Nonsingular cubics and complex tori

Abstract

The main goal of this project is to prove the correspondence between nonsingular projective cubic curves, also known as elliptic curves, and complex tori. The study is divided into two main parts: an algebraic part and an analytic part. We begin by introducing the necessary background in Riemann surfaces and algebraic curves. The algebraic part consists on the classification of projective cubic curves, and, in particular, the construction of a group structure on the nonsingular ones. In the analytic part, we first show that nonsingular projective cubic curves can be viewed as compact Riemann surfaces of genus one, and, subsequently, introduce complex tori and their group structure. Finally, we prove the equivalence between these two Riemann surfaces by constructing an explicit isomorphism that preserves the group structure.

Notation: We will work over the field of the complex numbers.