Modular curves: Riemann surfaces and transition maps

This lecture introduces modular curves as compact Riemann surfaces, denoted as X(1), obtained by compactifying the quotient space Y(1). The instructor explains the topology of modular curves, the definition of Riemann surfaces, and the construction of holomorphic charts around elliptic points and cusps. The lecture covers the properties of modular curves, including being Hausdorff, connected, and compact. Additionally, the transition maps between the charts defined around regular points, elliptic points, and cusps are shown to be holomorphic.