Messages - Page 2

Some people may want to attend the meeting for teachers, which unfortunately collides with the first lecture. If you already know some algebraic geometry, then this will cause no problems. In any case, I always make complete lecture notes, just drop by my office at any time if you would like to make copies.

The first lecture is tomorrow, tuesday 21.08 at 12:15-13:00, room B62 in NHA. I will provide some practical information and give a short overview of the course and elliptic curves in general.

I will probably also start talking about (a few things from) Ch. 1 in Silverman. This will be continued in the lecture on thursday. The material in Ch. 1 is more background stuff from Algebraic Geometry, but with a twist suitable for elliptic curves. Therefore, we shall discuss this somewhat briefly, both to set notation, and as a preparation for talking about elliptic curves.

Here is a basic overview the course. More detailed information will appear over the next few weeks, and during the semester. Some changes may occur, in particular towards the end of the semester, depending on the pace of the lectures and interests of the audience.

• Review of algebraic curves: Differentials, divisors. The Riemann-Roch Theorem and The Hurwitz Formula.

• Elliptic curves: Weierstrass equations, discriminant and j-invariant. Group law. Divisors. Isogenies and the endomorphism ring. The invariant differential. Points of finite order, Tate modules. Automorphisms.

• Elliptic curves over finite fields: Number of rational points, Hasse bound. Frobenius. Weil conjectures. Hasse invariant.

• Elliptic curves over the complex numbers: Elliptic functions. Complex tori. Uniformization....

Hi and welcome to the homepage of MAT4230 - Fall 2012!

The title of the course is Elliptic Curves and Abelian Varieties. The aim of the course is to give a thorough introduction to the theory of elliptic curves, and to cover the basic parts of the theory of abelian varieties. For this course, it is good to have a basic knowledge of Commutative Algebra and Algebraic Geometry (but it may be possible to follow the lectures with some extra effort without this background).

Elliptic curves are fundamental objects in algebraic geometry and number theory. The most important feature of an elliptic curve is the fact that its points form an abelian group. Because of this, elliptic curves enjoy rich arithmetic properties. In fact, some of the most famous conjectures in number theory concern, or have their origins in, elliptic curves. The interplay between the geometry and arithmetic of elliptic curves will surface many times during the semester.

Another import...