Seminar on Elliptic curves and Abelian Varieties, Spring 2024

    Description of the seminar

This is a learning seminar in preparation for the Arizona Winter School 2024.

The aim is to spend the six weeks preceding the AWS to review the lecture notes and problem sets from the Preliminary Arizona Winter School 2023 which was on Abelian varieties over finite fields and Elliptic curves with complex multiplication.

    Logistics Info

The seminar will be held on Wednesday from 6pm to 7:30pm in Room 622 in the Math Department at Columbia.

Please email me at mp3947 at columbia dot edu if you are interested in giving a talk and/or you want to be added to the mailing list.

    References

• Lassina Dembélé's notes on Abelian varieties on finite fields: Notes .
• Wanlin Li's notes on Elliptic curves with complex multiplication: Lecture 1 , Lecture 2 , Lecture 3 , Lecture 4 , Lecture 5 , Lecture 6 .
• Draft of Ben Moonen's book on Abelian varieties.
• Notes from Brian Conrad and Chris Skinner seminar.

    Schedule

1. Week 0
           Speaker: Morena Porzio
           Title: Organizational Meeting
           Abstract: We will skim through the material given by the PAWS23, relate it to the AWS24 and then outline the structure of the next six-weeks talks. Here are the notes.
2. Week 1
           Speaker: Morena Porzio
           Title: Introduction to abelian varieties over finite fields and elliptic curves with CM.
           Abstract: Recalls about definition of Abelian Variety, rigidity results, isogenies, dual, polarization (§ 1,2 of Dembélé's notes), examples and group law of Elliptic curves (§ 4 of Dembélé's notes). Definition of Pic^0(E), structure of End(\overline{E}), definition of Elliptic curve with CM (Lec1 of Li's notes). Here are the notes.
3. Week 2
           Speaker: Qiyao Yu
           Title: Endomorphism rings and Tate modules.
           Abstract: Properties of End(A)\otimes_Z Q, Hom(A,B)\otimes_Z Q, Tate's module T_l(A) for l\neq char(k) (§ 3, 5.1, 5.2, 5.3 of Dembélé's notes), propositions 5.12, 5.13, 5.14 (§ 5.4 of Dembélé's notes), Albert classification (§ 7.1 of Dembélé's notes). Here are the notes.
4. Week 3
           Speaker: Caleb Ji
           Title: Tate’s isogeny theorem and Moduli space of Abelian varieties.
           Abstract: Frobenius' maps, constructions and lemmas for Tate's isogeny theorem for l\neq char(k) (§ 6 of Dembélé's notes). Definition and properties of the Moduli spaces \mathcal{A}_g. Here are the notes.
5. Week 4
           Speaker: Amal Mattoo
           Title: The Weil conjectures.
           Abstract: Weil conjectures for Abelian varieties (§ 7.2 of Dembélé's notes), Jacobian of curves (§ 8 of Dembélé's notes), Weil conjectures for curves (§ 9.1 of Dembélé's notes). Here are the notes.
6. Week 5
           Speaker: Alan Zhao
           Title: Dieudonn´e modules and Serre–Tate deformation theory.
           Abstract: p-Divisible groups, Dieudonné ring D_k, anti-equivalence of categories G\mapsto M(G), Dieudonné modules attached to Abelian varieties and Tate's theorem for l=\char(k) (§ 10.1, 10.2, 10.4 of Dembélé's notes). Local invariants for Abelian varieties (§ 10.5 of Dembélé's notes). Here are the notes.
7. Week 6
           Speaker: Morena Porzio
           Title: Honda-Tate’s Theorem.
           Abstract: Recalls on Brauer Group (§ C of Dembélé's notes) and local invariants, proof of Honda-Tate's Theorem (§ 11 of Dembélé's notes) and implications on the characteristic polynomial of an abelian variety. Here are the notes.