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created: 2023-03-27T14:32
updated: 2023-03-27T14:32
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date: 2023-01-09 12:23
aliases: ["Winter 2022 reading group talk"]
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Last modified: `=this.file.mday`

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- Refs:
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# Winter 2022 reading group talk

References:

- [https://math.mit.edu/~poonen/papers/drinfeld.pdf](https://math.mit.edu/~poonen/papers/drinfeld.pdf)
- [http://personal.psu.edu/mup17/Research/ODM.pdf](http://personal.psu.edu/mup17/Research/ODM.pdf)
- Goss, Basic Structures of Function Field Arithmetic
- https://www.youtube.com/watch?v=kxxNoLl8h5s&list=PLQZfZKhc0kiC1X2PmKsWV0RnR5QSsPWYQ&index=2
- http://personal.psu.edu/mup17/Research/ODM.pdf MIHRAN PAPIKIAN ⭐

My notes:

[Books Basic Structures of Function Field Arithmetic Goss](Annotations/Books%20Basic%20Structures%20of%20Function%20Field%20Arithmetic%20Goss.md)
[[Projects/0000 Talks/2023 Function Fields and Drinfeld Modules/2023 Drinfeld 1/Notes 2]]

# Notes

Drinfeld modules generalize the Carlitz module, a module over $k(C)$ the ring of functions on a curve $C$ over a finite field $\FF_q$.
Give a function field analog of CM.
Generalize to shtukas (vector bundles over curves).

$\tau$ is a generaliation of Frobenius.


# Motivation
![2023-01-09-22](attachments/2023-01-09-22.png)

## Langlands
![2023-01-09-20](attachments/2023-01-09-20.png)
![2023-01-09-21](attachments/2023-01-09-21.png)

Explicit class field theory
![](attachments/2023-01-09-classfieldexpli.png)
## Analogies

![2023-01-09-13](attachments/2023-01-09-13.png)
![2023-01-09-16](attachments/2023-01-09-16.png)

# Carlitz


![2023-01-09-14](attachments/2023-01-09-14.png)

![2023-01-09-15](attachments/2023-01-09-15.png)


![2023-01-09-17](attachments/2023-01-09-17.png)

![2023-01-09-18](attachments/2023-01-09-18.png)

![2023-01-09-19](attachments/2023-01-09-19.png)



**Theorem 8.5.** For nonzero $M$ in $\mathbf{F}_p[T], \Lambda_M=\left\{e_C\left((A / M) \xi_p\right): A \in \mathbf{F}_p[T]\right\}$.

# Defining Drinfeld modules

![2023-01-09-23](attachments/2023-01-09-23.png)

![2023-01-09-24](attachments/2023-01-09-24.png)
![](attachments/2023-01-09-25.png)

![](attachments/2023-01-09-forma.png)