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MTH 405/505: Elliptic Curves


The class might be especially interesting to advanced high school students who are interested in research projects. If you are a high school student who has already taken Calculus and are interested in this class, then you should contact me at    tshaska@princeton.edu 

If you are an undergraduate student with strong computing skills (Python, Sage) and would like to get involved in some research project, then you should contact me asap. 

Course:
MTH 405/505 : Elliptic Curves
Instructor: T. Shaska
Semester: Fall, Winter, Summer


This course is intended as an introduction to elliptic curves and their applications in cryptography.  It is intended for students of mathematics, computer science, and engineering.  The course is offered either as a reading course or regular course depending on the number of students enrolled.

If you are one of the following

- an advanced high school student taking courses at OU, 
- a math major who is interested to going to graduate school,  
- a computer science major interested in computer security and cryptography

then you are strongly recommended to take this course. If you decide to do so, please fill the form at the end. 

Decsription: The course will be an introduction to elliptic curves covering the geometry of elliptic curves, minimal Weierstrass equation, elliptic curves over local fields, elliptic curves over global fields, Mordell-Weil theorem, the method of descent, torsion points, Selmer and Shafarevich-Tate groups, L-functions, and possible generalization of some of these topics to hyperelliptic curves. 


Prerequisites: 
MTH 263
MTH 275 or equivalent

The student is expected to have some mathematical maturity, knowledge of congruences and the arithmetic modulo n.


Textbook

http://www.amazon.com/Rational-Points-Elliptic-Undergraduate-Mathematics/dp/0387978259/ref=sr_1_6?s=books&ie=UTF8&qid=1396006443&sr=1-6&keywords=elliptic+curves
The following book is a good reference even though we will venture outside the topics of the book during the semester.

Title: Rational points on elliptic curves
Author: J. Silverman, J. Tate

Topics

Elliptic curves over C
  • Weierstrass equation
  • Discriminant and the j-invariant
Geometry of Elliptic curves
  • The group of points of elliptic curves
  • Subgroups E(Q), E(R)
  • Torsion and rank
Reduction modulo p
  • Primes of good reduction
  • Bad reduction
  • Conductor of elliptic curves
Elliptic curves over finite fields

Elliptic Curve Cryptography

Factoring large number with elliptic curves


Please fill the form below if interested for the course:



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