Birch–Swinnerton-Dyer Conjecture

The conjecture that relates the arithmetic of an elliptic curve E over Q to the behaviour of its L-function at s = 1: the algebraic rank (the rank of E(Q)) equals the order of vanishing of L(E,s) at s = 1, and the leading Taylor coefficient at that point is expressed in terms of arithmetic invariants (regulator, Tamagawa numbers, real period and the order of the Tate–Shafarevich group).