$p$-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

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.

18 pages

References in corpus (1)

• p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation