb{R})^{r_{1}}\times\mathrm{GL}_{n}(\mathbb{C})^{r_{2}}$ . Let $K_{\infty}^{\circ}$ be a maximal connected compact subgroup of $G_{\infty}$ . With $r_{1}$ and $r_{2}$ as earlier, $K_{\infty}^{\circ}$ will be isomorphic to $\mathrm{SO}(n)^{r_{1}}\times \mathrm{U}(n)^{r_{2}}$ .

Let $X$ be the quotient $G_{\infty}/\mathbb{R}_{>0}^{\times}K_{\infty}^{\circ}$ . This is an example of a symmetric space – for example, if $F=\mathbb{Q}$ and $n=2$ , $X$ is going to be $\mathbb{C}\setminus \mathbb{R}$ .

The space $X$ has an action of $G_{\infty}$ , and its subgroup $\mathrm{GL}_{n}(\mathcal{O}_{F})$ . Letting $N\geq 1$ , we may therefore take the quotient

$\displaystyle Y(N)=\mathrm{GL}_{n}(\mathcal{O}_{F})\backslash (X\times \mathrm{GL}_{n}(\mathcal{O}_{F}/N\mathcal{O}_{F}))$

For example, if $F=\mathbb{Q}$ and $n=2$ , then $Y(N)$ consists of copies of the (uncompactified) modular curve of level $N$ (the number of copies is equal to the number of primes less than $N$ ).

It is this space $Y(N)$ whose cohomology we are interested in. For instance $H^{i}(Y(N),\mathbb{C})$ is related to automorphic forms by a theorem of Jens Franke. However, it is complex, and not the p-adically varying one that we want. There is an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_{p}$ , but the important part of this cohomology comes from the cohomology with $\mathbb{Q}$ coefficients, which is unchanged when we do this isomorphism, and therefore does not really add anything.

This is now where we introduce completed cohomology. Let us require that $N$ and $p$ be mutually prime. We define the completed cohomology $\widetilde{H}^{i}$ as follows:

$\displaystyle \widetilde{H}^{i}:=\varprojlim_{s\geq 1}\varinjlim_{r\geq 0}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

The order of the limits here is important (we will see shortly what happens when they are interchanged). By first taking the direct limit we are essentially considering the union of $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ for all $r$ with $\mathbb{Z}/p^{s}\mathbb{Z}$ coefficients. This is a very big abelian group that might not even be finitely generated. Then the inverse limit means we are taking the $p$ -adic completion – having this as the last step guarantees that the result is something that is p-adically complete (hence the name p-adically completed cohomology). So the completed cohomology $\widetilde{H}^{i}$ is a p-adically complete module over $\mathbb{Z}_{p}$ , which again may not be finitely generated. Taking the tensor product of $\widetilde{H}^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ gives us a vector space $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ which moreover is a Banach space.

Let us consider now what happens if the order of the limits were interchanged. Let us denote the result by $H^{i}$ :

$\displaystyle H^{i}:=\varinjlim_{r\geq 0}\varprojlim_{s\geq 1}H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$

By taking the inverse limit first we are simply considering $H^{i}(Y(Np^{r}),\mathbb{Z}_p$ , and taking the direct limit means we are taking the union of $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ for all $r$ . If we take the tensor product of $H^{i}$ with $\mathbb{Q}_{p}$ over $\mathbb{Z}_{p}$ , then what we get is $H_{\mathbb{Q}_{p}}^{i}$ , the union of $H^{i}(Y(Np^{r}),\mathbb{Q}_p$ for all $r$ . Being the cohomology with characteristic zero coefficients, this may once again be related to the automorphic forms, as earlier.

Therefore, $H_{\mathbb{Q}_{p}}^{i}$ , via the Fontaine-Mazur-Langlands conjecture, should be related to the geometric Galois representations. Now it happens that we can actually embed $H_{\mathbb{Q}_{p}}^{i}$ into the completed cohomology $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ , because there is a map from $H^{i}(Y(Np^{r}),\mathbb{Z}_p)$ to $H^{i}(Y(Np^{r}),\mathbb{Z}/p^{s}\mathbb{Z})$ , and then we can take the direct limit over $r$ followed by the inverse limit over $r$ and then tensor over $\mathbb{Q}_{p}$ as previously.

