Abelian Varieties: Geometry, Parameter Spaces, and Arithmetic

In this article I wish to convey to beginners how analytic, algebro-geometric, and arithmetic techniques come together in the study of abelian varieties and their variation in families. For many modern arithmetic applications it is crucial that the entire theory admits an algebraic foundation (requiring the full force of the theory of schemes and beyond), but much of our geometric intuition and expectation is guided by the complex-analytic theory. It is for this reason that the first three sections §1-§3 are devoted to an overview of the analytic aspects of the theory, with an emphasis on those structures and examples that will be seen to admit natural analogues in the algebraic theory. With this experience behind us, §4-5 explain how to make the appropriate definitions in the algebraic setting and what sorts of results and interesting examples one obtains. The final two sections (§6 and §7) are devoted to the study of analytic and algebraic families of abelian varieties parameterized by some “modular” varieties, with the aim of explaining (via the Main Theorem of Complex Multiplication) how these varieties fit within the framework of Shimura varieties. It is assumed that the reader has some prior exposure to the basics of algebraic geometry over an algebraically closed field, and notions such as complex manifold and vector bundle over a manifold (as well as bundle operations, such as dual, tensor product, and pullback). We will certainly have to allow ourselves to work with algebro-geometric objects over a field that is not algebraically closed (so the reader unfamiliar with such things will have to take a lot on faith), and the proofs of many of the algebraic theorems that we state without proof require a solid command of the theory of schemes. Hence, these notes should be understood to be merely a survey of important notions, examples, and results for a reader who is taking their first steps into this vast and beautiful subject.