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


Victoria Gu


UNSW Sydney


Wed, 05/08/2020 - 12:00pm


Zoom link: https://unsw.zoom.us/j/92916731416


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.

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