$p$-adic multiple zeta values at roots of unity and $p$-adic pro-unipotent harmonic actions - IV-1 : $p$-adic multiple zeta values at roots of unity extended to sequences of integers of any sign

TL;DR

This paper extends the definition of $p$-adic multiple zeta values at roots of unity to include sequences of integers of any sign, using new $p$-adic harmonic actions and localization techniques.

Contribution

It introduces a method to define and analyze $p$-adic multiple zeta values with negative indices through localization and harmonic actions.

Findings

Extended $p$MZV$$'s to negative indices.

Developed $p$-adic analogues of elementary complex functions.

Provided new tools for studying $p$-adic periods and fundamental groupoids.

Abstract

This work is a study of $p$-adic multiple zeta values at roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $(\mathbb{P}^{1} - \{0,\mu_{N},\infty\})/ \mathbb{F}_{q}$. The main tool is new objects which we call $p$-adic pro-unipotent harmonic actions. In this part IV we define and study $p$-adic analogues of some elementary complex analytic functions which interpolate multiple zeta values at roots of unity such as the multiple zeta functions. The indices of $p$MZV$\mu_{N}$'s involve sequences of positive integers ; in this IV-1, by considering an operation which we call localization (inverting certain integration operators) in the pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,\mu_{N},\infty\}$, and by using $p$-adic pro-unipotent harmonic actions, we extend the definition of $p$MZV$\mu_{N}$'s to indices for which these integers can be negative, and we study these generalized $p$MZV$\mu_{N}$'s.