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 and p-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CMperiods. 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.

Note on Coleman's formula for the absolute Frobenius on Fermat curves

“…Coleman explicitly calculated the absolute Frobenius on Fermat curves [Co2]. The author rewrote his formula in [Ka1,Ka2] as follows.…”

“…Remark 1.1. Yoshida and the author formulated conjectures in [KY1, KY2,Ka2] which are generalizations of Coleman's formula, from cyclotomic fields to arbitrary CM-fields: Coleman's formula implies "the reciprocity law on cyclotomic units" [Ka1] and "the Gross-Koblitz formula on Gauss sums" [GK,Co1] simultaneously. The author conjectured a generalization [Ka2, Conjecture 4] of Coleman's formula which implies a part of Stark's conjecture and a generalization of (the rank 1 abelian) Gross-Stark conjecture simultaneously.…”