This chapter introduces modular forms and congruence subgroups, which
are central objects in this book. We first introduce the upper half
plane and the group then recall some definitions from
complex analysis. Next we define modular forms of level followed
by modular forms of general level. In Section *Remarks on Congruence Subgroups* we
discuss congruence subgroups and explain a
simple way to compute generators for them and determine element
membership. Section *Applications of Modular Forms* lists applications of
modular forms.

We assume familiarity with basic number theory, group theory, and
complex analysis. For a deeper understanding of modular forms, the
reader is urged to consult the standard books in the field, e.g.,
[**Lan95**, **Ser73**, **DI95**, **Miy89**, **Shi94**, **Kob84**].
See also [**DS05**],
which is an excellent first introduction to the theoretical
foundations of modular forms.

The group

acts on the *complex upper half plane*

by *linear fractional transformations*, as follows.

If , then for any we let

(1)

Definition 1.1

The *modular group* is the group of all matrices
with and .

For example, the matrices

(2)

are both elements of ; the matrix induces the function on , and induces the function .

Theorem 1.2

The group is generated by and .

Proof

See e.g. [**Ser73**, Section VII.1].

In Sage we compute the group and its generators as follows:

```
sage: G = SL(2,ZZ); G
Modular Group SL(2,Z)
sage: S, T = G.gens()
sage: S
[ 0 -1]
[ 1 0]
sage: T
[1 1]
[0 1]
```

Definition 1.3

Let be an open subset of . A function is
*holomorphic* if is complex differentiable at every point
, i.e., for each the limit

exists, where may approach along any path. A function
is *meromorphic* if it is holomorphic
except (possibly) at a discrete set of points in , and at
each there is a positive integer such that
is holomorphic at .

The function is a holomorphic function on ; in contrast, is meromorphic on but not holomorphic since it has a pole at . The function is not even meromorphic on .

Modular forms are holomorphic functions on that transform in a particular way under a certain subgroup of . Before defining general modular forms, we define modular forms of level .

Definition 1.4

A *weakly modular function* of *weight* is a
meromorphic function on such that for all
and all we have

(3)

The constant functions are weakly modular of weight .
There are no nonzero weakly modular functions of odd weight
(see *Exercise 1.4*), and it is
not obvious that there are any weakly modular functions
of even weight (but there are, as we will see!).
The product of two weakly modular functions of weights
and is a weakly modular function of weight
(see *Exercise 1.3*).

When is even, (3) has a possibly more conceptual interpretation; namely (3) is the same as

Thus (3) simply says that the weight “differential form” is fixed under the action of every element of .

By *Theorem 1.2*, the group is generated by
the matrices and of (2), so to show that a
meromorphic function on is a weakly modular function,
all we have to do is show that for all we have

(4)

Suppose is a weakly modular function of weight . A
*Fourier expansion* of , if it exists, is a representation of
as , for all
. Let , which we view as a holomorphic
function on . Let be the open unit disk with the origin
removed, and note that defines a map . By
(4) we have , so there is a function
such that . This function is a
complex-valued function on , but it may or may not be well behaved
at .

Suppose that is well behaved at , in the sense that for some and all in a neighborhood of we have the equality

(5)

If this is the case, we say that is
*meromorphic at *. If, moreover,
, we say
that is *holomorphic at *. We also
call (5) the *-expansion* of about .

Definition 1.5

A *modular function* of *weight* is a weakly modular function
of weight that is meromorphic at .

Definition 1.6

A *modular form* of *weight* (and
*level *) is a modular function of weight that is
holomorphic on and at .

If is a modular form, then there are numbers such that for all ,

(6)

Proposition 1.7

The above series converges for all .

Proof

The function is holomorphic on , so its Taylor series converges absolutely in .

Since as , we set .

Definition 1.8

A *cusp form* of *weight* (and *level *) is a
modular form of weight such that , i.e., .

