---
date: 2022-02-23 18:45
modification date: Tuesday 29th March 2022 17:06:05
title: "motivic L function"
aliases: [motivic L function, "Bloch-Kato", "Bloch-Kato conjecture"]
---

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


---

- Tags 
	- #NT #motivic 
- Refs:
	- Beilinson, Higher regulators and values of L-functions.
	- S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives.
	- Bernstein and S. Gelbart (editors), An Introduction to the Langlands Program
- Links:
	- [Galois representations](Unsorted/Galois%20representations.md)
	- [Unsorted/motive MOC](Unsorted/motive%20MOC.md)
	- [Beilinson conjecture](Beilinson%20conjecture)
	- [Bloch-Kato conjecture](Unsorted/Bloch-Kato.md)
	- [Milnor Conjecture](Unsorted/Milnor%20Conjecture.md)
	- [motive](Unsorted/motive%20MOC.md)


---

# motivic L function

An [L function](Unsorted/L%20function.md) associated to a Galois representation.
Generalizes the [Hasse-Weil L function](Unsorted/Hasse-Weil%20L%20function.md), conjectured to arise as [automorphic L functions](automorphic%20L%20function)

![](attachments/Pasted%20image%2020220430221713.png)

![](attachments/Pasted%20image%2020220430221801.png)

# Bloch-Kato

Gives an interpretation to the order of vanishing of $L(s, M)$ at $s=0$ for $M$ a [motive](Unsorted/motivic%20homotopy.md).

Related to [BSD](BSD) and finiteness of [Sha](Unsorted/Tate-Shafarevich%20group.md), involves [Selmer groups](Unsorted/Selmer%20group.md):
![Pasted%20image%2020211106014408.png](Pasted%20image%2020211106014408.png)

Relation to [automorphic representations](automorphic%20representations):
![Pasted%20image%2020211106014454.png](Pasted%20image%2020211106014454.png)

![](attachments/Pasted%20image%2020220202101846.png)

# Examples

![](attachments/Pasted%20image%2020220410154518.png)
![](attachments/Pasted%20image%2020220410154528.png)
![](attachments/Pasted%20image%2020220410154537.png)