$p$-adic Polylogarithms and $p$-adic Hecke $L$-functions for Totally Real Fields

TL;DR

This paper introduces a new definition of $p$-adic polylogarithms for totally real fields and relates their special values to $p$-adic Hecke L-functions, advancing the understanding of $p$-adic number theory.

Contribution

It defines $p$-adic polylogarithms as equivariant classes in cohomology and connects their special values to $p$-adic Hecke L-functions for totally real fields.

Findings

New $p$-adic polylogarithm definition as cohomology classes

Expression of $p$-adic L-function values in terms of polylogarithms

Enhanced understanding of $p$-adic number theory for totally real fields

Abstract

The purpose of this article is to newly define the $p$-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of $p$-adic $L$-functions interpolating nonpositive values of Hecke $L$-functions of the totally real field in terms of special values of these $p$-adic polylogarithms.