---
tag: GWKN5VG78A
kind: concept
name: Modular form
msc: [11F11, 11F03]
specializes: [3C73HND6XA]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_form
same_as:
  wikidata: Q870797
  wikipedia: Modular form
---

## Summary

A modular form of weight $k$ is a [[holomorphic function|0NAHXR1FRH]] on the [[upper half-plane|X6XGGH5FPW]].
It transforms in a fixed way under the [[modular group|VEKV1EM77B]] 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|CS1NVJBZMH]].

## Explanations

### L1 human

A modular form is a function with an enormous amount of symmetry.
It is periodic, $f(\tau + 1) = f(\tau)$, and it also changes in a simple way under $\tau \mapsto -1/\tau$.
These symmetries are so strong that, for each weight, the modular forms form a space of small dimension.
The [[Eisenstein series|EYREVY31BN]] $E_4(\tau) = 1 + 240 \sum_{n \ge 1} \sigma_3(n) q^n$ is an example of weight $4$.
Coefficients of modular forms count things, such as the number of ways to write $n$ as a sum of squares.