Seminar on Abelian Varieties
Week 1: Basic Properties of Abelian Varieties
We define an abelian varieties. We define isogenies of abelian varieties and give some properties of isogenies. We prove some basic properties including the following: abelian varieties are commutative, the relative dualizing sheaf is trivial and we prove that all rational maps from a nonsingular variety to an abelian variety define a morphism.
Week 2: Projectivity of Abelian Varieties
We show that all abelian varieties are projective. We discuss which n it is possible to ambed an abelian variety of dimension g into.
Week 3: Dual Abelian Varieties
We define the dual abelian variety of a given abelian variety, explain its universal property and sketch the proof of existence.
Week 4: The Tate Module of an Abelian Variety and Etale cohomology of abelian varieties
Precise topics to be determined.
Week 5: Endomorphisms of Abelian Varieties
Precise topics to be determined.
Week 6: Zeta Functions of Abelian Varieties
Precise topics to be determined.
Week 7: Elliptic Curves
Precise topics to be determined.
Week 8: Heights on Abelian Varieties and the Mordell-Weil Theorem
Precise topics to be determined.
Week 9: Jacobian Varieties
Precise topics to be determined.
Week 10: Neron Models
Precise topics to be determined.
Week 11: Moduli Space of Abelian Varieties Part 1
Precise topics to be determined.
Week 12: Moduli Space of Abelian Varieties Part 2
Precise topics to be deter