Modular Form

A function f is said to be an entire modular form of weight k if it satisfies

1. f is analytic in the Upper Half-Plane H,

2. $f\left({a\tau+b\over c\tau+d}\right)=(c\tau+d)^kf(\tau)$ whenever $\left[{\matrix{a & b\cr c & d\cr}}\right]$ is a member of the Modular Group Gamma,

3. The Fourier Series of f has the form

$$f(\tau)=\sum_{n=0}^\infty c(n)e^{2\pi in\tau}$$

Care must be taken when consulting the literature because some authors use the term ``dimension -k'' or ``degree -k'' instead of ``weight k,'' and others write 2k instead of k (Apostol 1997, pp. 114-115). More general types of modular forms (which are not ``entire'') can also be defined which allow poles in H or at $i\infty$. Since Klein's Absolute Invariant J, which is a Modular Function, has a pole at $i\infty$, it is a nonentire modular form of weight 0.

The set of all entire forms of weight k is denoted M_k, which is a linear space over the complex field. The dimension of M_k is 1 for k=4, 6, 8, 10, and 14 (Apostol 1997, p. 119).

c(0) is the value of f at $i\infty$, and if c(0)=0, the function is called a Cusp Form. The smallest r such that $c(r)\not=0$ is called the order of the zero of f at $i\infty$. An estimate for c(n) states that

c(n)=O(n^2k-1)

if $f\in M_{2k}$ and is not a Cusp Form (Apostol 1997, p. 135).

If $f\not=0$ is an entire modular form of weight k, let f have N zeros in the closure of the Fundamental Region $R_\Gamma$ (omitting the vertices). Then

$$k=12N+6N(i)+4N(\rho)+12N(i\infty),$$

where N(p) is the order of the zero at a point p (Apostol 1997, p. 115). In addition,

1. The only entire modular forms of weight k=0 are the constant functions.

2. If k is Odd, k<0, or k=2, then the only entire modular form of weight k is the zero function.

3. Every nonconstant entire modular form for weight $k\geq 4$, where k is Even.

4. The only entire Cusp Form of weight k<12 is the zero function.

(Apostol 1997, p. 116).

For f an entire modular form of Even weight $k\geq 0$, define $E_0(\tau)=1$ for all $\tau$. Then f can be expressed in exactly one way as a sum

$$f=\sum_{r=0\atop k-12r\not=2}^{\left\lfloor{k/12}\right\rfloor } a_rE_{k-12r}\Delta^r,$$

where a_r are complex numbers, E_n is an Eisenstein Series, and $\Delta$ is the Discriminant of the Weierstraß Elliptic Function. Cusp Forms of Even weight k are then those sums for which a₀=0 (Apostol 1997, pp. 117-118). Even more amazingly, every entire modular form f of weight k is a Polynomial in E₄ and E₆ given by

$$f=\sum_{a,b} c_{a,b}E_4^aE_6^a,$$

where the c_a,b are complex numbers and the sum is extended over all integers $a,b\geq 0$ such that 4a+6b=k (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.

See also Cusp Form, Dirichlet Series, Elliptic Curve, Elliptic Function, Fermat's Last Theorem, Hecke Algebra, Hecke Operator, Modular Function, Schläfli's Modular Form, Taniyama-Shimura Conjecture

References

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.