Number Theory – Mathematics 2540
Brown University – Spring, 2012
Professor Joseph Silverman

Topics In this course we will discuss two topics: Local Fields and Elliptic Curves. A thorough introduction to Local Fields would take most of the semester, while a thorough introduction to Elliptic Curves would take at least two semesters. In order to get to some significant theorems about elliptic curves, the coverage of local fields will be a rapid survey, with some proofs omitted.
Text There will not be an official textbook for local fields. You can find the material in any standard text, but I'll mostly be basing my lectures on Lang's Algebraic Number Theory (Chapter II) and Serre's Local Fields. The text for elliptic curves is:

The Arithmetic of Elliptic Curves
Joseph H. Silverman
Springer-Verlag, ISBN: 978-0-387-09493-9 – 2nd edition, © 2009
If you don't already have a copy, see me before you buy one.
Office Mathematics Department, Kassar House, Room 202
Phone 863-1124
Email jhs@math.brown.edu
Web Site www.math.brown.edu/~jhs/MA0254/MA0254HomePage.html
Office Hours Monday 2:30–3:15 PM
(Or send me an email to make an appointment. I tend to be in on MWF and not on TTh.)
Course Time MWF 10:00–10:50 AM (C hour)
Course Location Barus-Holley 165
Homework Homework assignments will be posted online. (I'll also try to announce them in class.)
Click here to go to the Math 254 Homework Page.
Note on Using Computers in Math 254 I tend to use a computer program called PARI-GP to do number theory and elliptic curve calculations. The good news about PARI is that it is free and very fast and powerful at doing number theoretic computations. The bad news is that it's not tremendouly user friendly. You can download PARI by clicking here. Another way to use PARI to do short calculations is to use the SAGE web site. You'll need to create a (free) account. Then you'll be able to type one or more PARI commands and type Shift-Return to perform the computation. Or you can use SAGE itself, which has a large collection of algorithms for working with number fields, local fields, and elliptic curves.

Tentative Syllabus:

Part I: Local Fields

  1. Valuations, Completions, and Absolute Values
  2. Hensel's Lemma
  3. The Multiplicative Group of a Complete Local Field
  4. Unramified and Tamely Ramified Extensions
  5. Introduction to Adeles and Ideles

Part II: The Arithmetic of Elliptic Curves

  1. Algebraic Geometry Overview
  2. The Geometry of Elliptic Curves
  3. The Formal Group of an Elliptic Curve
  4. Elliptic Curves over Finite Fields
  5. Elliptic Curves over Local Fields
  6. Elliptic Curves over Global Fields
  7. Introduction to Group Cohomology: H0 and H1
  8. Computing the Mordell–Weil Group
Further topics as time permits:

Go to Professor Silverman's Home Page.