A Note on Twisted Bernoulli Measures

TL;DR

This paper introduces twisted Bernoulli measures as a new family of p-adic measures related to polylogarithms, establishing their uniqueness and higher-order properties within p-adic analysis.

Contribution

It defines twisted Bernoulli measures parametrized by a specific p-adic domain and proves their uniqueness, extending the understanding of measures linked to p-adic polylogarithms.

Findings

Introduced twisted Bernoulli measures as higher-order p-adic measures.

Proved the uniqueness of these measures among polynomials over Q(y).

Connected these measures to existing measures used by Koblitz and Coleman.

Abstract

We introduce the twisted Bernoulli measures as a family of p-adic measures parametrized by the complement of the open disc with radius 1 and centered at 1 in the completion of an algebraic closure of p-adic numbers. These measures are the higher order versions of the measure used by Koblitz and Coleman to interpret (p-adic) polylogarithms. We also prove that these measures are the unique p-adic measures that can be obtained from polynomials over the field Q(y) which is similar to the uniqueness property of Bernoulli measures.