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

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 and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a 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 its 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, our conjecture follows from Coleman's formula on Fermat curves. We also prove some partial results in other cases.