The compactified level-one modular curve X(1)

Let ${\mathfrak{H}}^{\ast }=\mathfrak{H}\cup \mathbb{Q}\cup \{\mathrm{\infty }\}$ be the space of The cusp chart and compactness of X(1) with its cusp-neighbourhood topology and the action of $PS{L}_2(\mathbb{Z})$ (The modular group and its action on the upper half-plane, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection). The compactified level-one modular curve is the quotient

$X(1):=PS{L}_2(\mathbb{Z})\mathrm{\backslash }{\mathfrak{H}}^{\ast },$

with the quotient topology and the structure of a compact Riemann surface whose interior quotient map $\mathfrak{H}\to X(1)$ is holomorphic and whose cusp coordinate is $q=e^{2\pi i\tau }$ (The cusp chart and compactness of X(1), Riemann surfaces and holomorphic atlases); its single cusp is the class $[\mathrm{\infty }]$. The open modular curve is the dense open subset

$Y(1):=PS{L}_2(\mathbb{Z})\mathrm{\backslash }\mathfrak{H}=X(1)\setminus \{[\mathrm{\infty }]\},$

whose Riemann surface structure is the one supplied by the local chart lemma for the quotient of the upper half-plane (Local charts and the Riemann surface structure of a modular quotient). More generally, for a finite-index subgroup $\mathrm{\Gamma }\le PS{L}_2(\mathbb{Z})$ we write $X_{\mathrm{\Gamma }}:=\mathrm{\Gamma }\mathrm{\backslash }{\mathfrak{H}}^{\ast }$ and $Y_{\mathrm{\Gamma }}:=\mathrm{\Gamma }\mathrm{\backslash }\mathfrak{H}$ for the corresponding quotient spaces, with $Y_{\mathrm{\Gamma }}$ carrying the complex structure of Local charts and the Riemann surface structure of a modular quotient.