Month: September 2021

Automorphic Forms

/ 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)