nLab Birch and Swinnerton-Dyer conjecture

Idea

The Birch and Swinnerton-Dyer conjecture is a conjecture about the form of the first non-vanishing derivative of the Hasse-Weil L-function of an elliptic curve at $s= 1$, expressed in terms of a higher regulator, analogous to the class number formula for a Dedekind zeta function.

This is hence a conjecture about special values of L-functions. It influenced the more far-reaching Beilinson conjectures.

Related concepts

• Picard group, Tate-Shafarevich group

• Tamagawa number

References

• Wikipedia, Birch and Swinnerton-Dyer conjecture

• Frank Gounelas, The BSD cconjecture, regulators and special values of L-functions (pdf)

• Spencer Bloch, A note on height pairings, Tamagwawa numbers, and the Birch and Swinnerton-Dyer conjecture, Inventiones math. 58, 65-76 (1980) (pdf)

• Conrad, Venkatesh, et al., BSD Seminar Introduction to the BSD conjecture, All lecture notes