---
date: 2022-04-08 12:23
modification date: Friday 8th April 2022 12:23:51
title: "Tate-Shafarevich group"
aliases: [Tate-Shafarevich group, Sha]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [Galois cohomology](Unsorted/Galois%20cohomology.md)
	- [Mordell-Weil](Unsorted/Mordell-Weil.md)

---

# Tate-Shafarevich group


For $A$ an [abelian variety](Unsorted/abelian%20variety.md) or a [group scheme](group%20scheme.md) define over a field $k$, this is the group of 1-cocycles in $H^1(G_K; X)$ which become boundaries at every place:
$$\sha(X\slice k) = \intersect_{v\in \places{K}} \ker \qty{ H^1(G_k; X) \to H^1(G_{\kv}; X)}$$
Measures the extent to which the [Hasse principle](Unsorted/Hasse%20principle.md) holds for equations with coefficients in $k$.


$\sha(K)$ is precisely the group of $K$-torsors under $E$ that fail the local-global principle.

![](attachments/Pasted%20image%2020220414210330.png)

For $X = A[n]$ the $n\dash$torsion on an abelian variety, related to the [Selmer group](Unsorted/Selmer%20group.md):
![](attachments/Pasted%20image%2020220414210645.png)

# Conjecture

- Conjecture: $$\size \sha(X\slice k) < \infty$$
- Known for some $E \in \Ell$ with [CM](MOCs/elliptic%20curve.md) and $\rank E \leq 1$.


![](attachments/Pasted%20image%2020220414210756.png)

# Consequences

![](attachments/Pasted%20image%2020220408130030.png)

![](attachments/Pasted%20image%2020220414212225.png)