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).

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, Elliptic Curve, Elliptic Function, Fermat's Last Theorem, Hecke Algebra, Modular Function, Modular Function Multiplier, 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.

Knopp, M. I. Modular Functions, 2nd ed. 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.