The parity conjecture for abelian varieties

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Parity conjecture for abelian varieties

   Let $A$ be an abelian variety over a number field $K$. Write $\operatorname{rk}(A/K)$ for its Mordell–Weil rank, and let $w_{A/K}\in\{\pm1\}$ be the global root number of $A/K$.

   Parity conjecture. For every abelian variety $A$ over a number field $K$,

   $$(-1)^{\operatorname{rk}(A/K)}=w_{A/K}.$$

   The conjecture asserts that the global root number controls the parity of the Mordell–Weil rank. The supplied context says that it is currently out of reach, although the corresponding Selmer-rank version is more tractable.

   source: Vladimir Dokchitser and Celine Maistret, “Parity conjecture for abelian surfaces”, arXiv:1911.04626 (2023).

2. The parity conjecture for abelian varieties

   Let $A/K$ be an abelian variety over a number field $K$. For each place $v$ of $K$, let $w_{A/K_v}$ be the local root number of $A$. Parity conjecture. The parity of the Mordell–Weil rank is determined by the product of the local root numbers:

   $$(-1)^{\operatorname{rk}(A/K)}=\prod_v w_{A/K_v}.$$

   This is the parity prediction associated with the Birch–Swinnerton-Dyer conjecture and the functional equation of the $L$-function. It is not known in general.

   source: Jordan Docking, “2^-Selmer Rank Parities via the Prym Construction”, arXiv:2108.09564 (2023).

3. The parity conjecture for abelian varieties

   Let $A$ be an abelian variety over a number field $K$. Write $\mathrm{rk}(A/K)$ for its rank and $w_{A/K}$ for its global root number.

   Parity conjecture. For every abelian variety $A$ over a number field $K$,

   $$(-1)^{\mathrm{rk}(A/K)}=w_{A/K}.$$

   The conjecture links the parity of the algebraic rank to the sign in the functional equation of the $L$-function. No unconditional proof is known in general, although several cases follow from results on the $p$-parity conjecture under finiteness assumptions on Shafarevich–Tate groups.

   source: Holly Green and Celine Maistret, “The 2-parity conjecture for elliptic curves with isomorphic 2-torsion”, arXiv:2110.06718 (2022).