---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 20:45:32
title: p-adic Hodge theory
aliases: [p-adic Hodge theory]

---


---

- Tags 
	- #arithmetic-geometry #arithmetic-geometry/p-adic-hodge-theory 
- Refs:
	- <https://math.stanford.edu/~conrad/papers/notes.pdf>
	- Course notes: 
	  <https://www.dropbox.com/s/2gua6n5r0spx8i5/p-adicHodge.pdf?dl=0> #resources/course-notes 
	- A survey of comparison theorems and [period rings](period%20rings): 
	  <https://arxiv.org/pdf/2005.07919.pdf#page=1> #resources/summaries 
	  [A nice summary](https://arxiv.org/pdf/2005.07919.pdf#page=4) #resources/summaries 
	- 2020 UMich course: <http://www-personal.umich.edu/~ahorawa/math_679_p-adic_Hodge.pdf#page=2>
	  
- Links:
	- [Hodge theory MOC](Unsorted/Hodge%20theory%20MOC.md)
	- [kahler](Unsorted/Kahler.md)
	- [etale cohomology](Unsorted/etale.md)
	- [cyclotomic character](cyclotomic%20character)
	- [Hodge-Tate decomposition](Unsorted/Hodge-Tate%20conjecture.md)
	- Main theorems:
		- [Hodge-Tate conjecture](Unsorted/Hodge-Tate%20conjecture.md)
		- [de Rham conjecture](Unsorted/de%20Rham%20conjecture.md)
		- [crystalline conjecture](crystalline%20conjecture.md)
		- [semistable conjecture](semistable%20conjecture.md)
		- [potentially semistable conjecture](potentially%20semistable%20conjecture.md)
	- [syntomic](Unsorted/syntomic.md)
	- [period ring](Unsorted/period.md)
	- [alteration](Unsorted/alteration.md)
	- [Hodge F-crystal](Unsorted/Hodge%20F-crystal.md)
	- The many cohomology theories:
		- [etale cohomology](Unsorted/l-adic%20cohomology.md)
		- [l-adic cohomology](Unsorted/l-adic%20cohomology.md)
		- [de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md)
		- [Hodge-Tate cohomology](Hodge-Tate%20cohomology)
		- [crystalline cohomology](Unsorted/crystalline%20cohomology.md)
		- [de Rham-Witt](Unsorted/de%20Rham-Witt.md)

---

# p-adic Hodge theory

# Notation

![](attachments/Pasted%20image%2020220502145500.png)
![](attachments/Pasted%20image%2020220502145513.png)

## Motivations and Ideas

One of the best-known tools for studying [Galois representations](Unsorted/Galois%20representations.md) of the [absolute Galois group](Unsorted/absolute%20Galois%20group.md) of $\QQ$.

![](attachments/Pasted%20image%2020220503105252.png)
![](attachments/Pasted%20image%2020220503105341.png)
![](attachments/Pasted%20image%2020220503105412.png)

![](attachments/Pasted%20image%2020220318202122.png)

![](attachments/Pasted%20image%2020220318202217.png)

![](attachments/Pasted%20image%2020220502145626.png)

![](attachments/Pasted%20image%2020220502145657.png)


## Main theorems

![](attachments/Pasted%20image%2020220318201637.png)
![](attachments/Pasted%20image%2020220318202239.png)
![](attachments/Pasted%20image%2020220318202258.png)
![](attachments/Pasted%20image%2020220318202558.png)
![](attachments/Pasted%20image%2020220318203003.png)

# Motivations

![](attachments/Pasted%20image%2020220317212108.png)
![](attachments/Pasted%20image%2020220317212246.png)

![](attachments/Pasted%20image%2020220317212443.png)
![](attachments/Pasted%20image%2020220317212508.png)

See [period rings](Unsorted/period#The%20period%20rings).

# Main theorems

![](attachments/Pasted%20image%2020220317212833.png)
![](attachments/Pasted%20image%2020220317212848.png)


## Notes

![attachments/Pasted image 20210717011702.png](attachments/Pasted%20image%2020210717011702.png)
![Pasted image 20211106002648.png](Pasted%20image%2020211106002648.png)
![Pasted image 20211106002422.png](Pasted%20image%2020211106002422.png)
![Pasted image 20211106002457.png](Pasted%20image%2020211106002457.png)
![Pasted image 20211106002515.png](Pasted%20image%2020211106002515.png)
![Pasted image 20211106004153.png](Pasted%20image%2020211106004153.png)
![Pasted image 20211106012512.png](Pasted%20image%2020211106012512.png)
![Pasted image 20211106012855.png](Pasted%20image%2020211106012855.png)

On the importance of [tilting](Unsorted/tilting.md)
![Pasted image 20211106012609.png](Pasted%20image%2020211106012609.png)

![](attachments/Pasted%20image%2020220317210316.png)

# Applications

To [Galois cohomology](Unsorted/Galois%20cohomology.md):

![](attachments/Pasted%20image%2020220410155032.png)
![](attachments/Pasted%20image%2020220410155056.png)