### Introduction

In this blog post we define automorphic forms as meromorphic functions on the upper half plane . Automorphic forms generalize modular forms, which are holomorphic functions satisfying some special conditions on the upper half plane.

For more on modular forms and some of the standard notations used below, see an earlier post on modular forms here.

#### Meromorphic functions

Let denote the Riemann sphere i.e. . We say that a function from an open subset of to is **meromorphic** if it is identically zero or if its Laurent expansion is as follows:

for all in a disk around , with each and nonzero.

Additionally, the index is the **order of vanishing **of at , which we write as . For example, .

- If the order of is nonnegative, then is holomorphic at .
- If the order of is strictly positive, then is a zero of .
- Finally, if the order of is negative, then has a pole at .

The set of all such , i.e. the set of meromorphic functions defined on an open subset of , forms a field.

### Definition

Given a congruence subgroup of and an integer , we say that a complex-valued function on the upper half plane is an **automorphic form of weight k with respect to **if the conditions below are satisfied. We write .

- is meromorphic,
- is weight- invariant under , i.e., ], and
- ] is meromorphic at , which means either is identically or a Laurent series of on a certain disk about terminates from the left side. The Laurent series of is defined on the punctured disk centered at corresponding to a region where has no poles. Namely, if , where and is the smallest positive integer with

Note, the above definition can essentially be obtained from that of modular forms (see here) by replacing the condition of holomorphicity by that of meromorphicity.

The order of at any cusp is denoted by and is defined to be for and .

#### Example

We have , where is the modular invariant for the weight Eisenstein series and the weight cusp form ( i.e. and ). To see this, first observe .

For the other direction, consider i.e. meromorphic and nonconstant on with finitely many zeroes and finitely many poles counted with multiplicity. Here is the modular curve . Define by the ratio Then if and have the same zeroes and poles away from , then vanishes to the same order as at . The subscript indicates is a fixed point. Hence, the function is constant as it does not have any zeroes or poles. So, and the other containment follows.

#### The ring

** **Let be a congruence subgroup of . We have a graded ring . It can be shown that for any nonzero , i.e., . For example, when , is the field of meromorphic functions on written as .

#### Automorphic forms on the quotient

Observe that in general an automorphic form of weight with respect to a congruence subgroup is not well defined on the quotient . This follows from the transformation formula . However, the Reimann surface structure of allows us to define the order of vanishing of on the quotient.

Recall that and the points in are called **cusps***.* There is a natural definition of the order of at a non-cusp . Namely, with equal to the period of . Hence, the order of at is the same as that at except for at elliptic points.

More work is needed to define the order of at the cusps. Given a cusp for we pick some satisfying the equation . Then, the order of at is defined as follows: , where except when , in which case .

### Applications

Automorphic forms allow us to produce new modular forms. For example, consider , (where for )i.e., the Dedekind eta function and , i.e., the discriminant function. Then, one can show is a modular form provided the existence of a nonzero modular form of this type is guaranteed by another method. Further, one can compute dimensions for the spaces of modular forms by using the relation between automorphic forms of positive even weight with respect to and meromorphic differentials of degree on .

### Reference

F. Diamond and J. Shurman, *A First Course in Modular Forms*, *Graduate Texts in Math.* **228**, Springer-Verlag, New York, 2005.