Wednesdays, 7:30pm; Room 417, 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 
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 LernerBrecher 
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 jfunction 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 MordellWeil 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 padic field. Time permitting, I will also mention some of the famous open problem facing modern mathematics, such as the conjecture of Birch and SwinnertonDyer 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. 