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

TL;DR

This paper investigates the algebraic nature of products of special values of Barnes' multiple gamma function related to Stark units, providing new decompositions and algebraicity results in number theory.

Contribution

It introduces a decomposition of Stark units into multiple gamma function terms and proves that most of these terms are algebraic numbers, advancing understanding of Stark's conjecture.

Findings

Most decomposed terms are algebraic numbers

Explicit formulas relate Barnes' gamma function to Stark units

Connections between Yoshida's conjecture and Stark's conjecture

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.