
The AV Club (Abelian Varieties Reading Group)
Meeting times: Tuesday 4:306pm (Eckhart 202), Wednesday 2:304pm (Eckhart 207)
Highlights: (see more)
1. MordellWeil theorem for AVs + Mazur's criterion for rank zero,
2. Basic discussion of finite flat group schemes
3. Learn about reduction of elliptic curves and confront Neron models in the wild.
4. Raynaud's theorem
5. How to build Galois representations from weight 2 Hecke eigenforms.
6. EichlerShimura theorem.
7. Mazur's torsion theorem
Theory of the reading group (see more)
1. Two lectures per week.
2. No background assumed.
3. Speakers lecture on topics that they are already comfortable with.
4. Speakers come up with a three problem problem set that is uploaded a 5 days (or so) before their talk.
5. This course tracks Andrew Snowden's course pretty closely; so attendees should watch the relevant video before coming to lecture.
6. Hardcore attendees will also read the relevant reading and perhaps try the problems on the problem set prior to lecture too.
7. Pretty quickly the course will split into one day/week on group schemes/Neron models/reduction theory and another day/week on modular forms.
8. Associated subsequent drinking/social event to be announced.
References: (see more)

Tentative Schedule
Lecture 1: Elliptic Curves (see more; Paul Apisa, Tues. Oct. 13 )
Reference:
Snowden Lecture 2 , Silverman Ch. 2,3
Problem Set 1
Topics:
1. Embedding an elliptic curve into its class group is an isomorphism.
2. Define separability of a morphism.
3. Every isogeny factors as a separable map composed with an inseparable one.
4. Define dual isogeny.
5. Describe the Weierstrass form of an elliptic curve over C.
6. What is the Tate module?
7. What is the Weil pairing?
Lecture 2: Basic Abelian Varieties 1  Polarization and Line Bundles (see more; Jack Shotton; Wed. Oct. 14)
Reference:
Snowden Lecture 3 , Birkenhake and Lange (Chapter 2 and 3),
Donu Arapura's Notes
Problem Set 2
Topics:
1. State and prove the AppellHumbert theorem?
2. What are theta characteristics?
3. Let's use Kodaira to characterize projective group varieties!
4. Show that the dual abelian variety to X is naturally isomorphic to Pic^0(X).
5. What's a polarization?.
Lecture 3/4: Basic Abelian Varieties 2/3  MordellWeil (see more; MinhTam Trinh, Tues. Oct. 20 and Wed. Oct. 21)
Reference:
Snowden Lecture 4 , Milne or Mumford (first few sections), Silverman Chapter VII
MinhTam's Notes for Lecture 3
MinhTam's Notes for Lecture 4
Topics:
1. What is a complete variety?
2. State and prove the Theorem of the cube.
3. State and prove the Theorem of the square.
4. What are isogenies of abelian varieties?
5. What is the dual abelian variety?
6. What is Mumford's rigidity lemma?
7. State and prove MordellWeil for abelian varieties over number fields. (Is this analogous to Hermite's theorem on finiteness of number fields of bounded discriminant? This theorem should be stated and a proof sketched)
8. What is Poincare reducibility?
9. Classify onedimensional group varieties (not necessarily complete) over algebraically closed fields.
Lecture 5: What is a Group Scheme (and why care)? (see more; Yiwen Zhou, Tues. Oct. 27)
Reference:
Snowden Lecture 5
Tate's Finite Flat Group Scheme paper (
video synopsis )
Kevin Buzzard's Notes
Problem Set
Topics:
1. What is a scheme?
2. What is a group scheme?
3. What are examples of finite group schemes?
4. What is a Hopf algebra (and what is the connection to group schemes)?
5. What is a finite dimensional etale algebra? Classify the finite dimensional ones.
6. What is the connection between finite etale commutative group schemes over k and finite representations of Gal(k^sep/k)?
7. What is the connected etale sequence?
Lecture 6: Group Schemes 2 (see more; Yiwen Zhou, Wed. Oct. 28)
Reference:
Snowden Lecture 5
Tate's Finite Flat Group Scheme paper (
video synopsis )
Kevin Buzzard's Notes
Topics:
1. What is a projective module?
2. What is a flat scheme?
3. What is Cartier duality?
4. Classification of finite flat group schemes of order 2 over Spec(R).
Lecture 7: What is the modular curve over C? (see more; Paul Apisa, Tues. Nov. 3)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 8: Group Schemes 3 (see more; Yiwen Zhou, Wed. Nov. 4)
Lecture 9: What are modular forms on X(1)? (see more; Paul Apisa, Tues. Nov. 10)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 10: Raynaud's Theorem (see more; Yiwen Zhou, Wed. Nov. 11)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 11: What are modular curves over Q? (see more; Cancelled, Tues. Nov. 17)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 12: Reduction of Elliptic Curves (see more; Olivier Martin (tentatively), Wed. Nov. 18)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 13: How to compactify the modular curves (see more; Jack Sempliner, Tues. Nov. 24)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 14: Neron Models of Abelian Varieties (see more; Sean Howe, Tues. Dec. 1)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 15: Hecke Operators and Atkin Lehner involution (see more; Jeff Manning (tentatively) Wed. Dec. 2)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 16: When do the rational points of an AV have rank zero? (see more; Tues. Dec. 8)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 17: How to make a Galois representation from a weight 2 Hecke eigenform (see more; Jeff Manning (tentatively) Wed. Dec 9)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 18: Rational Points on ECs and Morphisms from X_0(N) to AVs. (see more; in Winter Quarter!)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 19: Reducing J_0(N) mod N (see more; in Winter Quarter!)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA
Lecture 20: Proof of Mazur (see more; in Winter Quarter!)
Reference: TBA
Problem Set: To be posted.
Topics:
1. TBA