Let be the ring of all *formal power series* in .
If , then , so . If is a cusp form of weight , then

Thus the differential is holomorphic at , since is a local parameter at .

In this section we define spaces of modular forms of arbitrary level.

Definition 1.9

A *congruence subgroup* of
is any subgroup of that contains

for some positive integer . The smallest such is the
*level of *.

The most important congruence subgroups in this book are

and

where means any element. Both groups have level (see
*Exercise 1.6*).

Let be an integer.
Define the *weight ** right action* of on
the set of all functions as follows.
If , let

(7)

Proof

See *Exercise 1.7*.

Definition 1.11

A *weakly modular function* of weight for a
congruence subgroup is a meromorphic
function such that
for all .

A central object in the theory of modular forms
is the *set of cusps*

Also, note that if the denominator or is above, then

An element acts on by

Also, note that if the denominator or is above, then

The set of *cusps for a congruence subgroup `\Gamma`* is the set
of -orbits of . (We
will often identify elements of with a representative
element from the orbit.) For example, the lemma below
asserts that if , then there is
exactly one orbit, so .

Lemma 1.12

For any cusps there exists such that .

Proof

This is *Exercise 1.8*.

Proposition 1.13

For any congruence subgroup , the set of cusps is finite.

Proof

This is *Exercise 1.9*.

See Section 3.8 of [**DS05**] and
*Algorithm 1.33* below for more discussion of cusps and results
relevant to their enumeration.

In order to define modular forms for general congruence subgroups, we
next explain what it means for a function to be holomorphic
on the *extended upper half plane*

See [**Shi94**, Section 1.3–1.5], for a detailed
description of the correct topology to consider on . In
particular, a basis of neighborhoods for is given by the
sets , where is an open disc in that is
tangent to the real line at .

Recall from Section *Modular Forms of Level * that a weakly modular
function on is holomorphic at if its
-expansion is of the form .

In order to make sense of holomorphicity of a weakly modular function for an arbitrary congruence subgroup at any , we first prove a lemma.

Lemma 1.14

If is a weakly modular function of weight for a congruence subgroup and if , then is a weakly modular function for .

Proof

If , then

Fix a weakly modular function of weight for a congruence
subgroup , and suppose . In
Section *Modular Forms of Level * we constructed the -expansion of by
using that , which held since
. There are
congruence subgroups such that . Moreover,
even if we are interested only in modular forms for ,
where we have for all , we will still have to
consider -expansions at infinity for modular forms on groups
, and these need not contain .
Fortunately, , so a congruence
subgroup of level contains . Thus we have
for some positive integer , e.g.,
always works, but there may be a smaller choice of .
The minimal choice of such that
,
where ,
is called the defn{width of the cusp}
relative to the group (see Section *Computing Widths of Cusps*).
When is meromorphic at infinity, we obtain
a Fourier expansion

(8)

in powers of the function . We say that is holomorphic at if in (8) we have .

What about the other cusps ? By
*Lemma 1.12* there is a such that
. We declare to be
*holomorphic at the cusp * if the weakly modular function is
holomorphic at .

Definition 1.15

A *modular form* of integer *weight* for a congruence subgroup
is a weakly modular function that
is holomorphic on . We let index{}
denote the space
of weight modular forms of weight for .

Proposition 1.16

If a weakly modular function is holomorphic at a set of representative elements for , then it is holomorphic at every element of .

Proof

Let be representatives for the set of cusps for . If , then there is such that for some . By hypothesis is holomorphic at , so if is such that , then is holomorphic at . Since is a weakly modular function for ,

(9)

But , so (9) implies that is holomorphic at .

