Modularity of Elliptic Curves

Abstract

This study explores the modularity of elliptic curves, a concept that connects algebraic geometry with analytic number theory. While Fermat’s Last Theorem, proven by Andrew Wiles in 1995, depends on this property, the focus here is on understanding the modularity of elliptic curves rather than the theorem’s proof. Elliptic curves are introduced as smooth cubic curves defined by Weierstrass equations, with key discussions on the projective plane, smoothness conditions, and the reduction of elliptic curves modulo a prime number. These topics introduce important quantities such as the conductor and coefficients that are critical for modularity. The study also delves into modular forms, holomorphic functions on the upper half-plane with specific transformation properties, and the essential role of Hecke operators in defining modularity. Finally, we connect what was discussed in Elliptic curves and modular forms to explain what it means for an elliptic curve over the rational numbers to be modular.