2015/02/16 by Tomokazu Kashio, Kashio, Tomokazu
Mathematics · #11M06 #11M35 #11R27 #11R42 #11S80 #14H45 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11M06 #msc:11M35 #msc:11R27 #msc:11R42 #msc:11S80 #msc:14H45
paper · pdf · doi:10.48550/arxiv.1502.04397
19 pages; typos corrected
openalex publication_date 2015/02/16 · arxiv created 2015/03/10 · arxiv updated 2015/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define a "period ring-valued beta function" and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark's conjecture implies that the exponential of the derivatives at s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler's formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative (and partial) proof by using the reciprocity law on the period ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.