---
date: 2022-04-05 23:42
modification date: Wednesday 27th April 2022 00:02:26
title: "adelic group"
aliases: [adelic group, adelic, adele, adelic analysis]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [modular curve](Unsorted/modular%20curve.md)

---

# adelic group

- **Finite adeles**: $\AA^\fin \da \QQ \tensor \prod_p\ZZpadic = \QQ \tensor \pro\ZZ$.
- **Adelic points**: for $G$ an [algebraic group](Unsorted/algebraic%20group.md), $G(\AA) \da \prod_{v}' G(k_v)$.
	- Interpretation: restricted direct product of local points $G(k_v)$ relative to the family $\ts{K_v}$ of open compact subgroups.
	- $G(\AA)$ is locally compact.
- **Automorphic quotient**: $$[G] \da \dcosetl{G(k)}{G(\AA)}$$
	- Fact: $G(\AA)\actson [G]$ by right translation and there is a unique-up-to-scaling $G(\AA)\dash$invariant measure on $[G]$.
	 

# Adelic analysis


![](attachments/Pasted%20image%2020220427000237.png)