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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

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