The parity conjecture for elliptic curves over the rationals

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. The parity conjecture for elliptic curves over the rationals

   Let $E$ be an elliptic curve over $\mathbb{Q}$. Write $r_E$ for the rank of the abelian group $E(\mathbb{Q})$, and let $W(E)\in\{\pm1\}$ be the global root number in the functional equation of its $L$-function. Parity conjecture. The parity of the arithmetic rank is determined by the global root number:

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

   This conjecture predicts the parity of the Mordell–Weil rank from the analytic functional equation and is used in the paper to infer conjectural rank-one cases; the source gives no resolution here.

   source: Xiumei Li and Jinxiang Zeng, “Factoring integer using elliptic curves over rational number field Q”, arXiv:1207.0274 (2013).

2. The parity conjecture for elliptic curves over the rationals

   Let $E$ be an elliptic curve defined over the rational numbers. Let $L(E/\mathbb{Q},s)$ be its $L$-function, and let $W(E/\mathbb{Q})\in\{\pm1\}$ be its global root number.

   Parity conjecture. The parity of the Mordell–Weil rank is determined by the global root number:

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

   The conjecture remains open in general; the paper uses it together with its results to predict many genus-$3$ curves that violate the Hasse local-global principle.

   source: Wade Hindes, “Rational points on certain families of symmetric equations”, arXiv:1403.0645 (2014).

3. The parity conjecture for elliptic curves over the rationals

   Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $w(E)$ denote the sign of its functional equation. Parity conjecture.

   $$(-1)^{\operatorname{rk}E(\mathbb{Q})}=w(E).$$

   This follows formally from the Birch–Swinnerton-Dyer conjecture and is not known for all elliptic curves over $\mathbb{Q}$; the paper proves the needed parity statement under finiteness of the $2$-primary part of $\Sha(E)$.

   source: Jerson Caro and Hector Pasten, “On the fibres of an elliptic surface where the rank does not jump”, arXiv:2210.14181 (2022).