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We consider partial zeta functions $\zeta(s,c)$ associated with ray classes $c$'s of a totally real field. Stark's conjecture implies that an appropriate product of $\exp(\zeta'(0,c))$'s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp(\zeta'(0,c))$ in terms of Barnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he defined the symbol $X(c,\iota)$ satisfying that $\exp(\zeta'(0,c))=\prod_{\iota} \exp(X(c,\iota))$ where $\iota$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb Q]$ terms. The main result is to show that $([F:\mathbb Q]-1)$ of them are algebraic numbers. We also study a relation between Yoshida's conjecture on CM-periods and Stark's conjecture., Comment: 30 pages

We provide an equivalent condition for the monogenity of the ring of integers of any cyclic cubic field. We show that if a cyclic cubic field is monogenic then it is a simplest cubic field $K_t$ which is the splitting field of a Shanks cubic polynomial $f_t(x):=x^3-tx^2-(t + 3)x-1$ with $t \in \mathbb Z$. Moreover we give an equivalent condition for when $K_t$ is monogenic, which is explicitly written in terms of $t$., 17 pages, title changed (old one: On the monogenity of the ring of integers of cyclic cubic fields), typos corrected(in particular, Theorem1.1-(iii), Corollary1.6-(iii)), references [Ar], [Gr1], [Gr2], [Gr3] [Ok] added

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

We study a relation between two refinements of the rank one abelian Gross-Stark conjecture: For a suitable abelian extension $H/F$ of number fields, a Gross-Stark unit is defined as a $p$-unit of $H$ satisfying some proporties. Let $\tau \in \mathrm{Gal}(H/F)$. Yoshida and the author constructed the symbol $Y_p(\tau)$ by using $p$-adic $\log$ multiple gamma functions, and conjectured that the $\log_p$ of a Gross-Stark unit can be expressed by $Y_p(\tau)$. Dasgupta constructed the symbol $u_T(\tau)$ by using the $p$-adic multiplicative integration, and conjectured that a Gross-Stark unit can be expressed by $u_T(\tau)$. In this paper, we give an explicit relation between $Y_p(\tau)$ and $u_T(\tau)$., Comment: 17 pages

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 $\zeta'(0,c)=\sum_{\iota}X(c,\iota)$, 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., Comment: 26pages, typos corrected, a reference added, a correction to the definition of alpha in Remark 6

The second author defined the absolute CM-period symbol using the multiple gamma function. This symbol is conjectured to give the same value as Shimura's period symbol up to multiplication by algebraic numbers. In this paper, we define a p-adic analogue of the absolute CM-period symbol, using the p-adic multiple gamma function studied by the first author. In the completely split case, we present a conjecture which predicts the exact value of this symbol as well as supporting evidences.

Shintani expressed the first derivative at $s = 0$ of a partial $\zeta$-function of an algebraic number field in terms of the multiple gamma function. Cassou-Noguès constructed a $p$-adic analogue of the partial $\zeta$-function and calculated the derivative at $s = 0$. In this paper, we will define a $p$-adic analogue of the multiple gamma function and give a $p$-adic analogue of Shintani’s formula. This formula has a strong resemblance to the original Shintani’s formula. Using this formula, we get a partial result toward Gross’ conjecture concerning the order at $s = 0$ of the $p$-adic $L$-function.

The absolute CM-period symbol was defined by the second author using the multiple gamma function. Conjecturally it coincides with Shimura’s period symbol up to multiplication by algebraic numbers. In this paper we define a p-adic analogue of the absolute CM-period symbol using the p-adic multiple gamma function studied by the first author. We present an explicit conjecture with solid evidence.