The parity conjecture for ranks of abelian varieties

Let $A$ be an abelian variety over a number field $K$. Write $\operatorname{rk} A$ for its Mordell–Weil rank, and let $w(A/K_v)$ denote the local root number at each place $v$ of $K$. Parity conjecture.

$$(-1)^{\operatorname{rk} A} = \prod_{v\textup{ place of }K} w(A/K_v).$$

This prediction is motivated by the Birch–Swinnerton-Dyer conjecture: the global root number is the sign in the functional equation of the $L$-function, and hence is expected to control the parity of the Mordell–Weil rank. The paper studies analogous local formulas for Jacobians with automorphisms; the conjecture itself is not resolved in the supplied text.