A function 
is said to be an entire modular form of weight 
if it satisfies
1. 
is analytic in the upper
half-plane 
,
2. 
whenever ![[a b; c d]](/images/equations/ModularForm/Inline6.svg)
is a member of the modular group Gamma,
3. The Fourier series of 
has the form
 |
(1)
|
Care must be taken when consulting the literature because some authors use the term "dimension 
"
or "degree 
"
instead of "weight 
,"
and others write 
instead of 
(Apostol 1997, pp. 114-115). More general types of modular forms (which are
not "entire") can also be defined which allow poles in 
or at 
. Since Klein's
absolute invariant 
,
which is a modular function, has a pole at 
, it is a nonentire modular form
of weight 0.
The set of all entire forms of weight 
is denoted 
, which is a linear space over the complex field. The dimension
of 
is 1 for 
, 6, 8, 10, and 14 (Apostol 1997, p. 119).
is the value of 
at 
,
and if 
,
the function is called a cusp form. The smallest 
such that 
is called the order of the zero of 
at 
.
An estimate for 
states that
 |
(2)
|
if 
and is not a cusp
form (Apostol 1997, p. 135).
If 
is an entire modular form of weight
, let 
have 
zeros in the closure of the fundamental
region 
(omitting the vertices). Then
 |
(3)
|
where 
is the order of the zero at a point 
(Apostol 1997, p. 115). In addition,
1. The only entire modular forms of weight 
are the constant functions.
2. If 
is odd, 
, or 
, then the only entire modular form of weight 
is the zero function.
3. Every nonconstant entire modular form has weight 
, where 
is even.
4. The only entire cusp form of weight 
is the zero function.
(Apostol 1997, p. 116).
For 
an entire modular form of even
weight 
,
define 
for all 
.
Then 
can be expressed in exactly one way as a sum
 |
(4)
|
where 
are complex numbers, 
is an Eisenstein series, and 
is the modular discriminant
of the Weierstrass elliptic function.
cusp forms of even weight
are then those sums for which 
(Apostol 1997, pp. 117-118). Even more amazingly,
every entire modular form 
of weight 
is a polynomial in 
and 
given by
 |
(5)
|
where the 
are complex numbers and the sum is extended over all integers 
such that 
(Apostol 1998, p. 118).
Modular forms satisfy rather spectacular and special properties resulting from their surprising array of internal symmetries. Hecke discovered an amazing connection between each modular form and a corresponding Dirichlet L-series. A remarkable connection between rational elliptic curves and modular forms is given by the Taniyama-Shimura conjecture, which states that any rational elliptic curve is a modular form in disguise. This result was the one proved by Andrew Wiles in his celebrated proof of Fermat's last theorem.
