Columbia Undergraduate Math Society

Spring 2017« Summer 2017 Lectures »Fall 2017

Wednesdays, 7:30pm; Room 417, Mathematics
Topic: Elliptic Curves 
Texts:  J.S. Milne, Elliptic Curves,
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
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 Noah Miller Bosonic String Theory
in 26 Dimensions
In this talk, I will tell you why bosonic string theory works best in 26 dimensions. I will say the words "elliptic curve" exactly once in the talk and it will blow your mind.    
July 26 David Grabovsky Galois Cohomology
and Way Too Many
Exact Sequences
It was an ancient problem posed by the Greeks, to find integer solutions to polynomial equations; or, in more modern terminology, to find rational points on algebraic curves. To that end, we will study elliptic curves over the rational numbers and endeavor to prove a weak version of the Mordell-Weil Theorem: over a number field, an elliptic curve forms a finitely generated abelian group. Our weapons of choice will be the cohomology of Galois groups and the algebra of elliptic curves over the p-adic field. Time permitting, I will also mention some of the famous open problem facing modern mathematics, such as the conjecture of Birch and Swinnerton-Dyer and the question of computing the rank of an elliptic curve.    
August 2 Zach Davis The Riemann Hypothesis
on Elliptic Curves
over Finite Fields
We generalize the Riemann zeta function and Riemann hypothesis to a statement on global fields. We briefly discuss the Weil conjectures. Then, we state and prove the analogous Riemann hypothesis for elliptic curves defined over finite fields.    
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