Birch and Swinnerton-Dyer conjecture

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 - modularity - L function - motivic L function - Application: congruent number problem - Mazur’s theorem - Faltings theorem


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, the Néron-Tate height pairing. From this one can construct a regulator 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 \(E{\hbox{-}}\)-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:

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 image 20211106014015.png Pasted image 20211106014643.png

Misc

attachments/Pasted%20image%2020220414212913.png Pasted image 20211105130242.png Pasted image 20211106014111.png Pasted image 20211106015517.png

Pasted image 20211106015545.png Pasted image 20211106015621.png Pasted image 20211106015531.png attachments/Pasted%20image%2020220209095333.png

#AG/elliptic-curves #open/conjectures