Fermat curves and the reciprocity law on cyclotomic units

TL;DR

This paper introduces a new reciprocity law for a period ring-valued beta function, linking Fermat curves, cyclotomic units, and Stark's conjecture, providing a refined understanding of algebraic properties of special values.

Contribution

It establishes a novel reciprocity law for a period ring-valued beta function, refining existing laws on cyclotomic units and offering an alternative proof related to Stark's conjecture.

Findings

Reciprocity law for the period ring-valued beta function

Connection between Fermat curves and algebraic number theory

Partial proof of algebraicity of special zeta function derivatives

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.