# Modular form

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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, ${\displaystyle \mathrm {SL} _{2}(\mathbf {Z} ))}$, 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 ${\displaystyle \mathbf {H} }$, consisting of those complex numbers with a strictly positive imaginary part, as follows:

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

where

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

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