On a common refinement of Stark units and Gross–Stark units

Abstract

The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then, we formulate a conjecture on the reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is a real place. It also implies a refinement of the Gross–Stark conjecture under a certain assumption. When the base field is the rational number field, we see that our conjecture follows from Coleman's formula on Fermat curves. We also provide some partial results in other cases.