Recall that a congruence subgroup is a subgroup of
,that contains for some . Any congruence subgroup has
finite index in , since does. What about the
converse: is every finite index subgroup of
a congruence subgroup? This is the *congruence subgroup problem*. One can ask about the congruence subgroup problem with
replaced by many similar groups. If is a prime, then one can
prove that every finite index subgroup of is a
congruence subgroup (i.e., contains the kernel of reduction modulo
some integer coprime to ), and for any , all finite index
subgroups of are congruence subgroups (see
[**Hum80**]). However, there are numerous finite index
subgroups of that are not congruence subgroups. The paper
[**Hsu96**] contains an *algorithm* to decide if certain
finite index subgroups are congruence subgroups and gives an example
of a subgroup of index 12 that is not a congruence subgroup.

One can consider modular forms even for noncongruence subgroups. See,
e.g., [**Tho89**] and the papers it references for work on
this topic. We will not consider such modular forms further in this
book.
Note that modular symbols (which we define later in this book)
*are* computable for noncongruence subgroups.

Finding coset representatives for , and in is straightforward and will be discussed at length later in this book. To make the problem more explicit, note that you can quotient out by first. Then the question amounts to finding coset representatives for a subgroup of (and lifting), which is reasonably straightforward.

Given coset representatives for a finite index subgroup of , we can compute generators for as follows. Let be a set of coset representatives for . Let be the matrices denoted by and in (2). Define maps as follows. If , then there exists a unique such that . Let . Likewise, there is a unique such that and we let . Note that and are in for all . Then is generated by .

Proposition 1.17

The above procedure computes generators for .

Proof

Without loss of generality, assume that represents the coset of . Let be an element of . Since and generate , it is possible to write as a product of powers of and . There is a procedure, which we explain below with an example in order to avoid cumbersome notation, which writes as a product of elements of times a right coset representative . For example, if

then for some . Continuing,

for some . Again,

The procedure illustrated above (with an example) makes sense for arbitrary and, after carrying it out, writes as a product of elements of times a right coset representative . But and is the right coset representative for , so this right coset representative must be .

Remark 1.18

We could also apply the proof of *Proposition 1.17* to
write any element of in terms of the given generators.
Moreover, we could use it to write any element
in the form , where and
, so we can
decide whether or not .

Let be a congruence subgroup of level . Suppose
is a cusp, and choose such
that . Recall that the minimal such that
is called the
*width of the cusp* for the group . In this
section we discuss how to compute .

Algorithm 1.19

Given a congruence subgroup of level and a cusp for , this algorithm computes the width of . We assume that is given by congruence conditions, e.g., or

[Find ]: Use the extended Euclidean algorithm to find such that , as follows.

If , set ; otherwise, write , find such that , and set .

[Compute Conjugate Matrix] Compute the following element of :

Note that the entries of are constant or linear in .

[Solve] The congruence conditions that define give rise to four linear congruence conditions on . Use techniques from elementary number theory (or enumeration) to find the smallest simultaneous positive solution to these four equations.

Example 1.20

Suppose and or . Then has the property that . Next, the congruence condition is

Thus the smallest positive solution is , so the width of is .

Suppose where are distinct primes, and let . Then sends to . The congruence condition for is

Since , we see that is the smallest solution. Thus has width , and symmetrically has width .

Remark 1.21

For , once we enforce that the bottom left entry is and use that the determinant is 1, the coprimality from the other two congruences is automatic. So there is one congruence to solve in the case. There are two congruences in the case.

The above definition of modular forms might leave the impression that
modular forms occupy an obscure corner of complex analysis. This is
*not* the case! Modular forms are highly geometric,
arithmetic, and topological objects that are of extreme interest all
over mathematics:

