Columbia Undergraduate Math Society

Spring 2017« Summer 2017 Lectures »Fall 2017

Wednesdays, 7:30pm; Room TBD, Mathematics
Topic: Elliptic Curves 
Texts:  J.S. Milne, Elliptic Curves, http://www.jmilne.org/math/Books/ectext5.pdf
Joseph H. Silverman, The Arithmetic of Elliptic Curves
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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        
         
         
     
 
 
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