---
tag: T5W0CMS0DM
kind: concept
name: Modular curve
msc: [11G18, 14G35, 11F06]
sources:
  - url: https://en.wikipedia.org/wiki/Modular_curve
same_as:
  wikidata: Q3001220
  wikipedia: Modular curve
---

## Summary

A modular curve is the quotient of the [[upper half-plane|X6XGGH5FPW]] by a [[congruence subgroup|1H6R856P3C]] of $\mathrm{SL}_2(\Z)$, compactified by adding finitely many cusps.
The result is a compact [[Riemann surface|5K0N8QMV6Q]] and an algebraic curve defined over a number field.
Modular curves parametrize [[elliptic curves|427N6S6QQW]] with extra level structure, and their holomorphic differentials correspond to cusp [[modular forms|GWKN5VG78A]] of weight $2$.
They are central in the proof of Fermat's Last Theorem through the modularity of elliptic curves.

## Explanations

### L1 human

The group $\mathrm{SL}_2(\Z)$ moves points of the upper half-plane by fractional linear maps $\tau \mapsto (a\tau + b)/(c\tau + d)$.
Gluing together points that are moved into each other produces a surface.
For the full group the surface, with one cusp added, is a sphere, and each point other than the cusp stands for one elliptic curve over $\C$ up to isomorphism.
For smaller groups, such as matrices with $c$ divisible by $N$, one gets curves of higher genus that also remember some points of finite order on the elliptic curve.