This embedding of $H_{\mathbb{Q}_{p}}^{i}$ into $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should now bring to mind the picture with the geometric Galois representations which sit inside the rigid analytic space of Galois representations which may not necessarily be geometric, as discussed earlier. It is in fact a conjecture that $\widetilde{H}_{\mathbb{Q}_{p}}^{i}$ should know about the rigid analytic space of Galois representations.

In the case $F=\mathbb{Q}$ and $n=2$ , the completed cohomology is some space of p-adic modular forms, and there is much that is known via the work of Matthew Emerton, who also showed that the p-adic local Langlands correspondence appears inside the completed cohomology. This has led to a proof of many cases of the Fontaine-Mazur conjecture for $2$ -dimensional odd Galois representations.

We have only provided a rough survey of the motivations behind the theory of completed cohomology in this post. We will discuss further deeper aspects of it, and its relations to the p-adic local Langlands correspondence and the Fontaine-Mazur conjecture in future posts.

References:

Completed cohomology and the p-adic Langlands correspondence by Matthew Emerton on YouTube

Completed cohomology and the p-adic Langlands program by Matthew Emerton

Completed cohomology – a survey by Frank Calegari and Matthew Emerton

Automorphic Forms

September 15, 2021September 15, 2021 / Anton Hilado / 2 Comments

An automorphic form is a kind of function of the adelic points (see also Adeles and Ideles) of a reductive group (see also Reductive Groups Part I: Over Algebraically Closed Fields), that can be used to investigate its representation theory. Choosing an important kind of function of a group that will be helpful in investigating its representation theory was also discussed in Representation Theory and Fourier Analysis, where we found the square-integrable functions on the circle to be useful in studying its representations (or that of the real line) since it decomposed into a direct sum of irreducible representations. In fact, the cuspidal automorphic forms we will introduce later on in this post will also have this property (called semisimple) of decomposing into a direct sum of irreducible representations.

Remark: We have briefly mentioned, in the unramified case, cuspidal automorphic forms as certain functions on $\mathrm{Bun}_{G}$ in The Global Langlands Correspondence for Function Fields over a Finite Field. The function field (over a finite field) version of the cuspidal automorphic forms we define here are actually obtained as linear combinations of translates of such functions hence why it is enough to study them. In this post, we will discuss automorphic forms in more detail, beginning with the version over the field of rational numbers before generalizing to more general global fields.

Defining modular forms as functions on $\mathrm{GL}_{2}(\mathbb{A})$

In a way, automorphic forms can also generalize modular forms (see also Modular Forms), and this will give us a way to connect the two theories. We shall take this route first and recast modular forms in a new language – instead of functions on the upper half-plane, we shall now look at them as functions on the group $\mathrm{GL}_{2}(\mathbb{A})$ (here $\mathbb{A}$ denotes the adeles of $\mathbb{Q}$ ).

Let $K_{f}$ be a compact open subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ whose elements all have determinants in $\widehat{\mathbb{Z}}^{\times}$ . Here $\mathbb{A}_{f}$ stands for the finite adeles, which are defined in the same way as the adeles, except we don’t include the infinite primes in the restricted product. We have that

$\displaystyle \mathrm{GL_{2}}(\mathbb{A})=\mathrm{GL_{2}}(\mathbb{Q})\mathrm{GL}_{2}(\mathbb{R})^{+}K_{f}$

where $\mathrm{GL}_{2}(\mathbb{R})^{+}$ is the subgroup of $\mathrm{GL}_{2}(\mathbb{R})$ consisting of elements that have positive determinant. Now let us take the double quotient $\mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}$ . By the above expression for $\mathrm{GL}_{2}(\mathbb{A})$ as a product, we have

$\displaystyle \mathrm{GL_{2}}(\mathbb{Q})\backslash\mathrm{GL}_{2}(\mathbb{A})/K_{f}\simeq \Gamma\backslash\mathrm{GL_{2}}(\mathbb{R})$

where $\Gamma$ is the subgroup of $\mathrm{GL}_{2}(\mathbb{R})$ given by projecting $\mathrm{GL}_{2}(\mathbb{Q})\cap \mathrm{GL}_{2}(\mathbb{R})^{+}K_{f}$ into its archimedean component. Now suppose we are in the special case that $K_{f}$ is given by $\displaystyle \prod_{p} \mathrm{GL}_{2}(\mathbb{Z}_{p})$ . Then it turns out that $\Gamma$ is just $\mathrm{SL}_{2}(\mathbb{Z})$ ! Using appropriate choices of $K_{f}$ , we can also obtain congruence subgroups such as $\Gamma_{0}(N)$ (see also Modular Forms).

