Abelian Varieties

v1.1. (July 27, 1998). First version on the web; 110 pages.
v2.0. (March 16, 2008). Corrected, revised, and expanded; 172 pages.

Contents

Chapter I: Abelian Varieties: Geometry

1. Definitions; basic properties.
2. Abelian varieties over the complex numbers.
3. Rational maps into abelian varieties.
4. Review of cohomology.
5. The theorem of the cube.
6. Abelian varieties are projective.
7. Isogenies
8. The dual abelian variety
9. The dual exact sequence.
10. Endomorphisms.
11. Polarizations and invertible sheaves.
12. The etale cohomology of an abelian variety.
13. Weil pairings.
14. The Rosati involution.
15. Geometric finiteness theorems.
16. Families of abelian varieties.
17. Neron models; semistable reduction.
18. Abel and Jacobi.

Chapter II: Abelian Varieties: Arithmetic

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

Chapter III: Jacobian varieties.

1. Definitions
2. The canonical maps from C to its Jacobian variety
3. The symmetric powers of a curve
4. The construction of the Jacobian variety
5. The canonical maps from the symmetric powers of C to its Jacobian variety
6. The Jacobian variety as Albanese variety; autoduality
7. Weil's construction of the Jacobian variety
8. Generalizations
9. Obtaining the coverings of curve from its Jacobian
10. Abelian varieties are quotients of Jacobian varieties
11. The zeta function of a curve
12. Torelli's theorem: statement and applications
13. Torelli's theorem: the proof
14. Bibliographic notes

Chapter IV: Finiteness Theorems.

1. Introduction.
2. The Tate conjecture; semisimplicity.
3. Finiteness I implies Finiteness II.
4. Finiteness II implies the Shafarevich conjecture.
5. Shafarevich's conjecture implies Mordell's conjecture.
6. The Faltings height.
7. The modular height.
8. The completion of the proof of Finiteness I.

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

Errata

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