### 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

- Definitions
- Rigidity
- Rational maps into abelian varieties
- Review of the cohomology of schemes
- The seesaw principle
- The theorems of the cube and square
- Abelian varieties are projective
- Isogenies
- The dual abelian variety: definition
- The dual abelian variety: construction
- The dual exact sequence
- Endomorphisms
- Polarizations and the cohomology of invertible sheaves
- A finiteness theorems
- The étale cohomology of an abelian variety
- Pairings
- The Rosati involution
- Two more finiteness theorems
- The zeta function of an abelian variety
- Abelian schemes

pdf (first posted 06.08.12, 46 pages).