The group $\mathrm{GL}_{2}(\mathbb{R})^{+}$ acts on the upper half-plane by fractional linear transformations, i.e. if we have $\displaystyle g_{\infty}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in\mathrm{GL}_{2}(\mathbb{R})^{+}$ , then $g_{\infty}$ sends $\tau$ in the upper half-plane to $\displaystyle g_{\infty}(\tau)=\frac{az+b}{cz+d}$ . Let

$\displaystyle j(g_{\infty},\tau)=\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d)$ .

Now given a modular form $f$ of weight $m$ and level $\Gamma_{0}(N)$ , we may associate to it a function $\phi_{\infty}(g_{\infty})$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ as follows:

$\displaystyle f\mapsto \phi_{\infty}(g_{\infty})=f(g_{\infty}(i))j(g_{\infty},i)^{-m}$

We can also go the other way, recovering $f$ from such a $\phi_{\infty}$ :

$\displaystyle \phi_{\infty}\mapsto f(g_{\infty}(i))=\phi_{\infty}(g_{\infty})j(g_{\infty},i)^{m}$

for any $g_{\infty}$ such that $g_{\infty}(i)=\tau$ . Ultimately we want a function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$ , and we achieve this by setting $\phi(g)=\phi(\gamma g_{\infty} k_{f})$ to just have the same value as $\phi_{\infty}(g_{\infty})$ .

Translating properties of modular forms into properties of functions on $\mathrm{GL}_{2}(\mathbb{A})$

Invariance under $\mathrm{GL}_{2}(\mathbb{Q}$ and $K_{f}$

Now we want to know what properties $\phi$ must have, so that we can determine which functions on $\mathrm{GL}_{2}(\mathbb{A})$ come from modular forms. We have just seen that we must have

$\displaystyle \phi(g)=\phi(\gamma g_{\infty} k_{f})=\phi_{\infty}(g)$ .

The action of $Z_{\infty}^{+}$ and $K_{\infty}^{+}$

Let us now consider the action of the center of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ (which we denote by $Z_{\infty}$ ) and the action of $\mathrm{SO}(2)$ , which is a maximal compact subgroup of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ (and therefore we shall also denote it by $K_{\infty})^{+}$ . The center $Z_{\infty}$ is composed of the matrices of the form $z_{\infty}$ times the identity matrix, and it acts trivially on the upper half-plane. Therefore we will have

$\displaystyle j(z_{\infty}g_{\infty},\tau)=\mathrm{sgn}(z_{\infty})\mathrm{det}(g_{\infty})^{-\frac{1}{2}}(c\tau+d)$

Now for the maximal compact subgroup $K_{\infty}^{+}$ . As previously mentioned, this is the group $\mathrm{SO}(2)$ , and may be expressed as matrices of the form

$\displaystyle k_{\theta}=\begin{pmatrix}\mathrm{cos}(\theta) & \mathrm{sin}(\theta)\\-\mathrm{sin}(\theta) & \mathrm{cos}(\theta)\end{pmatrix}$ .

Then in the action of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ on the upper half-plane, $Z_{\infty}K_{\infty}^{+}$ is the stabilizer of $i$ . We will also have

$\displaystyle j(z_{\infty}k_{\theta},i)=\mathrm{sgn}(z_{\infty})e^{i\theta}$

This leads us to the second property our function $\phi$ must satisfy. First we consider $\phi_{\infty}$ . For $z_{\infty}k_{\theta}\in Z_{\infty}K_{\infty}^{+}$ , we must have

$\displaystyle \phi_{\infty}(g_{\infty}z_{\infty}k_{\theta})=\phi_{\infty}(g_{\infty})\mathrm{sgn}(z)^{m}(e^{i\theta})^{m}$ .

Note the appearance of the weight $m$ . Now when we extend this function $\phi_{\infty}$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ to a function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$ , we must replace $Z_{\infty}$ by its connected component $Z_{\infty}^{+}$ .

Holomorphicity and the action of the Lie algebra

Next we must translate the property that the modular form $f$ is holomorphic into a property of $\phi$ . For this we shall introduce certain “raising” and “lowering” operators.

