Abelian Varieties - J.S. Milne Top

Group Theory

Fields and Galois Theory

Algebraic Geometry

Algebraic Number Theory

Modular Functions and Modular Forms

Elliptic Curves -- see books.

Abelian Varieties

Lectures on Etale Cohomology

Class Field Theory

Complex Multiplication

Basic Theory of Affine Group Schemes

Lie Algebras, Algebraic Groups, and Lie Groups

Reductive Groups

Errata

pdf (current version 2.00)
### Contents

Introduction
#### Chapter I: Abelian Varieties: Geometry

#### Chapter II: Abelian Varieties: Arithmetic

#### Chapter III: Jacobian varieties.

#### Chapter IV: Finiteness Theorems.

### History

v1.1. (July 27, 1998). First version on the web; 110 pages.

v2.0. (March 16, 2008). Corrected, revised, and expanded; 172 pages.

An introduction to both the geometry and the arithmetic of abelian varieties. It includes a discussion of the theorems of Honda and Tate concerning abelian varieties over finite fields and the paper of Faltings in which he proves Mordell's Conjecture. Warning: These notes are less polished than the others.

- Definitions; basic properties.
- Abelian varieties over the complex numbers.
- Rational maps into abelian varieties.
- Review of cohomology.
- The theorem of the cube.
- Abelian varieties are projective.
- Isogenies
- The dual abelian variety
- The dual exact sequence.
- Endomorphisms.
- Polarizations and invertible sheaves.
- The etale cohomology of an abelian variety.
- Weil pairings.
- The Rosati involution.
- Geometric finiteness theorems.
- Families of abelian varieties.
- Neron models; semistable reduction.
- Abel and Jacobi.

- The zeta function of an abelian variety.
- Abelian varieties over finite fields.
- Abelian varieties with complex multiplication.

- Definitions
- The canonical maps from C to its Jacobian variety
- The symmetric powers of a curve
- The construction of the Jacobian variety
- The canonical maps from the symmetric powers of C to its Jacobian variety
- The Jacobian variety as Albanese variety; autoduality
- Weil's construction of the Jacobian variety
- Generalizations
- Obtaining the coverings of curve from its Jacobian
- Abelian varieties are quotients of Jacobian varieties
- The zeta function of a curve
- Torelli's theorem: statement and applications
- Torelli's theorem: the proof
- Bibliographic notes

- Introduction.
- The Tate conjecture; semisimplicity.
- Finiteness I implies Finiteness II.
- Finiteness II implies the Shafarevich conjecture.
- Shafarevich's conjecture implies Mordell's conjecture.
- The Faltings height.
- The modular height.
- The completion of the proof of Finiteness I.

Appendix: Review of Faltings 1983 (MR 85g:11026)

v2.0. (March 16, 2008). Corrected, revised, and expanded; 172 pages.