On the algebraicity of some products of special values of Barnes' multiple gamma function

Abstract

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.