A bound on the norm of overconvergent $p$-adic multiple polylogarithms

Abstract

We generalize the definition of overconvergent $p$-adic multiple polylogarithms and of $p$-adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized $p$-adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent $p$-adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on $p$-adic cyclotomic multiple zeta values.

25 pages. This is Part I-1 of "$p$-adic cyclotomic multiple zeta values and $p$-adic pro-unipotent harmonic actions"

References in corpus (6)

• p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation
• Adjoint cyclotomic multiple zeta values and cyclotomic multiple harmonic values
• Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values
• An explicit theory of ${\pi }_1^{\mathrm{un},\mathrm{crys}}({\mathbb{P}}^1-\{0,{\mu }_N,\infty \})$ - II-3 : Sequences of multiple harmonic sums viewed as periods
• Depth reductions for associators
• An explicit theory of $π_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,μ_{N},\infty\})$ - II-2 : From standard algebraic relations of weighted multiple harmonic sums to those of cyclotomic $p$-adic multiple zeta values

Cited by in corpus (7)

• Pro-unipotent harmonic actions and a computation of $p$-adic cyclotomic multiple zeta values
• Depth reductions for associators
• Non-vanishing of certain cyclotomic multiple harmonic sums and application to the non-vanishing of certain $p$-adic cyclotomic multiple zeta values
• $p$-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
• Pro-unipotent harmonic actions and dynamical properties of $p$-adic cyclotomic multiple zeta values
• $p$-adic multiple zeta values and $p$-adic pro-unipotent harmonic actions : summary of parts I and II
• On generic double shuffle relations, localized multiple polylogarithms and algebraic functions