The p-parity conjecture for abelian varieties and self-dual representations

Let $A/K$ be an abelian variety over a number field, and let $p$ be a prime. Define the $p$-Selmer vector space

$${\mathcal X}_p(A/K)=(\text{Pontryagin dual of the }p^{\infty}\text{-Selmer group of }A/K)\otimes_{{\mathbb Z}_p}{\mathbb Q}_p.$$

If $F/K$ is a Galois extension and $\tau$ is a self-dual representation of $\operatorname{Gal}(F/K)$, write $\langle\tau,{\mathcal X}_p(A/F)\rangle$ for the corresponding multiplicity and let $w(A/K,\tau)$ denote the associated twisted root number. The $p$-parity conjecture. For an abelian variety $A/K$,

$$(-1)^{\dim {\mathcal X}_p(A/K)}=w(A/K).$$

Moreover,

$$(-1)^{\langle\tau,{\mathcal X}_p(A/F)\rangle}=w(A/K,\tau).$$

Here $w(A/K)$ is the global root number. The dimension of ${\mathcal X}_p(A/K)$ equals the Mordell--Weil rank when the Tate--Shafarevich group is finite; the conjecture predicts the corresponding parity statement without assuming that finiteness.