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  "body": "\n## Summary\n\nA 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.\nThe result is a compact [[Riemann surface|5K0N8QMV6Q]] and an algebraic curve defined over a number field.\nModular curves parametrize [[elliptic curves|427N6S6QQW]] with extra level structure, and their holomorphic differentials correspond to cusp [[modular forms|GWKN5VG78A]] of weight $2$.\nThey are central in the proof of Fermat's Last Theorem through the modularity of elliptic curves.\n\n## Explanations\n\n### L1 human\n\nThe group $\\mathrm{SL}_2(\\Z)$ moves points of the upper half-plane by fractional linear maps $\\tau \\mapsto (a\\tau + b)/(c\\tau + d)$.\nGluing together points that are moved into each other produces a surface.\nFor 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.\nFor 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.\n",
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