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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Contact UMS
| Date | Speaker | Title | Abstract | Notes |
| June 7 |
Adam Block
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Introduction to
Elliptic Curves
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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.
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Talk Notes |
| June 14 |
Theo Coyne
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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.
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Talk Notes |
| June 21 |
Willie Dong
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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.
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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.
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| July 5 |
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No meeting |
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| 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. | Talk Notes |
| 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. |