For every abelian variety $A$ over a number field $K$ and for a prime number $p$ ,$$\begin{equation*} (-1)^{\operatorname{rk}_p (A/K)} = w_{A/K}. \end{equation*}$$
2-parity conjecture holds for all principally polarised abelian surfaces over number fields $A/K$ such that $\operatorname{Gal}(K(A[2])/K)$ is a 2-group that are
Jacobians of semistable genus 2 curves with good ordinary reduction at primes above 2, or not isomorphic to the Jacobian of a genus 2 curve. Assuming the finiteness of Tate–Shafarevich groups, the $p$ -parity conjecture clearly implies the parity conjecture. applied to $F\!=\! K(A[2])$ . The first expresses the parity of the $2^{\infty }$ -Selmer rank of $A/K$ as a product of some local terms $\lambda _{A/K_v}$ , $$\begin{equation*} (-1)^{\operatorname{rk}_2 A/K} = \prod _v \lambda _{A/K_v}, \end{equation*}$$ analogously to the formula for the global root number as a product of local root numbers $w_{A/K}=\prod w_{A/K_v}$ , the products taken over all the places of $K$ . This makes crucial use of a Richelot isogeny on $A$ whose existence is guaranteed by the restriction on $\operatorname{Gal}(K(A[2])/K)$ . The second part is the proof that this expression for the parity of the rank is compatible with root numbers. In other words, that $\lambda _{A/K_v}w_{A/K_v}$ satisfies the product formula $$\begin{equation*} \prod _v \lambda _{A/K_v}w_{A/K_v} = 1, \end{equation*}$$ which leads to the desired expression $(-1)^{\operatorname{rk}_2 A/K} \!=\! w_{A/K}$ . This product formula is more delicate than one might expect, because one often has $\lambda _{A/K_v}\!\ne \! w_{A/K_v}$ . However, rather miraculously, $\lambda _{A/K_v}\in \lbrace \pm 1\rbrace$ always differs from $w_{A/K_v}\in \lbrace \pm 1\rbrace$ at an even number of places $v$ . below gives an explanation for this phenomenon, by describing an explicit relation between the two local invariants. The key point of the conjecture is that it reduces the global problem of controlling the parity of 2-Selmer ranks to the purely local one of proving an identity between various invariants of genus 2 curves defined over local fields. We prove this conjecture for all semistable curves with good ordinary reduction at primes above 2 The proof relies on explicit formulas and the study of genus 2 curves over local fields, paper. We note that recently Docking has managed to prove an analogue of the parity formula $(-1)^{\operatorname{rk}_2 A/K}=\prod \lambda _{A/K_v}$ for Jacobians of curves of genus 3 with $\operatorname{Gal}(K(A[2])/K)$ a 2-group, A proof of the product formula $\prod _v \lambda _{A/K_v} w_{A/K_v}=1$ in his setting, combined, would give an analogue for Jacobians of curves of genus 3.