**Fermat’s last theorem:**Wiles’ proof [**Wil95**] of Fermat’s last theorem uses modular forms extensively. The work of Wiles et al. on modularity also massively extends computational methods for elliptic curves over , because many elliptic curve algorithms, e.g., for computing -functions, modular degrees, Heegner points, etc., require that the elliptic curve be modular.**Diophantine equations:**Wiles’ proof of Fermat’s last theorem has made available a wide array of new techniques for solving certain diophantine equations. Such work relies crucially on having access to tables or software for computing modular forms. See, e.g., [**Dar97**], [**Mer99**], [**Che05**], [**SC03**]. (Wiles did not need a computer, because the relevant spaces of modular forms that arise in his proof have dimension !) Also, according to Siksek (personal communication) the paper [**BMS06**] would “have been entirely impossible to write without [the algorithms described in this book].”**Congruent number problem:**This ancient open problem is to determine which integers are the area of a right triangle with rational side lengths. There is a potential solution that uses modular forms (of weight ) extensively (the solution is conditional on truth of the Birch and Swinnerton-Dyer conjecture, which is not yet known). See [**Kob84**].**Topology:**Topological modular forms are a major area of current research.**Construction of Ramanujan graphs:**Modular forms can be used to construct almost optimal expander graphs, which play a role in communications network theory.**Cryptography and Coding Theory:**Point counting on elliptic curves over finite fields is crucial to the construction of elliptic curve cryptosystems, and modular forms are relevant to efficient algorithms for point counting (see [**Elk98**]). Algebraic curves that are associated to modular forms are useful in constructing and studying certain error-correcting codes (see [**Ebe02**]).**The Birch and Swinnerton-Dyer conjecture:**This central open problem in arithmetic geometry relates arithmetic properties of elliptic curves (and abelian varieties) to special values of -functions. Most deep results toward this conjecture use modular forms extensively (e.g., work of Kolyvagin, Gross-Zagier, and Kato). Also, modular forms are used to compute and prove results about special values of these -functions. See [**Wil00**].**Serre’s Conjecture on modularity of Galois representation:**Let be the Galois group of an algebraic closure of . Serre conjectured and many people have (nearly!) proved that every continuous homomorphism , where is a finite field and , “arises” from a modular form. More precisely, for almost all primes the coefficients of a modular (eigen-)form are congruent to the traces of elements , where are certain special elements of called Frobenius elements. See [**Rib01**] and [**DS05**, Ch. 9].**Generating functions for partitions:**The generating functions for various kinds of partitions of an integer can often be related to modular forms. Deep theorems about modular forms then translate into results about partitions. See work of Ramanujan, Gordon, Andrews, and Ahlgren and Ono (e.g., [**AO01**]).**Lattices:**If is an even unimodular lattice (the basis matrix has determinant and for all ), then the theta seriesis a modular form of weight . The coefficient of is the number of lattice vectors with squared length . Theorems and computational methods for modular forms translate into theorems and computational methods for lattices. For example, the 290 theorem of M. Bharghava and J. Hanke is a theorem about lattices, which asserts that an integer-valued quadratic form represents all positive integers if and only if it represents the integers up to ; it is proved by doing many calculations with modular forms (both theoretical and with a computer).

Exercise 1.1

Suppose has positive determinant. Prove that if is a complex number with positive imaginary part, then the imaginary part of is also positive.

Exercise 1.2

Prove that every rational function (quotient of two polynomials) is a meromorphic function on .

Exercise 1.3

Suppose and are weakly modular functions for a congruence subgroup with .

- Prove that the product is a weakly modular function for .
- Prove that is a weakly modular function for .
- If and are modular functions, show that is a modular function for .
- If and are modular forms, show that is a modular form for .

Exercise 1.4

Suppose is a weakly modular function of odd weight and level for some . Show that .

Exercise 1.5

Prove that .

Exercise 1.6

- Prove that is a group.
- Prove that has finite index in (Hint: It contains the kernel of the homomorphism .)
- Prove that has finite index in .
- Prove that and have level .

Exercise 1.7

Let be an integer, and for any function and , set . Prove that if , then for all we have

Exercise 1.8

Prove that for any , there exists such that .

Exercise 1.9

Prove *Proposition 1.13*, which asserts that the set
of cusps , for any congruence subgroup , is
finite.

Exercise 1.10

Use *Algorithm 1.19* to give an
example of a group and cusp
with width .