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date: 2021-10-27 19:35
modification date: Saturday 6th November 2021 00:27:31
title: Galois representations
aliases: [Galois representation]

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

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- Tags: 
	- #todo/untagged
- Refs:
	- Harvard notes <https://people.math.harvard.edu/~smarks/mod-forms-tutorial/mf-notes/galois-reps.pdf> #resources/notes 
- Links:
	- [Artin L function](Unsorted/Artin%20L%20function.md)

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# Galois representations


> A possible conceptual explanation for the importance of Galois representations delivers the Tannaka-Krein theorem. Roughly, this states that knowing the representation theory is equivalent to knowing the group. The group you want to understand is the absolute Galois groups (with a profinite topology) via its Galois representations, and understand the Galois representation via automorphic forms.
> Perhaps one famous example is the Taniyama Shimura conjecture and consequently Fermat's last theorem: A certain construction with the elliptic curve gave a Galois representation, and the later was then shown to correspond to an automorphic form.

![Pasted image 20211106002737.png](Pasted%20image%2020211106002737.png)
![Pasted image 20211106002807.png](Pasted%20image%2020211106002807.png)

# Semistable/admissible

See [semistable](Unsorted/semistable.md) and [admissible representation](Unsorted/admissible%20representation.md),[cyclotomic character](Unsorted/cyclotomic%20character.md), [crystalline](Unsorted/crystalline%20cohomology.md)
![](attachments/Pasted%20image%2020220318211519.png)

# Unramified

![](attachments/Pasted%20image%2020220502183116.png)
![](attachments/Pasted%20image%2020220323170346.png)