---
date: 2022-07-26 14:26
modification date: Tuesday 26th July 2022 14:26:40
title: "2022-07-26"
aliases: [2022-07-26]
---

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

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Tate twist](Unsorted/l-adic%20cohomology.md)
	- [supersingular](supersingular)

---

# 2022-07-26

![](attachments/Pasted%20image%2020220726142643.png)

# Shafarevich and why it implies Mordell

![](attachments/Pasted%20image%2020220726152410.png)
![](attachments/Pasted%20image%2020220726152333.png)

# Torelli

![](attachments/Pasted%20image%2020220726152544.png)

Punchline: nice curves are determined by their Jacobians
![](attachments/Pasted%20image%2020220726152608.png)

# High-level motivations

Tate Conjecture
![](attachments/Pasted%20image%2020220726153607.png)

![](attachments/Pasted%20image%2020220726160202.png)
![](attachments/Pasted%20image%2020220726160210.png)
![](attachments/Pasted%20image%2020220726160228.png)
![](attachments/Pasted%20image%2020220726160248.png)

![](attachments/Pasted%20image%2020220726153621.png)
![](attachments/Pasted%20image%2020220726160311.png)
![](attachments/Pasted%20image%2020220726160353.png)
![](attachments/Pasted%20image%2020220726160441.png)
![](attachments/Pasted%20image%2020220726160653.png)

Etale cohomology carries interesting extra structure as a $G_k$ module:

![](attachments/Pasted%20image%2020220726153738.png)

![](attachments/Pasted%20image%2020220726153849.png)
![](attachments/Pasted%20image%2020220726153914.png)

Relation to Hodge conjecture:

![](attachments/Pasted%20image%2020220726153957.png)
![](attachments/Pasted%20image%2020220726154024.png)
![](attachments/Pasted%20image%2020220726154107.png)
![](attachments/Pasted%20image%2020220726154150.png)
![](attachments/Pasted%20image%2020220726154202.png)

Relation to Shimura varieties:
![](attachments/Pasted%20image%2020220726154321.png)

# Jacobians

For $C$ an algebraic curve:
![](attachments/Pasted%20image%2020220726164901.png)
![](attachments/Pasted%20image%2020220726164942.png)
![](attachments/Pasted%20image%2020220726165016.png)
![](attachments/Pasted%20image%2020220726165049.png)

![](attachments/Pasted%20image%2020220726161551.png)
![](attachments/Pasted%20image%2020220726161620.png)

![](attachments/Pasted%20image%2020220726162157.png)
![](attachments/Pasted%20image%2020220726162227.png)

![](attachments/Pasted%20image%2020220726162309.png)
![](attachments/Pasted%20image%2020220726162314.png)
![](attachments/Pasted%20image%2020220726162331.png)
![](attachments/Pasted%20image%2020220726162458.png)

Uses of Jacobians: Chabauty-Coleman

![](attachments/Pasted%20image%2020220726163631.png)

Applications of Shaferevich:
![](attachments/Pasted%20image%2020220726163828.png)
![](attachments/Pasted%20image%2020220726163855.png)
![](attachments/Pasted%20image%2020220726163907.png)
![](attachments/Pasted%20image%2020220726163917.png)
![](attachments/Pasted%20image%2020220726163927.png)
![](attachments/Pasted%20image%2020220726163940.png)
![](attachments/Pasted%20image%2020220726164124.png)
![](attachments/Pasted%20image%2020220726164216.png)

# Complex Abel-Jacobi

![](attachments/Pasted%20image%2020220726165221.png)
![](attachments/Pasted%20image%2020220726165341.png)
![](attachments/Pasted%20image%2020220726165751.png)
![](attachments/Pasted%20image%2020220726170000.png)
![](attachments/Pasted%20image%2020220726170012.png)

## Intermediate Jacobians

![](attachments/Pasted%20image%2020220726165826.png)
![](attachments/Pasted%20image%2020220726165849.png)

# Schottky problem

![](attachments/Pasted%20image%2020220727094043.png)

![](attachments/Pasted%20image%2020220727091601.png)
![](attachments/Pasted%20image%2020220727091635.png)


# Motivating divisors

![](attachments/Pasted%20image%2020220727092906.png)
![](attachments/Pasted%20image%2020220727092916.png)
![](attachments/Pasted%20image%2020220727092926.png)
![](attachments/Pasted%20image%2020220727093011.png)
![](attachments/Pasted%20image%2020220727093102.png)
![](attachments/Pasted%20image%2020220727093127.png)

![](attachments/Pasted%20image%2020220727093148.png)
![](attachments/Pasted%20image%2020220727093202.png)
![](attachments/Pasted%20image%2020220727093208.png)
![](attachments/Pasted%20image%2020220727093235.png)
![](attachments/Pasted%20image%2020220727093420.png)
![](attachments/Pasted%20image%2020220727093432.png)

# Theta functions

![](attachments/Pasted%20image%2020220727095011.png)