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On the algebraicity of some products of special values of Barnes' multiple gamma function
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 140, Number 3, June 2018
- pp. 617-651
- 10.1353/ajm.2018.0015
- Article
- Additional Information
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 $[F:\Bbb{Q}]$ terms. The main result is to show that $([F:{\Bbb Q}] - 1)$ of them are algebraic numbers. We also study a relation between Yoshida's conjecture on CM-periods and Stark's conjecture.