Let $\mathfrak{g}_{0}$ be the (real) Lie algebra of $\mathrm{GL}_{2}(\mathbb{R})^{+}$ . An element $X\in\mathfrak{g}_{0}$ acts on the space of smooth functions on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ as follows:

$\displaystyle X\phi(g_{\infty})=\frac{d}{dt}\phi(g_{\infty}\mathrm{exp}(tX))\bigg\vert_{t=0}$

We can extend this to an action of the complexified Lie algebra $\mathfrak{g}$ , defined to be $\mathfrak{g}\otimes_{\mathbb{R}} \mathbb{C}$ , by setting

$\displaystyle (X+iY)\phi=X\phi+iY\phi$

We now look at two special elements of $\mathfrak{g}$ . They are

$\displaystyle X_{+}=\frac{1}{2}\begin{pmatrix}1 & i\\i & -1\end{pmatrix}$

and

$\displaystyle X_{-}=\frac{1}{2}\begin{pmatrix}1 & -i\\-i & -1\end{pmatrix}$ .

Let us now look at how these special elements act on the smooth functions on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ . We have

$\displaystyle X_{+}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m+2}$

and

$\displaystyle X_{-}\phi(g_{\infty}k_{\theta})=\phi(g_{\infty})(e^{i\theta})^{m-2}$

In other words, the action of $X_{+}$ raises the weight by $2$ , while the action of $X_{-}$ lowers the weight by $2$ . Now it turns out that the condition that the function $f$ on the upper half-plane is holomorphic is the same condition as the function $\phi$ on $\mathrm{GL}_{2}(\mathbb{R})^{+}$ satisfying $X_{-}\phi=0$ !

Holomorphicity at the cusps

Now we have expressed the holomorphicity of our modular form $f$ as a condition on our function $\phi$ on $\mathrm{GL}_{2}(\mathbb{A})$ . However not only do we want our modular forms to be holomorphic on the upper half-plane, we also want them to be “holomorphic at the cusps”, i.e. they do not go to infinity at the cusps. This is going to be accomplished by requiring the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ to be “slowly increasing” for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$ . This means that for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$ , we have

$\displaystyle \vert \phi(g_{\infty}g_{f})\geq C\Vert g_{\infty}\Vert^{N}$

where $C$ and $N$ are some positive constants and the norm on the right-hand side is given by, for $g_{\infty}=\begin{pmatrix}a & b\\c & d\end{pmatrix}$ ,

$\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=(a^{2}+b^{2}+c^{2}+d^{2})(1+\mathrm{det}(g_{\infty}^{-2}))=\mathrm{Tr}(g_{\infty}^{T}g_{\infty})+\mathrm{Tr}((g_{\infty}^{-1})^{T}g_{\infty}^{-1})$ .

Summary of the properties

Let us summarize now the properties we want our function $\phi$ to have in order that it come from a modular form $f$ :

For all $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$ , we have $\phi(\gamma g)=\phi(g)$ .
For all $k_{f}\in K_{f}$ , we have $\phi(gk_{f})=\phi(g)$ .
For all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$ , the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is smooth.
For all $k_{\theta}\in K_{\infty}$ we have $\phi(gk_{\theta})=\phi(g)e^{i\theta}$ .
The function $\phi$ is invariant under $Z_{\infty}^{+}$ .
We have $\displaystyle X_{-}\phi=0$ .
The function given by $g_{\infty}\mapsto\phi(g_{\infty}g_{f})$ is slowly increasing.

Cuspidality

Now let us consider the case where $f$ is a cusp form. We want to translate the cuspidality condition to a condition on $\phi$ , and we do this by noting that this means that the Fourier expansion of $f$ has no constant term. Given that Fourier coefficients can be expressed using Fourier transforms, we make use of the measure theory on the adeles to express this cuspidality condition as

$\displaystyle \int_{\mathbb{Q}\setminus\mathbb{A}}\phi\left(\begin{pmatrix}1 & x\\0&1\end{pmatrix}\right)dx=0$ .

Automorphic forms

We have now defined modular forms as functions on $\mathrm{GL}_{2}(\mathbb{A})$ , and enumerated some of their important properties. Modular forms, as functions on $\mathrm{GL}_{2}(\mathbb{A})$ , turn out to be merely be specific examples of more general functions on $\mathrm{GL}_{2}(\mathbb{A})$ that satisfy similar, but more relaxed, properties. These are the automorphic forms.

