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 ,
- 3. The Fourier Series of has the form

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.

**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.

© 1996-9

1999-05-26