Dear MO community,

let $F$ be a totally real field and $K$ be an imaginary quadratic field of class number one, and $M=FK$ be the quadratic CM extension of $F$ obtained by adjoining $K$ to $F$. Let $p$ be a prime that splits in $K$ and fix an embedding of $K$ into $\mathbb{C}$. It will determine a $p$-ordinary CM type $\Sigma$ of $M$. With these assumptions, Katz and Hida-Tilouine showed that there is a period \begin{align}\Omega_\infty=(\Omega_{\infty,\sigma})_{\sigma \in \Sigma} \in (M\otimes\mathbb{C})^\times \end{align} so that for Hecke character $\tau$ with infinity type $m_0\Sigma +2d$ in certain range, the number \begin{align} L(0,\tau)\frac{\pi^d}{\Omega_{\infty}^{m_0+2d}} \end{align} is algebraic. On the other hand, we know from theory of CM elliptic curves that \begin{align} \frac{L(1,\psi)}{\Omega_{E/K}^+} \end{align} is algebraic where $\psi$ is the Hecke character associated to and elliptic curve $E/\mathbb{Q}$ with CM by $\mathcal O_K$ and $\Omega_{E/K}^{\pm}$ is the periods of Neron differential of minimal Weierstrass model of $E$. My question:

Is there any relations between $\Omega_\infty$ and $\Omega_{E/K}^{\pm}$? Is the ratio $\pi$ times a $p$-adic unit always?

I suspect there should be some relation because the periods of $L(s,\psi\circ N_{M/K})$ are powers of $\Omega_{E/K}^{\pm}$, but do not see how to prove the claims in my question above. One related result about periods was shown by Bouganis and Dokchitser[Algebraicity of L-values for elliptic curves in a false Tate curve tower, 2007), but it seems they do not consider the integrality.