Franz Lemmermeyer

Topics in Algebraic Geometry: Elliptic Curves


John Baez, a well known mathematical physicist, writes an enlightening weekly column about what is happening in his area. Here's week 201.


    Monday     11:40 - 12:30, SBZ-11
    Wednesday  10:40 - 12:30, seminar room 
You will be graded on take home exams and homework.

Content and Preliminaries

From the quiz I gather that all of you have some background in complex analysis, and that the majority is interested in abelian varieties. As for the other topics: I will review some of them briefly in class. Of course I will get nowhere if I do everything from scratch, so here's my suggestion: I will discuss affine and projective spaces, the snake lemma, p-adic numbers, and eventually a simple case of Riemann-Roch in Algebraic Geometry; you don't have to attend, but probably should read the notes. In this class, I will explain these concepts more briefly, but maybe also more informally.
You will also need a little bit of Galois theory; let me advertise my own seminar on Galois theory this semester: if more than one student is interested, we'll do it. It will also give you a short introduction to algebraic number theory. Email me if you're interested.


  1. J. Silverman, J. Tate: Rational points on elliptic curves
    The most elementary introduction.
  2. J.W.S. Cassels: Elliptic Curves.
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    Actually it's quite a good book, even for men (Cassels' book, I mean).
  3. L. Washington: Elliptic Curves
    This one has just appeared. It is very elementary.
  4. A. Knapp: Elliptic Curves
    Gives a nice introduction to the analytic aspects
  5. J. Silverman: Arithmetic of Elliptic Curves
    The standard reference
  6. Husemöller: Elliptic Curves
    There will be a new edition in 2004


Homework Problems


Here's a windows executable of pari. If you type in ?, you'll get a list of chapters; ?4 lists e.g. the number theoretical functions, and ?gcd tells you what gcd does. You can find a more detailed manual at the pari homepage in Bordeaux.
John Voight gives a description of how to use pari (and other programs) for computing with elliptic curves.


Here are Nigel Boston's notes on Wiles' proof of Fermat's Last Theorem. And here is a page with several low-brow introductions to the proof, along with the original articles.