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

    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: Proof of Honda-Tate’s Theorem and reductions of CM elliptic curves.
            Abstract: Examples and proof of Honda-Tate's Theorem (§ 11 of Dembélé's notes). Everywhere potentially good reduction of CM Elliptic curves (Lec3 of Li's notes) and set of supersingular primes (Lec6 of Li's notes). Here are the notes.