---
created: 2023-03-14T19:17
updated: 2024-04-13T20:50
aliases:
  - Satake-Baily-Borel
  - Baily-Borel compactification
tags:
  - dissertation
  - compactifications
---

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [level structure](Unsorted/level%20structure.md)
	- [Hilbert modular forms](Hilbert%20modular%20forms)
	- [[Borel-Serre]]

---

# Baily-Borel

![](attachments/2023-03-14-6.png)
![](attachments/2023-03-14satake.png)
![](attachments/2023-03-14bb.png)


# Construction

![](2024-04-13-8.png)

Minimality:

![](2024-04-13-9.png)


# Boundary components

For $\SO_{2, n}$, see explicit constructions here: <https://www-users.cse.umn.edu/~kwlan/articles/intro-sh-ex.pdf#page=44&zoom=180,-45,704>

![](attachments/2023-03-14ratl.png)

# For curves

See [Double coset spaces](Unsorted/automorphic%20form.md#Double%20coset%20spaces):
![](attachments/2023-03-14bb-1.png)

![](attachments/2023-03-14-15.png)

# For $\Ag$

![](2024-04-13-10.png)


# For orthogonal type modular varieties

![](2024-04-13-11.png)