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

TL;DR

This paper extends the definition of overconvergent $p$-adic multiple polylogarithms, establishes a norm bound, and characterizes these objects via regularized $p$-adic iterated integrals, aiding future research on $p$-adic zeta values.

Contribution

It introduces a generalized definition involving Frobenius iterations and proves a key norm bound, advancing the understanding of $p$-adic multiple polylogarithms.

Findings

Bound on the norm of overconvergent $p$-adic multiple polylogarithms.

Characterization of these objects via regularized $p$-adic iterated integrals.

Foundation for future work on $p$-adic cyclotomic multiple zeta values.

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.