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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.
Chapter I: Abelian Varieties: Geometry
- 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.
- The dual abelian variety
- The dual exact sequence.
- 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.
Chapter II: Abelian Varieties: Arithmetic
- The zeta function of an abelian variety.
- Abelian varieties over finite fields.
- Abelian varieties with complex multiplication.
Chapter III: Jacobian varieties.
- 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
- 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
Chapter IV: Finiteness Theorems.
- 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)
v1.1. (July 27, 1998). First version on the web; 110 pages.
v2.0. (March 16, 2008). Corrected, revised, and expanded; 172 pages.
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