Date Speaker Title Abstract Notes June 7 Adam Block Introduction to Elliptic Curves 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. Talk Notes June 14 Theo Coyne Elliptic Curves and Complex Tori 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. Talk Notes June 21 Willie Dong Reduction of an Elliptic Curve Mod p 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. Talk Notes June 28 Matthew Lerner-Brecher Elliptic Curves and their Formal Groups In this talk, I will introduce formal groups and some of their basic properties. Using the Weierstrass equation, I will then show how we can construct the formal group associated with an elliptic curve E. Time permitting, I will also discuss the height of elliptic curves. July 5 No meeting July 12 George Drimba Heegner Numbers and Almost Integers We will survey the theory of elliptic curves with complex multiplication and explore the j-function in order to find answers to arithmetic questions. July 19 July 26 August 2