---
tags: research
---

# RZ space learning seminar

- Location: HU1018
- Time: Wed 13:00-14:00
## Timetable

Time|Speaker|Notes
-|-|-
Oct.9|Qirui Li|[Catier Modules](/hYuVO_pyQaKuzKDJn_Ticw)
Oct. 16|Qirui Li|[Local Catier Theory](/4AeJG87TQLOLHSSa3HpxKw)
Oct. 23|Qirui Li|[Dieudonne Modules and de-Rham cohomology](/Z_TJrOUGQAm0FjnU37kZPA)

## Learning Goal

In this seminar, we want to understand part of the book [RZ] by Rapoport and Zink.

We will study the definition and properties of the period morphism
$$
\breve π_1
: \breve M^{rig} \rightarrow \breve F^{wa}
$$

which is a map from generic fiber of a certain moduli of p-divisible groups to weakly admissible locus arises as a subspace form flag variety.

We would like to see the theory as explicitly as possible. An explicit construction of Dieudonne Module would be displayed.

## Tentative Schedule
<!--
### Introduction

#### An example of period morphism for formal modules

A formal module of height 2. Say a formal module defined over the thickending. The map from tangent space to the Dieudonne module by reduction.

- Define a Dieudonne Module of a formal group explicitly,

G formal module over k, the functions on it can be described by the ring

k[[X]]

The group law is given by some power-series

X[+]Y = X + Y +...

This is say that for any moving particles on $G$, say with the time $X=f(t)$, we can add moving particles, adding the position of the two particle we obtain a thrid particle moving on the formal module.

To define Dieudonne Module, we consider the set of elements

$$1+tk[[X]][[t]]$$

elements of it looks like
$$
1+Xt+(X^2+X^3+\cdots)t^2+\cdots
$$

This element can thinking as Deformations of moving particles.

Those elements can be written as in the form

$$
(1+f_1(X)t)(1+f_2(X)t^2)(1+f_3(X)t^3)\cdots
$$

Take primitive elemnts and form a Witt module,

- Translate elements in Dieudonne Module in various ways.

- as p-typical curves. as group invariant $H^1$ class.

Period morphisms. Define a map from Tangent space of formal group to Dieudonne Module, it is an analogue of Hodge-Tate period morphisms.

Hodge Tate period map.

Say something about Dieudonne Module can recover p-divisible groups.
-->

### Explicit understanding of p-divisible groups(4 weeks)

1. Breif introduction to p-divisible groups([[2]](#References))
2. Catier Theory for finite flat group schemes ([[1]](#References))
3. Local Catier-Dieudonne Theory([[1]](#References))
4. Grothendick Messing Theory

### Understanding the RZ space (4weeks)

6. Moduli space of quasi-isogenies
7. Rapporport Zink space
8. Local Models
9. Some rigid geometry

This seminar may extend to the next semester

### Period morphisms (4 weeks)

10. The period morphisms
11. The image of period morphisms
12. Admissible locus

# References

[[1]](https://pdfs.semanticscholar.org/b86c/9270ab92c0cca0cc1ff562cf578c421f9a99.pdf?_ga=2.144141877.126651306.1569957241-393231388.1569957241) Notes on Catier theory, C.Chai

[[2]](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.295.7566&rep=rep1&type=pdf) p-Divisible groups, J.T. Tate

[[3]](https://www.math.leidenuniv.nl/scripties/wang.pdf) Moduli space of p-divisible groups and Period Morphisms

# Paticipants:

[e-mail];
[e-mail];
[e-mail];
Ali Cheraghi <[e-mail]>;
[e-mail]