The Birch and Swinnerton-Dyer conjecture and a geometric analog(ue)

by John Tate

What it is about

The first part of the paper described the BSD conjecture in general functorial terms, and is the first place where it was stated for abelian varieties. This more general formulation is due to Tate, but he shrugs off credit for it. Tate then discusses why if the conjecture is true for an abelian variety A, it is automatically true for any abelian variety isogenous to A, a result that inspired his global duality theorem. Next he gives an analogue of the Shafarevich-Tate group in a geometric context and relates it to the Brauer group of a curve.

Electronic Version

Here is a 12MB scan of the paper in PDF format.