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

2017/12/28 by Jarossay, David
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1712.09976

Abstract

This work is a study of p-adic multiple zeta values at roots of unity (pMZVμ_N's), the p-adic periods of the crystalline pro-unipotent fundamental groupoid of (ℙ¹ - \0,μ_N,∞\)/ \mathbbF_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 pMZVμ_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 ℙ¹ - \0,μ_N,∞\, and by using p-adic pro-unipotent harmonic actions, we extend the definition of pMZVμ_N's to indices for which these integers can be negative, and we study these generalized pMZVμ_N's.

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