Elliptic curves over the complex field: lattice-elliptic curve correspondence

Abstract: 

An elliptic curve over a field k is a smooth projective curve of genus 1 with a distinguished point O. With Riemann-Roch theorem and assuming that char(K) $\neq$ 2,3 ,  every elliptic curve can be represented by $y^2 = x^3 + Ax + B,$ for $A , B \in k$. In this talk,  we give an overview of the uniformisation theorem, which states that any elliptic curve defined over $\mathbb{C}$ is isomorphic to $\mathbb{C}/L$ for some lattice $L$. We start by looking at maps that carry $\mathbb{C}/L$ to $E(\mathbb{C})$ isomorphically.  Then we look at two proofs of the inversion problem. The first proof uses the fact that j-invariant classifies elliptic curves over C up to isomorphism. The second proof is more concrete and is achieved via construction of a Riemann surface.