The $p$-parity conjecture for abelian varieties

Equivalent formulations 3

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

1. $p$-Parity conjecture for abelian varieties

   Let $A$ be an abelian variety over a number field $K$, and let $p$ be a prime number. Write $\operatorname{rk}_p(A/K)$ for the $p^\infty$-Selmer rank of $A/K$, and let $w_{A/K}\in\{\pm1\}$ be the global root number.

   $p$-Parity conjecture. For every abelian variety $A$ over a number field $K$ and every prime number $p$,

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

   The $p^\infty$-Selmer rank differs from the Mordell–Weil rank by the contribution of the divisible part of the Tate–Shafarevich group, so this formulation is more tractable than the ordinary parity conjecture. The supplied text does not state that it has been resolved in general.

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

2. The $p$-parity conjecture for abelian varieties

   Let $A/K$ be an abelian variety over a number field $K$, and let $p$ be prime. Write $\operatorname{rk}_p(A/K)$ for the $p^\infty$-Selmer rank. For each place $v$ of $K$, let $w_{A/K_v}$ be the local root number. $p$-parity conjecture. The parity of the $p^\infty$-Selmer rank satisfies

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

   When the Tate–Shafarevich group of $A/K$ is finite, the $p^\infty$-Selmer rank equals the Mordell–Weil rank. The conjecture is open in general.

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

3. The $p$-parity conjecture for abelian varieties

   Let $A$ be an abelian variety over a number field $K$, let $p$ be a prime, and write $\mathrm{rk}_p(A/K)$ for the $p^{\infty}$-Selmer rank and $w_{A/K}$ for the global root number.

   $p$-parity conjecture. For every abelian variety $A$ over a number field $K$ and every prime $p$,

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

   The $p$-parity conjecture gives parity information for Selmer ranks and implies instances of the ordinary parity conjecture when the relevant Shafarevich–Tate groups are finite. It remains open for elliptic curves over number fields in general, although the paper proves it over totally real fields.

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