Modular form

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, $\mathrm {SL} _{2}(\mathbf {Z} ))$, consisting of 2 by 2 matrices with integer entries with determinant 1, is referred to as the modular group. An action of the modular group may be defined on the upper half-plane $\mathbf {H}$, consisting of those complex numbers with a strictly positive imaginary part, as follows:

$\gamma (\tau )={\frac {a\tau +b}{c\tau +d}}$,

where

$\gamma =\left[{\begin{array}{cc}a&b\\c&d\end{array}}\right]\in \mathrm {SL} _{2}(\mathbf {Z} )$

and $\tau \in \mathbf {H}$. 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.

Weak modularity

A function $f:\mathbf {H} \to \mathbf {\hat {C}}$, where $\mathbf {\hat {C}}$ denotes the Riemann sphere, is said to be weakly modular of weight $k$ if $f(\gamma (\tau ))=(c\tau +d)^{k}f(\tau )$ for all $\gamma \in \mathrm {SL} _{2}(\mathbf {Z} )$.