On the ratios of Barnes' multiple gamma functions to the $p$-adic analogues

Let $F$ be a totally real field. For each ideal class $c$ of $F$ and each real embedding $\iota$ of $F$, Hiroyuki Yoshida defined an invariant $X(c,\iota )$ as a finite sum of log of Barnes’ multiple gamma functions with some correction terms. Then the derivative value of the partial zeta function $\zeta (s,c)$ has a canonical decomposition ${\zeta }^{\mathrm{\prime }}(0,c)={\sum }_{\iota }X(c,\iota )$, where $\iota$ runs over all real embeddings of $F$. Yoshida studied the relation between $\exp (X(c,\iota ))$'s, Stark units, and Shimura’s period symbol. Yoshida and the author also defined and studied the $p$-adic analogue $X_p(c,\iota )$: In particular, we discussed the relation between the ratios $[\exp (X(c,\iota )):{\exp }_p(X_p(c,\iota ))]$ and Gross-Stark units. In a previous paper, the author proved the algebraicity of some products of $\exp (X(c,\iota ))$'s. In this paper, we prove its $p$-adic analogue. Then, by using these algebraicity properties, we discuss the relation between the ratios $[\exp (X(c,\iota )):{\exp }_p(X_p(c,\iota ))]$ and Stark units.