The elliptic-curve parity conjecture for irreducible representations

Let $k$ be a number field, let $E$ be an elliptic curve over $k$, and let $\tau$ be an irreducible complex representation of $\operatorname{Gal}(\overline{k}/k)$. Put $F=\overline{k}^{\operatorname{ker}(\tau)}$, and let $E(F)^\tau$ denote the $\tau$-isotypic component of $E(F)\otimes\mathbb{C}$. Assume that $L(E/k,\tau,s)=L(E/k,\overline{\tau},s)$, and define

r(\tau)=\frac{\text{dim}_{_}{\mathbb{C}}E(F)^\tau}{\text{dim}_{(}\tau)}.

Parity conjecture. Under these assumptions,

$$W(E/k,\tau)=(-1)^{r(\tau)}.$$

This is presented as a generalization of the parity conjecture, arising from an extension of the Birch and Swinnerton-Dyer conjecture related to the Deligne–Gross conjecture. Its resolution status is not specified in the supplied text.