I will lay much of the mathematical groundwork we will need moving forward in the summer. I will discuss intersection numbers, Bezout’s Theorem, projective geometry, and such. I will also introduce the group law on an elliptic curve and, if I have time, prove that we do, indeed, get a group.

We will introduce lattices in the complex plane and realize complex tori as quotients of the complex plane by lattices. Using the Weierstrass $\wp$-function, we will see that elliptic curves over $\mathbb{C}$ are algebraically and analytically the same as complex tori. We may also study the endomorphism group of complex elliptic curves, time permitting.

In this talk, I will discuss the reduction of an elliptic curve mod p, and, time permitting, go back to the contents of Theo’s talk and draw a relation between elliptic curves over $\mathbb{C}$ and the KdV equation.