The parity conjecture for Artin twists of elliptic curves

Let $E/K$ be an elliptic curve over a number field, let $F/K$ be a finite extension, and let $\rho$ be a self-dual Artin representation of $\operatorname{Gal}(\overline K/K)$ factoring through $\operatorname{Gal}(F/K)$. Write $\langle\rho,E(F)\otimes_{\mathbb Z}\mathbb C\rangle$ for the multiplicity of $\rho$.

Parity conjecture for Artin twists.

$$(-1)^{\langle\rho,E(F)\otimes_{\mathbb Z}\mathbb C\rangle}=w(E/K,\rho),$$

where $w(E/K,\rho)$ is the global root number of the twist of $E$ by $\rho$.

This refines the ordinary parity conjecture by predicting the parity of each self-dual Artin-isotypic contribution to the Mordell–Weil group. The source uses it conditionally throughout its examples, so it remains open in general.