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  "body": "\n## Definition\n\nLet $\\mathbb{H} = \\set{\\tau \\in \\C : \\operatorname{Im} \\tau \u003e 0}$ be the [[upper half-plane|X6XGGH5FPW]], on which $\\mathrm{SL}_2(\\Z)$ acts by $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix} \\tau = \\frac{a\\tau + b}{c\\tau + d}$.\nLet $\\Gamma \\subseteq \\mathrm{SL}_2(\\Z)$ be a [[congruence subgroup|1H6R856P3C]], that is, a subgroup that contains $\\Gamma(N) = \\set{\\gamma \\in \\mathrm{SL}_2(\\Z) : \\gamma \\equiv I \\pmod N}$ for some $N \\ge 1$.\nThe *open modular curve* of $\\Gamma$ is the quotient $Y(\\Gamma) = \\Gamma \\backslash \\mathbb{H}$.\nThe *modular curve* of $\\Gamma$ is\n$$X(\\Gamma) = \\Gamma \\backslash \\big( \\mathbb{H} \\cup \\mathbb{P}^1(\\Q) \\big),$$\nwhere $\\mathbb{P}^1(\\Q) = \\Q \\cup \\set{\\infty}$ is acted on by the same formula; the finitely many points of $X(\\Gamma) \\setminus Y(\\Gamma)$ are the cusps.\n\n## Remarks\n\nWith the standard charts at elliptic points and at cusps, $X(\\Gamma)$ is a compact [[Riemann surface|5K0N8QMV6Q]], hence a projective algebraic curve over $\\C$.\nThe standard families are $X(N) = X(\\Gamma(N))$, $X_0(N)$ for $\\Gamma_0(N) = \\set{\\gamma : c \\equiv 0 \\pmod N}$ and $X_1(N)$ for $\\Gamma_1(N) = \\set{\\gamma : c \\equiv 0,\\ a \\equiv d \\equiv 1 \\pmod N}$.\nThe curves $X_0(N)$ and $X_1(N)$ have models over $\\Q$.\nThe open curve $Y_0(N)$ parametrizes pairs $(E, C)$ of an elliptic curve over $\\C$ and a cyclic subgroup $C$ of order $N$, up to isomorphism.\n\nExample: for $\\Gamma = \\mathrm{SL}_2(\\Z)$, the [[modular group|VEKV1EM77B]] up to sign, the $j$-invariant identifies $Y(1)$ with $\\C$ and $X(1)$ with $\\mathbb{P}^1(\\C)$, of genus $0$.\nThe curve $X_0(11)$ has genus $1$ and is an elliptic curve over $\\Q$.\n\nNon-example: a subgroup of finite index in $\\mathrm{SL}_2(\\Z)$ that contains no $\\Gamma(N)$ is not a congruence subgroup.\nIts quotient is still a compact Riemann surface, but it is not a modular curve in the sense above.\n",
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