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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 important feature of elliptic curves is that they are in many cases the simplest and most accessible examples of interesting classes of varieties. For instance, elliptic curves are the only 1-dimensional abelian varieties, that is, smooth projective group varieties. In fact, we will see that many properties of abelian varieties are visible already in dimension 1, and that a good understanding of elliptic curves makes the step to general abelian varieties easier and more logical. Elliptic curves are also the only 1-dimensional Calabi-Yau varieties, a class of varieties of great importance both in mathematics and in modern physics.