The Birch and Swinnerton-Dyer conjecture can be formulated for elliptic curves over any number field, and generalizations for more general…

The Birch and Swinnerton-Dyer conjecture can be formulated for elliptic curves over any number field, and generalizations for more general arithmetic objects have been proposed. The Millennium Prize is offered for resolving the conjecture for elliptic curves over rational numbers.