---
date: 2023-03-16 00:05
aliases: ["Untitled"]
---

Last modified: `=this.file.mday`

# Drinfeld Talk 2 Draft 2

- What is $d_\infty$ and $v_\infty$?
- What is $\gens{\alpha}$ and $\gens{\alpha}^y$?
- RH for Drinfeld modules: 4.12.8.5?
- L for EC with CM factors via Hecke characters?

# Setup


- Basic comparison: $\zeta_A(s) \da \sum_{a\in A^{\mathrm{mon}} } a^{-s}$ compared to $\zeta_k(s) \da \sum_{D\in \Div^+(X)} \Norm(D)^{-s}$.


## Zeta functions (7.8)

For $X$ a curve:

![](attachments/2023-03-16.png)
![](attachments/2023-03-16-1.png)
![](attachments/2023-03-16-2.png)

## Partial zetas

![](attachments/2023-03-16partial.png)
![](attachments/2023-03-16-3.png)

## The complex plane $S_\infty$ (8.1)

![](attachments/2023-03-16-4.png)

![](attachments/2023-03-16-5.png)


Motivating these definitions:
![](attachments/2023-03-16-6.png)

Behaves like exponentials:
![](attachments/2023-03-16exp.png)

Prototypical L-function:
![](attachments/2023-03-16-7.png)

## Exponentiating ideals (8.2)

Issue: Dedekind domains are PIDs iff factorial, so for $\ZZ$ we identify $(a)$ with $a$, but we can't generally do this for $A$ and need extend exponentiation to ideals.

![](attachments/2023-03-16-8.png)

![](attachments/Pasted%20image%2020230316003113.png)
![](attachments/2023-03-16-10.png)

## Functional analysis (8.4)

![](attachments/2023-03-16cts.png)

![](attachments/2023-03-16-11.png)
![](attachments/2023-03-16-12.png)

Coefficient extraction:

![](attachments/2023-03-16-13.png)

Periodicity:

![](attachments/2023-03-16-14.png)

![](attachments/2023-03-16-16.png)

![](attachments/2023-03-16-17.png)

## Entire functions (8.5)

![](attachments/2023-03-16-18.png)

![](attachments/2023-03-16-19.png)

![](attachments/Pasted%20image%2020230316005041.png)

![](attachments/2023-03-16-21.png)
![](attachments/2023-03-16-22.png)
![](attachments/2023-03-16-23.png)

![](attachments/2023-03-16-24.png)
![](attachments/2023-03-16-25.png)
![](attachments/2023-03-16-26.png)

### Algebraicity

![](attachments/2023-03-16-27.png)

![](attachments/2023-03-16-28.png)

![](attachments/2023-03-16-29.png)
![](attachments/2023-03-16-30.png)

![](attachments/2023-03-16-31.png)

## L-series (8.6)

![](attachments/2023-03-16-32.png)

![](attachments/2023-03-16-33.png)


![](attachments/2023-03-16-34.png)

![](attachments/2023-03-16-35.png)

![](attachments/2023-03-16-36.png)

![](attachments/2023-03-16-37.png)
![](attachments/2023-03-16-38.png)

![](attachments/2023-03-16-39.png)

![](attachments/2023-03-16-40.png)
![](attachments/2023-03-16-41.png)
![](attachments/2023-03-16-42.png)

Half-planes of convergence:

![](attachments/2023-03-16-43.png)

Use a notion of purity to get convergence for $L\dash$functions of $T\dash$modules.

![](attachments/2023-03-16-44.png)

## Dirichlet series (8.7)

![](attachments/Pasted%20image%2020230316010554.png)

Determined by special values:
![](attachments/2023-03-16-46.png)

![](attachments/2023-03-16-47.png)

8.8 proves that $L(\hat\rho, s)$ is entire for certain families of compatible representations $\hat\rho$.
Example:
![](attachments/2023-03-16-48.png)
![](attachments/2023-03-16-49.png)

8.9 shows that $L\dash$series associated to finite characters are entire:

![](attachments/2023-03-16-50.png)
![](attachments/2023-03-16-51.png)
![](attachments/2023-03-16-52.png)

![](attachments/2023-03-16-53.png)

## Teich. character

![](attachments/2023-03-16-54.png)

![](attachments/2023-03-16-55.png)

![](attachments/2023-03-16-56.png)

![](attachments/2023-03-16-57.png)

## Special values (8.12)

![](attachments/2023-03-16-58.png)
![](attachments/2023-03-16-59.png)

![](attachments/2023-03-16-60.png)

## Trivial zeros

![](attachments/2023-03-16-61.png)
![](attachments/2023-03-16-62.png)

![](attachments/2023-03-16-63.png)


## Class groups

![](attachments/2023-03-16-64.png)


![](attachments/2023-03-16-65.png)

![](attachments/2023-03-16-66.png)

![](attachments/2023-03-16-67.png)

![](attachments/2023-03-16-68.png)

## Arith/geom, Kummer-Vandiver

![](attachments/2023-03-16-69.png)

![](attachments/2023-03-16-70.png)

![](attachments/2023-03-16-71.png)

![](attachments/2023-03-16-72.png)

![](attachments/2023-03-16-73.png)

![](attachments/2023-03-16-74.png)

![](attachments/2023-03-16-75.png)

![](attachments/2023-03-16-76.png)

![](attachments/2023-03-16-77.png)

## Special values

![](attachments/2023-03-16-79.png)

![](attachments/2023-03-16-78.png)

![](attachments/2023-03-16-80.png)

![](attachments/2023-03-16-81.png)

## Functional equation

![](attachments/2023-03-16-82.png)

![](attachments/2023-03-16-83.png)

![](attachments/2023-03-16-84.png)

![](attachments/2023-03-16-85.png)

![](attachments/2023-03-16-86.png)

Measure
![](attachments/2023-03-16-87.png)

![](attachments/2023-03-16-88.png)
![](attachments/2023-03-16-89.png)

Real zeros
![](attachments/2023-03-16-90.png)

![](attachments/2023-03-16-92.png)
![](attachments/2023-03-16-91.png)
![](attachments/2023-03-16-93.png)