An elliptic curve over the complex field is isomorphic to the quotient of the complex plane by a lattice \(\Lambda\). The quotient of the upper half plane by an action of subgroups of \(\mathrm{SL}_2(\mathbb{Z})\) define moduli spaces of the isomorphism classes of elliptic curves and their N-torsion points. These quotient spaces, known as modular curves, are compactified by gluing in a finite number of points, known as cusps. In this paper we will explore the isomorphism classes of elliptic curves under varying group actions, study interesting properties of cusps and their width, and examine how cusps behave under maps between modular curves.

This talk is in-person in Hylan 1104 but might also be streamed live on zoom meeting ID: 986 2892 7517