MATH 7390 - Abelian Varieties

Farbod Shokrieh, spring 2016.

This is an introductory course on the analytic and algebraic theory of Abelian (and Jacobian) varieties. We will start with the classical complex-analytic case to build some intuition. Then we will discuss the general theory over other fields. More advanced topics might include:

• Non-Archimedean uniformization,
• Heights and metrized line bundles,
• Degenerating families,
• Theta functions,
• Néron Models.

(The choice of the more advanced topics will depend on the background and interest of the audience.)

Prerequisite

Familiarity with algebraic geometry, especially the language of schemes.

References

We will not follow any specific books. I will use the following resources while preparing for the lectures:

• Mumford - Abelian Varieties (book)
• Milne - Abelian Varieties (notes available online)
• Birkenhake, Lange - Complex Abelian Varieties (book)
• Faltings, Chai - Degeneration of Abelian Varieties (book)
• Bosch, Lütkebohmert, Raynaud - Néron Models (book)
• Cornell, Silverman - Arithmetic Geometry (book)
• Mumford - Tata Lectures on Theta (I,II,III) (books)
• Various papers and notes by Mumford, B. Conrad, Bosch-Lütkebohmert.