The first few properties are the same, for instance for all $\gamma\in\mathrm{GL}_{2}(\mathbb{Q})$ , we want $\phi(\gamma g)=\phi(g)$ , and for all $k_{f}\in K_{f}$ , where $K_{f}$ is a compact subgroup of $\mathrm{GL}_{2}(\mathbb{A}_{f})$ , we want $\phi(g k)=\phi(g)$ . We will also want the function given by $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ to be smooth for all $g_{f}\in\mathrm{GL}_{2}(\mathbb{A}_{f})$ .

What we want to relax a little bit is the conditions on the actions of $K_{\infty}$ , $Z_{\infty}^{+}$ , and the Lie algebra $\mathfrak{g}$ , in that we want the space we get by having them act on some function $\phi$ to be finite-dimensional. Instead of looking at the action of the Lie algebra $\mathfrak{g}$ , it is often convenient to instead look at the action of its universal enveloping algebra $U(\mathfrak{g})$ . The universal enveloping algebra is an honest to goodness associative algebra that contains the Lie algebra (and is in fact generated by its elements) such that the commutator of the universal enveloping algebra gives the Lie bracket of the Lie algebra. We shall denote the center of $U(\mathfrak{g})$ by $Z(\mathfrak{g})$ . Now it turns out that $Z(\mathfrak{g})$ is generated by the Lie algebra of $Z_{\infty}^{+}$ and the Casimir operator $\Delta$ , defined to be

$\displaystyle \Delta=H^{2}+2X_{+}X_{-}+2X_{-}X_{+}$

where $H$ is the element given by $\begin{pmatrix}0&-i\\i &0\end{pmatrix}$ . Therefore, the action of the center of the universal enveloping algebra encodes the action of $Z_{\infty}^{+}$ and the Lie algebra $\mathfrak{g}$ at the same time.

Let us now define automorphic forms in general. Even though the focus on this post is on $\mathrm{GL}_{2}$ and over the rational numbers $\mathbb{Q}$ , we can just give the most general definition of automorphic forms now, even for more general reductive groups and more general global fields. So let $G$ be a reductive group and let $F$ be a global field. The space of automorphic forms on $G$ , denoted $\mathcal{A}$ , is the space of functions $\phi:G(\mathbb{A}_{F})\to\mathbb{C}$ satisfying the following properties:

For all $\gamma\in G(F)$ , we have $\phi(\gamma g)=\phi(g)$ .
For all $k_{f}\in K_{f}$ , $K_{f}$ a compact open subgroup of $G(\mathbb{A}_{f})$ , we have $\phi(gk_{f})=\phi(g)$ .
For all $g_{f}\in G(\mathbb{A}_{F,f})$ , the function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is smooth.
The function $\phi$ is $K_{\infty}$ -finite, i.e. the space $\mathbb{C}[K_{\infty}]\cdot\phi$ is finite dimensional.
The function $\phi$ is $Z(\mathfrak{g})$ -finite, i.e. the space $Z(\mathfrak{g})\cdot\phi$ is finite dimensional.
The function $g_{\infty}\mapsto \phi(g_{\infty}g_{f})$ is slowly increasing.

Here slowly increasing means that for all embeddings $\iota:G_{\infty}\to\mathrm{GL}_{n}(\mathbb{R})$ of the infinite part of $G(\mathbb{A}_{F})$ , we have

$\displaystyle \Vert \phi(g_{\infty}g_{f})\Vert=\mathrm{Tr}(\iota(g_{\infty})^{T}\iota(g_{\infty}))+\mathrm{Tr}((\iota(g_{\infty}^{-1})^{T}\iota(g_{\infty})^{-1})$ .

Furthermore, we say that the automorphic form $\phi$ is cuspidal if, for all parabolic subgroups $P\subseteq G$ , $\phi$ satisfies the following additional condition:

$\displaystyle \int_{N(\mathbb{F})\setminus N(\mathbb{A}_{F})}\phi(ng)dn=0$

where $N$ is the unipotent radical (the unipotent part of the maximal connected normal solvable subgroup) of the parabolic subgroup $P$ .

