# relation between period of Katz and Hida-Tilouine and period of CM elliptic curves

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.

• I'm not sure about general $F$ but for $F = \mathbb{Q}$ this comes up in the paper by Bouganis and Venjakob, "On the noncommutative main conjecture for elliptic curves with complex multiplication" (1006.1490 on the Arxiv). In the proof of lemma 2.10 they show that the ratios $\Omega_+ / \Omega_\infty$ and $\Omega_- / \Omega_{\infty}$ are in $K$. (I think their normalisation for $\Omega_\infty$ is probably different from yours by a factor of $\pi$ -- compare proof of thm 2.3 of their paper.) – David Loeffler Sep 2 '11 at 11:49
• PS: Sorry, I was thinking about algebraicity and you wanted integrality, but in fact B + V address this as well -- they show that the ratios of the periods are in $\mathbb{Z}_p^\times$ (at least for $p \ne 2$). – David Loeffler Sep 2 '11 at 11:51
• @Loeffler Thanks a lot for pointing out that B+V addressed this special case in their paper. I wasn't aware of that. Still, I am still interested whether similar statement if true in the general case $F\not=\mathbb{Q}$. – Dohyeong Kim Sep 2 '11 at 13:01