Fermat curves and a refinement of the reciprocity law on cyclotomic units

Abstract

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.

p-Adic measures associated with zeta values and p-adic log multiple gamma functions

“…We also studied a relation between the same ratios and Stark units associated with real places in [Ka3]. We found a more significant relation between the ratios [ p-adic gamma function : gamma function ] and cyclotomic units in [Ka2].…”

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.…”

On the ratios of Barnes' multiple gamma functions to the p-adic analogues

“…In [Ka2] we proved a reciprocity law (5) on a period-ring valued beta function, which is a refinement of (4) in the following sense: For simplicity, we assume that p | m, p = 2 here. We use the following notation.…”

“…To clarify the meaning of the main results in this paper, we briefly describe the results in [Ka2]. Let F = Q, v the unique real place of Q.…”