1. Course Description

 

The main topic of this course is moduli spaces of abelian varieties

and the Hecke symmetries on them, especially the fine structures

on these moduli spaces in positive characteristic p.

 

2. Outline

1. Review of abelian varieties, commutative finite group schemes and p-divisible groups, deformation abelian varieties and p-divisible groups

 

2. Overview: moduli spaces of abelian varieties and Hecke symmetries

 

3. Examples of fine structures on moduli spaces of abelian varieties:

stratifications and foliations

 

4. Sustained p-divisible groups and leaves on moduli spaces of abelian varieties in characteristic p

 

5. Action of local stabilizer subgroups on deformation spaces

 

6. Tate-linear structures on sustained deformation spaces

 

7. Rigidity of Tate-linear formal varieties

 

 

3. References and suggested readings

 

I. Books.

Abelian varieties: Mumford's book "Abelian Varieties".

 

p-divisible group: Zink's book "Cartiertheorie kommutativer formaler Gruppen" (English translation available from Zink's website).

 

Moduli of abelian varieties: Mumford's book "Geometric Invariant Theory", and "Degeneration of Abelian Varieties" by Faltings and Chai.

 

Sustained p-divisible groups and Tate-linear structures: "Hecke Orbits", draft by C.-L. Chai and F. Oort (with a comprehensive bibliography).

 

II. Survey articles

 

P. Deligne, Travaux de Shimura, Seminaire Bourbaki 1971. This article is the best introduction to Shimura varieties.

 

C.-L. Chai and F. Oort, Moduli of abelian varieties, chap. 5 of the book "Open problems in arithmetic algebraic geometry", edited by F. Oort.

 

C.-L. Chai, Hecke orbits as Shimura varieties in positive characteristic, Proc. ICM 2006.

 

Tate-linear formal varieties: C.-L. Chai, Tate-linear formal varieties, Taiwanese Journal of Math. 2024

 

4. Registration

https://forms.gle/vGS5igQqjVWCBfhT9