Modular form

Summary

A modular form of weight $k$ is a holomorphic function on the upper half-plane. It transforms in a fixed way under the modular group and stays bounded at infinity. Its Fourier expansion $\sum_{n \ge 0} a_n q^n$ in $q = e^{2 \pi i \tau}$ often has arithmetic coefficients. The spaces of modular forms of a given weight are finite-dimensional, so identities between them can be proved by comparing a few coefficients. Modular forms are central in modern number theory, for example in the proof of Fermat's Last Theorem through the modularity theorem.