Abelian varieties are projective algebraic groups, that is, projective algebraic varieties with a group action that's compatible with the algebraic structure. They appear regularly in several parts of mathematics besides algebraic geometry, for example in number theory, differential geometry and topology. They are also useful for demonstrating complicated general ideas, because some of their structure simplifies certain details, but they are complicated enough to make the applications interesting. A huge percentage of explicit examples in general involve abelian varieties at some stage of the construction, even if the end result does not.

One dimensional abelian varieties are better known as "elliptic curves". These have played a central role in number theory for a long time. In particular, the most difficult part of Wiles' celebrated proof of Fermat's Last Theorem involved elliptic curves.

Prerequisites:

• 506 (Algebra) and 546 (Manifolds), or instructor's permission.
• Basic knowledge of algebraic geometry (e.g., on the level of 507) is helpful, but not mandatory.

The class will not follow one single textbook. Nevertheless, the following references may be useful.

• David Mumford, Abelian Varieties, Oxford University Press, 1985.
• V. Kumar Murty, Introduction to Abelian Varieties, American Mathematical Society, 1993.  ISBN: 0821811797.
• Herbert Lange and Christina Birkenhake, Complex Abelian Varieties,Springer-Verlag, 1992.  ISBN: 0387547479.