Abelian Varieties - J.S. Milne   Top
Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Algebraic Groups; Lie Algebras; Lie Groups; Reductive Groups
Errata
pdf (current version 2.00)

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.

Contents

Introduction

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.

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

History

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

pdf (old version 1.10)