A Bloch–Kato nonvanishing implication for polarized Hecke characters of weight minus one

Let $E$ be a quadratic imaginary field, let $\chi$ be an algebraic Hecke character of $E$, and let $\chi_p:G_E\to K$ be its $p$-adic realization. Suppose that $\chi$ is polarized and has weight $-1$, meaning

$$\chi^\perp=\chi |.|^{-1},$$

where $\chi^\perp(z)=\chi^{-1}(czc)$ and $c\in G_{\mathbb Q}$ induces complex conjugation in $E$. Let $H^1_f(E,\chi_p)$ denote the Bloch–Kato Selmer group. The Bloch–Kato nonvanishing conjecture.

$$\operatorname{ord}_{s=0}L(\chi,s)\neq 0 \Rightarrow \dim H^1_f(E,\chi_p)\geq 1.$$

This is a special case of the Bloch–Kato conjecture, relating the order of vanishing of an $L$-function to the dimension of a Selmer group. The source presents this implication in the context of the theorem of Rubin on the Iwasawa main conjecture for CM elliptic curves; its resolution status is not specified in the supplied text.