These cuspidal automorphic forms, which we denote by $\mathcal{A}_{0}$ , form a subspace of the automorphic forms $\mathcal{A}$ .

Automorphic forms and representation theory

As stated earlier, automorphic forms give us a way of understanding the representation theory of $G(\mathbb{A}_{F})$ where $G$ is a reductive group. Let us now discuss these representation-theoretic aspects.

We will actually look at automorphic forms not as representations of $G(\mathbb{A})$ , but as $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$ -modules. This means they have actions of $\mathfrak{g}$ , $K_{\infty}$ , and $G(\mathbb{A}_{F,f})$ all satisfying certain compatibility conditions. A $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$ -module is called admissible if any irreducible representation $K_{\infty}\times K_{f}$ shows up inside it with finite multiplicity, and irreducible if it has no proper subspaces fixed by $\mathfrak{g}$ , $K_{\infty}$ , and $G(\mathbb{A}_{F,f})$ . Recall from The Local Langlands Correspondence for General Linear Groups, irreducible admissible representations of $G(F_{v})$ , where $F_{v}$ is some local field, are precisely the kinds of representations that show up in the automorphic side of the local Langlands correspondence.

In fact the global and the local picture are related by Flath’s theorem, which says that, for an irreducible admissible $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$ -module $\pi$ , we have the following factorization

$\displaystyle \pi=\bigotimes'_{v\not\vert\infty}\pi_{v}\otimes \pi_{\infty}$

into a restricted tensor product (explained in the next paragraph) of irreducible admissible representations $\pi_{v}$ of $G(F_{v})$ , running over all places $v$ of $F$ . At the infinite place, $\pi_{v}$ is an irreducible admissible $(\mathfrak{g}, K_{\infty})$ -module.

The restricted tensor product is a direct limit over $S$ of $V_{S}=\bigotimes_{s\in S} \pi_{s}$ where for $S\subset T$ we have the inclusion $V_{S}\hookrightarrow V_{T}$ given by $x_{S}\mapsto x_{S}\otimes\bigotimes_{v\in T\setminus S}\xi_{v}^{0}$ , where $\xi_{v}$ is a vector fixed by a certain maximal compact open subgroup (called hyperspecial) $K_{v}$ of $G(F_{v})$ (a representation of $G(F_{v})$ containing such a fixed vector is called unramified).

We have that $\mathcal{A}$ and $A_{0}$ are $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{f})$ -modules. An automorphic representation of a reductive group $G$ is an indecomposable $(\mathfrak{g},K_{\infty})\times G(\mathbb{A}_{F,f})$ -module that is isomorphic to a subquotient of $\mathcal{A}$ . A cuspidal automorphic representation is an automorphic representation that is isomorphic to a subquotient of $\mathcal{A}_{0}$ . It is a property of the space of cuspidal automorphic forms that it is semisimple, i.e. it decomposes into a direct product of cuspidal automorphic representations (a property that is not necessarily shared by the bigger space of automorphic forms!).

An automorphic form generates such an automorphic representation, and a theorem of Harish-Chandra states that such a representation is admissible. In fact, automorphic representations are always admissible. Again recalling that irreducible admissible representations of $G(F_{v})$ make up the automorphic side of the local Langlands correspondence, we therefore expect that automorphic representations of $G$ will make up the automorphic side of the global Langlands correspondence.

However, to state the original global Langlands correspondence in general is still quite complicated, as it involves an as of-yet hypothetical object called the Langlands group, which plays a somewhat analogous role as the Weil group in the local Langlands correspondence. Instead there are certain variants of it that are considered easier to approach, for instance by imposing conditions on the representations such as being “algebraic at infinity“. These variants of the global Langlands correspondence will hopefully be discussed in future posts.

References:

Automorphic form on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

MSRI Summer School on Automorphic Forms and the Langlands Program by Kevin Buzzard

The Automorphic Project

An Introduction to the Langlands Program by Daniel Bump, James W. Cogdell, Ehud de Shalit, Dennis Gaitsgory, Emmanuel Kowalski, and Stephen S. Kudla (edited by Joseph Bernstein and Stephen Gelbart)

Theories and Theorems

Math and Physics for Everyone

automorphic forms

The Theta Correspondence

Completed Cohomology

Automorphic Forms

Defining modular forms as functions on $\mathrm{GL}_{2}(\mathbb{A})$