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 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.

Apostol, T. M. "Modular Forms with Multiplicative Coefficients." Ch. 6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 113-141, 1997.

Hecke, E. "Über Modulfunktionen und die Dirichlet Reihen mit Eulerscher Produktentwicklungen. I." Math. Ann. 114, 1-28, 1937.

Knopp, M. I. Modular Functions in Analytic Number Theory. New York: Chelsea, 1993.

Koblitz, N. Introduction to Elliptic Curves and Modular Forms. New York: Springer-Verlag, 1993.

Rankin, R. A. Modular Forms and Functions. Cambridge, England: Cambridge University Press, 1977.

Sarnack, P. Some Applications of Modular Forms. Cambridge, England: Cambridge University Press, 1993.

CITE THIS AS:

Weisstein, Eric W. "Modular Form." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ModularForm.html