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


 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 exponentials 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 proof of a weaker result 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.