Birch–Swinnerton-Dyer parity conjecture for elliptic curves

Let $E$ be an elliptic curve over a number field $K$. Write $w(E/K)=\prod_v w(E/K_v)\in\{+1,-1\}$ for its global root number, and let $\operatorname{rk} E(K)$ denote its Mordell–Weil rank.

Parity conjecture. The Mordell–Weil rank $\operatorname{rk} E(K)$ is even if and only if the root number $w(E/K)$ is $+1$.

The conjecture relates the parity of the algebraic rank to the sign in the expected functional equation. The source notes that, apart from rank $0$ and $1$ cases over $\mathbb{Q}$, little was known at the time.