ON THE ALGEBRAICITY OF SOME PRODUCTS OF SPECIAL VALUES OF BARNES' MULTIPLE GAMMA FUNCTION

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