---
date: 2022-03-29 17:13
modification date: Tuesday 29th March 2022 17:13:58
title: "automorphic L function"
aliases: [automorphic L functions]

---

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


---

- Tags 
	- #arithmetic-geometry/Langlands 
- Refs:
	- #todo/add-references
- Links:
	- [automorphy](Unsorted/automorphic%20form.md)

---

# automorphic L function

Setup:

- $G$ a [reductive](Unsorted/reductive.md) group over a [global field](Unsorted/global%20field.md) $F$
- $\pi$ an [automorphic representation](Unsorted/automorphic%20representation.md) of $G$, which decomposes as $\tensor_v \pi_v$ for $v\in \Places(F)$.
- $\psi$ a $\CC\dash$representation of the [Langlands dual](Langlands%20dual) group $G\dual$.
  
To this data there is an associated [L function](Unsorted/L%20function.md) $L(s, \pi, \psi)$.
Should decompose as a product of local $L$ functions at places $v$ of $F$.