http://www.math.tamu.edu/~map/courses/662sp10/

Topics of Study: Elliptic curves make up an important class of geometric objects, which have connections in many areas of mathematics, especially number theory and algebraic geometry, as well as several applications, including to cryptography and integer factorization. At the outset elliptic curves are merely solutions of certain cubic polynomial equations, but the points on these curves possess a natural abelian group structure, which leads to many mesmerizing problems. This course will focus on the study of elliptic curves over various fields, including the rational numbers, finite fields, the padic numbers, and the complex numbers. The two main goals of the course will be to prove the MordellWeil theorem, which states that the group of points over the rational numbers is finitely generated, and to discuss the Birch and SwinnertonDyer conjecture, which relates the rank of the MordellWeil group to the vanishing of the Lfunction of the elliptic curve. Students interested in doing research in Number Theory or Arithmetic Geometry are especially encouraged to attend this course. The course will cover the following topics as time permits:
Prerequisites: The course prerequisites are Math 653 (1st semester graduate algebra), or consent of the instructor. Otherwise the course will be fairly selfcontained, and necessary elements of algebra, number theory, and algebraic geometry will be covered during the semester. 

Instructor:  Dr. Matthew Papanikolas 


Office Hours:  Tues. 3:004:00, Wed. 11:0012:00; also by appointment 




Office:  321 Milner 

Office Phone:  8451615 

Email:  map@math.tamu.edu 




Textbook:  Elliptic Curves, 2nd Ed., by Dale Husemöller, SpringerVerlag, 2004, ISBN 0387954902. 




Prerequisites:  Math 653 or equivalent 




Course Webpage:  http://www.math.tamu.edu/~map/courses/662sp10/ 




Course Work and Grades:  There will be regular homework assignments, as well as a final project. These will serve as the basis for grades in the course. 