ELLIPTIC CURVES OVER COMPLEX NUMBERS

Abstract

The goal of this project is to form an understanding of the theory of elliptic curves over complex numbers and study an application of the same. We first introduce elliptic curves, elliptic (or doubly-periodic) functions and a very useful example of the latter: The Weierstrass ‘℘’ function. The relation between elliptic curves, lattices and complex torus C/L and via the ℘ function is studied. Elliptic integrals are introduced and their application in computing periods is explained. Finally an application of elliptic integrals in constructing a conformal mapping from a rectangle to the upper half plane is explained.