---
date: 2021-11-05 13:02
modification date: Friday 5th November 2021 13:02:35
title: Birch and Swinnerton-Dyer conjecture
aliases: [Birch and Swinnerton-Dyer, BSD, weak BSD]

---

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

---
- Tags: 
	- #AG/elliptic-curves #open/conjectures 
- Refs:
	- Swinnerton-Dyer, Notes on elliptic curves. II
	- J. T. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog.
- Links:
	- [elliptic curve](MOCs/elliptic%20curve.md)
	- [modularity](Unsorted/modular%20form.md)
	- [L function](Unsorted/L%20function.md)
	- [motivic L function](motivic%20L%20function.md)
	- Application: [congruent number problem](congruent%20number%20problem.md)
	- [Mazur's theorem](Unsorted/Mordell-Weil.md)
	- [Faltings theorem](Unsorted/Faltings%20theorem.md)


---

# Birch and Swinnerton-Dyer conjecture

![](attachments/Pasted%20image%2020220502100709.png)

On the free abelian group $E(K) / E(K)_{\text {tor }}$, there is areal-valued positive definite [bilinear form](bilinear%20form), the [Néron-Tate height pairing](Unsorted/Neron-Tate%20height.md). From this one can construct a [regulator](regulator.md) Reg ${ }_{E} / K$, which is the determinant of the matrix of this bilinear form with respect to a basis.

The **Tate-Shafarevich group** of $E$ is a certain abelian torsion group $\mathrm{\Pi}_{E / K}$ attached to $E$, classifying locally solvable $E\dash$-torsors. This group is conjectured to be finite for every $E$, but this has not been proved in general. If $\Pi_{E / K}$ is finite, then its order is a square.

# Weak BSD

Relation to finiteness of [Sha](Unsorted/Tate-Shafarevich%20group.md):

![](attachments/Pasted%20image%2020220408193342.png)
![](attachments/Pasted%20image%2020220408193457.png)
![](attachments/Pasted%20image%2020220408193857.png)

# Birch and Swinnerton-Dyer conjecture

#open/conjectures

![](attachments/Pasted%20image%2020220430220015.png)
![](attachments/Pasted%20image%2020220430220023.png)
![](attachments/Pasted%20image%2020220430221554.png)


![](attachments/Pasted%20image%2020220217213805.png)

![Pasted image 20211106013954.png](Pasted%20image%2020211106013954.png)
![Pasted image 20211106014015.png](Pasted%20image%2020211106014015.png)
![Pasted image 20211106014643.png](Pasted%20image%2020211106014643.png)

# Misc
![](attachments/Pasted%20image%2020220414212913.png)
![Pasted image 20211105130242.png](Pasted%20image%2020211105130242.png)
![Pasted image 20211106014111.png](Pasted%20image%2020211106014111.png)
![Pasted image 20211106015517.png](Pasted%20image%2020211106015517.png)

![Pasted image 20211106015545.png](Pasted%20image%2020211106015545.png)
![Pasted image 20211106015621.png](Pasted%20image%2020211106015621.png)
![Pasted image 20211106015531.png](Pasted%20image%2020211106015531.png)
![](attachments/Pasted%20image%2020220209095333.png)