Last time we are talking about \(X(1)\) and \(X_1(N)\), they are fairly easy to define over \(\mathbb{C}\) as Riemann surfaces.
\[\mathbb{H}^* = \mathbb{H}\cup \mathbb{P}^1(\mathbb{Q}) = \mathbb{H}\cup \mathbb{Q}\cup \{\infty\}\]
\[X(1)(\mathbb{C}) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}^*, \quad Y(1)(\mathbb{C}) = \mathrm{SL}_2(\mathbb{Z}) \backslash \mathbb{H}\]
bacause \(\mathrm{SL}_2(\mathbb{Z})\) acts transitively on \(\mathbb{P}^1\), we have \(X(1)(\mathbb{C}) = Y(1)(\mathbb{C})\cup \{\infty\}\), a single cusp.