A modular form is a type of function in complex analysis, with connections to algebraic geometry and number theory. Modular forms played a key rôle in Andrew Wiles' highly-publicized proof of Fermat's last theorem.
The modular group
The special linear group of dimension 2 over the integers, , consisting of 2 by 2 matrices with integer entries and determinant 1, is referred to as the modular group. An action of the modular group may be defined on the upper half-plane , consisting of those complex numbers with a strictly positive imaginary part, as follows:
and . The proof that this is indeed an action, respecting the group operation and inverses, is beyond the scope of this article, though it is easy to verify that the half-plane is closed under it.
A function , where denotes the Riemann sphere, is said to be weakly modular of weight if for all .