The parity conjecture for elliptic curves

Equivalent formulations 4

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

1. The parity conjecture for elliptic curves

   Let $E$ be an elliptic curve over a number field $K$. Its Mordell–Weil rank is the rank of $E(K)$ modulo torsion, and let $w(E/K)\in\{\pm1\}$ denote its global root number.

   Parity conjecture.

   $$(-1)^{\operatorname{rk} E/K}=w(E/K).$$

   This is the parity prediction associated with the Birch–Swinnerton-Dyer conjecture. The paper explains that the Shafarevich–Tate conjecture implies this equality for all elliptic curves over number fields.

   source: Tim Dokchitser and Vladimir Dokchitser, “Root numbers and parity of ranks of elliptic curves”, arXiv:0906.1815 (2009).

2. The parity conjecture for elliptic curves

   Let $E$ be an elliptic curve over $\mathbb{Q}$. Write $W(E)$ for its root number and $r(E)$ for its geometric rank. Parity conjecture.

   $$W(E)=(-1)^{r(E)}.$$

   This predicts that the root number determines the parity of the geometric rank and is a weak form of the Birch and Swinnerton-Dyer conjecture, which predicts equality of analytic and geometric ranks. In the paper it is used to expect rank elevation in integer fibres when the generic rank and root number have the appropriate parity; no resolution is stated here.

   source: Rena Chu and Julie Desjardins, “Constant root number on integer fibres of elliptic surfaces”, arXiv:2011.02386 (2022).

3. The parity conjecture for elliptic curves

   For an elliptic curve $E/\Q$, let $W(E)\in\{-1,1\}$ be the root number in the functional equation of its $L$-function, and let $\operatorname{rank} E(\Q)$ denote the rank of its Mordell–Weil group. The parity conjecture. For every elliptic curve $E/\Q$,

   $$W(E)=(-1)^{\operatorname{rank} E(\Q)}.$$

   This is a consequence of the Birch and Swinnerton-Dyer conjecture. It is known when the Tate–Shafarevich group is finite and unconditionally for analytic rank $0$ or $1$, but remains open in general.

   source: Jonathan Love, “Root numbers of a family of elliptic curves and two applications”, arXiv:2201.04708 (2023).

4. The parity conjecture for elliptic curves

   Let $E$ be an elliptic curve over a number field $K$. Its Mordell–Weil rank is denoted by $\operatorname{Rank}(E)$, and let $w(E/K)$ be its global root number. Parity conjecture.

   $$(-1)^{\operatorname{Rank}(E)}=w(E/K).$$

   This conjecture relates the parity of the Mordell–Weil rank to the global root number and is used in the paper to infer rank parity from local root-number computations. Its general validity remains open.

   source: Roberto Hernandez, “Rational Points on a Family of Genus 3 Hyperelliptic Curves”, arXiv:2510.17791 (2025).