MATH 732: The Arithmetic of Elliptic Curves

(Official) Course Description: 

MATH 732: The Arithmetic of Elliptic Curves

This course will be an introduction to the theory of elliptic curves. We will use the standard reference in the subject, "The Arithmetic of Elliptic Curves", by J. Silverman. The goal of the course will be to understand and calculate the set of all rational points on a given elliptic curve (i.e. calculate the torsion and the rank), and a number of refined invariants (such as the order of the Tate-Shafarevich group). The prerequisite for this course is a basic understanding of algebraic number theory and algebraic geometry, although I will adjust the material to the audience background as much as I can.

Books

I will be following the main reference: Joseph H. Silverman, “The Arithmetic of Elliptic Curves”, Springer. However, there are other references that may be very useful.

1. J. Tate, “The arithmetic of elliptic curves”, Invent. Math. 23 (1974), 179-206 – The main reference was heavily influenced by this key survey article.

2. J. H. Silverman, J. Tate, “Rational Points on Elliptic Curves”, Springer – This book is an elementary version of the main reference.

3. J. H. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Springer.

4. J. S. Milne, “Elliptic Curves”, Kea Books - this book is freely available at
   http://www.jmilne.org/math/CourseNotes/math679.html

5. N. Koblitz, “Introduction to Elliptic Curves and Modular Forms”.
   
   

ASSIGNMENTS

During the semester I will propose assignments to be handed in or presented in class. Silverman's book contains an excellent collection of exercises. I will also propose exercises to be solved with Magma (or Sage, or PARI).