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Meeting on Tuesdays at 5:30-6:45 Room 622

Seminar on Abelian Varieties

Week 1: Basic Properties of Abelian Varieties
01.28.25 Sofia Wood

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 extend to a morphism.​

Week 2: Projectivity of Abelian Varieties
02.04.25 Anna Abasheva

We show that all abelian varieties are projective, not just proper. The proof can be thought of as a generalization of the fact that every elliptic curve embeds into a 2-dimensional projective space via a Weierstrass equation.

 

In the notes and time permitting, we discuss which n it is possible to embed an abelian variety of dimension g into projective space of dimension n. It turns out that in general, for an abelian variety of dimension g, the smallest such value of n is 2g+1. Note that the case of elliptic curves, all of which can be embedded in 2-dimensional projective space via a Weierstrass equation are an exception.

Week 3: Dual Abelian Varieties
02.11.25 Nicolás Vilches Reyes

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
02.18.25 Carlos Alvarado

Precise topics to be determined.

Week 5: Endomorphisms of Abelian Varieties
02.25.25 Sofia Wood

Precise topics to be determined.

Week 6: Zeta Functions of Abelian Varieties
03.04.25 Lisa Faulkner Valiente

Precise topics to be determined.

Week 7: Elliptic Curves
03.25.25 Hechen Hu

Precise topics to be determined.

Week 8: Heights on Abelian Varieties and the Mordell-Weil Theorem
04.01.25 Vidhu Adhihetty

Precise topics to be determined.

Week 9: Jacobian Varieties
04.08.25 Amal Mattoo

Precise topics to be determined.

Week 10: Neron Models
04.15.25 Matthew Hase-Liu

Precise topics to be determined.

Week 11: Moduli Space of Abelian Varieties Part 1
04.22.25 Rafah Hajjar Muñoz

Precise topics to be determined.

Week 12: Moduli Space of Abelian Varieties Part 2
04.29.25 Sofia Wood

Precise topics to be determined

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