pdf (first posted 06.08.12, 46 pages).

Abstract

This the original TeX file for my article Abelian Varieties, published as Chapter V of Arithmetic geometry (Storrs, Conn., 1984), 103--150, Springer, New York, 1986. The table of contents has been restored, some corrections have been made, there are minor improvements to the exposition, and an index has been added. The numbering is unchanged.

The article reviews the theory of abelian varieties emphasizing those points of particular interest to arithmetic geometers. In the main it follows Mumford's book (1970) except that most of the results are stated relative to an arbitrary base field, some additional results are proved, and étale cohomology is included. Many proofs have had to be omitted or only sketched. The reader is assumed to be familiar with Hartshorne 1977, Chaps. II, III, and (for a few sections that can be skipped) some étale cohomology. The last section of my article, Jacobian Varieties, contains bibliographic notes for both articles.

Contents

1. Definitions
2. Rigidity
3. Rational maps into abelian varieties
4. Review of the cohomology of schemes
5. The seesaw principle
6. The theorems of the cube and square
7. Abelian varieties are projective
8. Isogenies
9. The dual abelian variety: definition
10. The dual abelian variety: construction
11. The dual exact sequence
12. Endomorphisms
13. Polarizations and the cohomology of invertible sheaves
14. A finiteness theorems
15. The étale cohomology of an abelian variety
16. Pairings
17. The Rosati involution
18. Two more finiteness theorems
19. The zeta function of an abelian variety
20. Abelian schemes