The parity conjecture for elliptic curves over number fields

From papers

Let $E$ be an elliptic curve over a number field $K$ . Let $w(E/K)\in\{\pm 1\}$ be the root number, defined as the conjectural sign in the functional equation for $L(E/K,s)$ under $s\leftrightarrow 2-s$ , and let $\sigma(E/K,p)=(-1)^{\text{$ p $-Selmer rank of$ E/K $}}$ for a prime $p$ . Parity conjecture. For any (some) prime $p$ , the root number agrees with the parity of the $p$ -Selmer rank, so

w(E/K)=\sigma(E/K,p).

This conjecture compares the analytic parity predicted by the functional equation with the arithmetic parity of the Selmer rank. The paper's abstract states that it confirms the parity conjecture for elliptic curves with a cyclic $p$ -isogeny under the hypotheses treated there, while the general statement remains open.

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Equivalent formulations 2

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

The parity conjecture for elliptic curves over number fields
Let $E$ be an elliptic curve over a number field $K$ . Its Mordell–Weil rank is denoted by $\operatorname{rk}(E/K)$ , and $w(E/K)\in\{\pm1\}$ is its global root number.

Parity conjecture.
$(-1)^{\operatorname{rk}(E/K)}=w(E/K).$
This predicts the parity of the Mordell–Weil rank from local arithmetic data and, when the root number is $-1$ , predicts the existence of a point of infinite order. It follows formally from the Birch–Swinnerton-Dyer and Hasse–Weil conjectures, but remains open in general.
source: Lilybelle Cowland Kellock and Vladimir Dokchitser, “Root numbers and parity phenomena”, arXiv:2303.07883 (2023).
The parity conjecture for elliptic curves over number fields
Let $E$ be an elliptic curve defined over a number field $K$ . The global root number $w(E/K)$ is the sign predicted by the functional equation of the $L$ -function of $E$ over $K$ , and $\operatorname{rank}(E/K)$ denotes the Mordell–Weil rank. Parity conjecture.
$w(E/K)=(-1)^{\operatorname{rank}(E/K)}.$
This conjecture relates the global root number to the parity of the Mordell–Weil rank and is a central prediction concerning elliptic curves and their $L$ -functions. The supplied context does not state whether it is known in this generality.
source: Tim Evink, “Imaginary quadratic fields F with X_0(15)(F) finite”, arXiv:2405.09337 (2024).

Sources & referencesView supporting material

Primary source

Tim Dokchitser and Vladimir Dokchitser, “Parity of ranks for elliptic curves with a cyclic isogeny”, arXiv:math/0604149 (2007).

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