Arithmetic BSD invariant conjecture for Artin twists of elliptic curves

Let $E/{\mathbb Q}$ be an elliptic curve. For every Artin representation $\rho$ over ${\mathbb Q}$, suppose there is an invariant $\BSD(E,\rho)\in{\mathbb C}^{\times}$. If $\rho$ and $\tau$ factor through $\operatorname{Gal}(K/{\mathbb Q})$, and $r=\langle\rho,E(K)_{\mathbb C}\rangle$, the invariant is required to satisfy

$$\BSD(E,F)=\BSD(E,\operatorname{Ind}_{F/{\mathbb Q}}{\mathbf 1})$$

for every number field $F$, with $\text{\tencyr\cyracc{Sh}}_{E/F}$ finite,

$$\BSD(E,\rho\oplus\tau)=\BSD(E,\rho)\BSD(E,\tau),$$ $$\BSD(E,\rho)=\BSD(E,\rho^*)\cdot(-1)^r w_{E,\rho}w_\rho^{-2},$$

and, if $\rho$ is self-dual, $\BSD(E,\rho)\in{\mathbb R}$ and $\operatorname{sign}\BSD(E,\rho)=\operatorname{sign}w_\rho$. If $\langle\rho,E(K)_{\mathbb C}\rangle=0$, it must additionally satisfy

$$\BSD(E,\rho)\in{\mathbb Q}(\rho)^\times, \qquad \BSD(E,\rho^{\mathfrak g})=\BSD(E,\rho)^{\mathfrak g}$$

for every ${\mathfrak g}\in\operatorname{Gal}({\mathbb Q}(\rho)/{\mathbb Q})$; and if $\rho$ is a non-trivial primitive Dirichlet character of order $d$, with either the conductors of $E$ and $\rho$ coprime or $E$ semistable with no non-trivial isogenies over ${\mathbb Q}$, then $\BSD(E,\rho)\in{\mathbb Z}[\zeta_d]$. Arithmetic BSD invariant conjecture. Such invariants exist with all of the stated properties. The conjectured invariant is intended to provide an Artin-representation decomposition of the Birch–Swinnerton-Dyer quotient and yields predictions for Selmer groups, Tate–Shafarevich groups, and ranks. The paper does not provide a resolution of this conjecture.