A period-ring-valued gamma function and a refinement of the reciprocity law on Stark units (Algebraic Number Theory and Related Topics 2017)

Abstract

This is an announcement of the results of the paper "On a common refinement of Stark units and Gross-Stark units". We study a relation between CM-periods, multiple gamma functions, the rank one abelian Stark conjecture, and their p-adic analogues. The main results are as follows. First we construct two kinds of period-ring valued functions under a slight generalization of Hiroyuki Yoshida's conjecture on "Absolute CM-periods". Here the period ring is in the sense of p-adic Hodge theory. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action on Fontaine's period ring Bcris. We show that our conjecture implies a part of Stark's conjecture and a refinement of Gross' p-adic analogue simultaneously. We also provide some partial results for our conjecture.