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 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

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 (here denotes the adeles of ).

Let be a compact open subgroup of whose elements all have determinants in . Here 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

where is the subgroup of consisting of elements that have positive determinant. Now let us take the double quotient . By the above expression for as a product, we have

where is the subgroup of given by projecting into its archimedean component. Now suppose we are in the special case that is given by . Then it turns out that is just ! Using appropriate choices of , we can also obtain congruence subgroups such as (see also Modular Forms).

The group acts on the upper half-plane by fractional linear transformations, i.e. if we have , then sends in the upper half-plane to . Let

.

Now given a modular form of weight and level , we may associate to it a function on as follows:

We can also go the other way, recovering from such a :

for any such that . Ultimately we want a function on , and we achieve this by setting to just have the same value as .

### Translating properties of modular forms into properties of functions on

#### Invariance under and

Now we want to know what properties must have, so that we can determine which functions on come from modular forms. We have just seen that we must have

.

#### The action of and

Let us now consider the action of the center of (which we denote by ) and the action of , which is a maximal compact subgroup of (and therefore we shall also denote it by . The center is composed of the matrices of the form times the identity matrix, and it acts trivially on the upper half-plane. Therefore we will have

Now for the maximal compact subgroup . As previously mentioned, this is the group , and may be expressed as matrices of the form

.

Then in the action of on the upper half-plane, is the stabilizer of . We will also have

This leads us to the second property our function must satisfy. First we consider . For , we must have

.

Note the appearance of the weight . Now when we extend this function on to a function on , we must replace by its connected component .

#### Holomorphicity and the action of the Lie algebra

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

Let be the (real) Lie algebra of . An element acts on the space of smooth functions on as follows:

We can extend this to an action of the *complexified* Lie algebra , defined to be , by setting

We now look at two special elements of . They are

and

.

Let us now look at how these special elements act on the smooth functions on . We have

and

In other words, the action of raises the weight by , while the action of lowers the weight by . Now it turns out that the condition that the function on the upper half-plane is holomorphic is the same condition as the function on satisfying !

#### Holomorphicity at the cusps

Now we have expressed the holomorphicity of our modular form as a condition on our function on . 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 to be “**slowly increasing**” for all . This means that for all , we have

where and are some positive constants and the norm on the right-hand side is given by, for ,

.

#### Summary of the properties

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

- For all , we have .
- For all , we have .
- For all , the function is smooth.
- For all we have .
- The function is invariant under .
- We have .
- The function given by is slowly increasing.

#### Cuspidality

Now let us consider the case where is a cusp form. We want to translate the cuspidality condition to a condition on , and we do this by noting that this means that the Fourier expansion of 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

.

### Automorphic forms

We have now defined modular forms as functions on , and enumerated some of their important properties. Modular forms, as functions on , turn out to be merely be specific examples of more general functions on that satisfy similar, but more relaxed, properties. These are the automorphic forms.

The first few properties are the same, for instance for all , we want , and for all , where is a compact subgroup of , we want . We will also want the function given by to be smooth for all .

What we want to relax a little bit is the conditions on the actions of , , and the Lie algebra , in that we want the space we get by having them act on some function to be finite-dimensional. Instead of looking at the action of the Lie algebra , it is often convenient to instead look at the action of its **universal enveloping algebra** . 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 by . Now it turns out that is generated by the Lie algebra of and the **Casimir operator** , defined to be

where is the element given by . Therefore, the action of the center of the universal enveloping algebra encodes the action of and the Lie algebra at the same time.

Let us now define automorphic forms in general. Even though the focus on this post is on and over the rational numbers , 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 be a reductive group and let be a global field. **The space of automorphic forms on **, denoted , is the space of functions satisfying the following properties:

- For all , we have .
- For all , a compact open subgroup of , we have .
- For all , the function is smooth.
- The function is -finite, i.e. the space is finite dimensional.
- The function is -finite, i.e. the space is finite dimensional.
- The function is slowly increasing.

Here slowly increasing means that for all embeddings of the infinite part of , we have

.

Furthermore, we say that the automorphic form is **cuspidal** if, for all parabolic subgroups , satisfies the following additional condition:

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

These **cuspidal automorphic forms**, which we denote by , form a subspace of the automorphic forms .

### Automorphic forms and representation theory

As stated earlier, automorphic forms give us a way of understanding the representation theory of where is a reductive group. Let us now discuss these representation-theoretic aspects.

We will actually look at automorphic forms not as representations of , but as **-modules**. This means they have actions of , , and all satisfying certain compatibility conditions. A -module is called **admissible** if any irreducible representation shows up inside it with finite multiplicity, and **irreducible** if it has no proper subspaces fixed by , , and . Recall from The Local Langlands Correspondence for General Linear Groups, irreducible admissible representations of , where 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 -module , we have the following factorization

into a **restricted tensor product** (explained in the next paragraph) of irreducible admissible representations of , running over all places of . At the infinite place, is an irreducible admissible -module.

The restricted tensor product is a direct limit over of where for we have the inclusion given by , where is a vector fixed by a certain maximal compact open subgroup (called **hyperspecial**) of (a representation of containing such a fixed vector is called **unramified**).

We have that and are -modules. An **automorphic representation** of a reductive group is an indecomposable -module that is isomorphic to a subquotient of . A **cuspidal automorphic representation** is an automorphic representation that is isomorphic to a subquotient of . 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 make up the automorphic side of the local Langlands correspondence, we therefore expect that automorphic representations of 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

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)