In mathematics, a modular form is a holomorphic function on the complex upper half-plane, ${\displaystyle {\mathcal {H}}}$, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory.

Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group ${\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )}$. Every modular form is attached to a Galois representation.

The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously represented in a possibly apocryphal quote attributed to Martin Eichler describing modular forms as being the fifth fundamental operation in mathematics, after addition, subtraction, multiplication and division.

In general, given a subgroup ${\displaystyle \Gamma <{\text{SL}}_{2}(\mathbb {Z} )}$ of finite index (called an arithmetic group), a modular form of level ${\displaystyle \Gamma }$ and weight ${\displaystyle k}$ is a holomorphic function ${\displaystyle f:{\mathcal {H}}\to \mathbb {C} }$ from the upper half-plane satisfying the following two conditions:

In addition, a modular form is called a cusp form if it satisfies the following growth condition:

Note that ${\displaystyle \gamma }$ is a matrix

identified with the function ${\textstyle \gamma (z)=(az+b)/(cz+d)}$. The identification of functions with matrices makes function composition equivalent to matrix multiplication.

Modular forms can also be interpreted as sections of a specific line bundle on modular varieties. For ${\displaystyle \Gamma <{\text{SL}}_{2}(\mathbb {Z} )}$ a modular form of level ${\displaystyle \Gamma }$ and weight ${\displaystyle k}$ can be defined as an element of