2015/10/05 by Tomokazu Kashio, Kashio, Tomokazu
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.1510.01141
We consider partial zeta functions \ζ(s,c) associated with ray classes\nc's of a totally real field. Stark's conjecture implies that an appropriate\nproduct of \exp(\ζ'(0,c))'s is an algebraic number which is called a Stark\nunit. Shintani gave an explicit formula for \exp(\ζ'(0,c)) in terms of\nBarnes' multiple gamma function. Yoshida ``decomposed'' Shintani's formula: he\ndefined the symbol X(c,\ι) satisfying that\n\exp(\ζ'(0,c))=\∏\ι \exp(X(c,\ι)) where \ι runs over all\nreal embeddings of F. Hence we can decompose a Stark unit into a product of\n[F: mathbb Q] terms. The main result is to show that ([F: mathbb Q]-1) of\nthem are algebraic numbers. We also study a relation between Yoshida's\nconjecture on CM-periods and Stark's conjecture.\n