---
date: 2022-02-14 10:01
modification date: Monday 14th February 2022 10:01:21
title: number field function field analogy
aliases: [function field analogy]

---

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# number field/function field analogy

- $\CC \injects \CC\fps{t} \injects \CC\fls{t}$ is $\mu_{p-1} \injects \ZZpadic \injects \QQpadic$.

| Number field                                            | Function field                                                                                                                                            |
| ------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------- |
| $\ZZ$                                                   | $\FF_q[C]$                                                                                                                                   |
| $\QQ$                                                   | $\FF_q(C)$                                                                                                                                   |
| $\ZZpadic$                                              | $\hat{\OO_x}$                                                                                                                                |
| $\dcoset{\GL_n(\hat\ZZ)}{\GL_n(\AA_{\QQ})}{\GL_n(\QQ)}$ | $\dcoset{ \displaystyle\prod_{x\in \abs{X}}\GL_n(\hat \OO_x)} {\displaystyle\prod_{x\in \abs{X}}' \GL_n(\ff\hat{\OO_x})} {\GL_n(\FF_q(C) )}$ |


![](attachments/Pasted%20image%2020220214100138.png)
![](attachments/Pasted%20image%2020220214100218.png)
![](attachments/Pasted%20image%2020220214100322.png)
![](attachments/Pasted%20image%2020220422235936.png)