p-adic iterated integration on semistable curves

TL;DR

This paper advances the understanding of p-adic iterated integrals on semistable curves by reformulating the theory through the unipotent log rigid fundamental group, linking it to Frobenius and monodromy operators, and clarifying the relationship between Berkovich--Coleman and Vologodsky integration.

Contribution

It introduces a new formulation of p-adic iterated integrals using the unipotent log rigid fundamental group and clarifies the connection between different integration theories on semistable curves.

Findings

Identification of Frobenius-invariant subgroup with the dual graph's fundamental group

Characterization of Berkovich--Coleman integration as Frobenius-invariant path integration

Equivalence of Vologodsky's path-independent integration with canonical path integration

Abstract

We reformulate the theory of p-adic iterated integrals on semistable curves using the unipotent log rigid fundamental group. This fundamental group carries Frobenius and monodromy operators whose basic properties are established. By identifying the Frobenius-invariant subgroup of the fundamental group with the fundamental group of the dual graph, we characterize Berkovich--Coleman integration, which is path-dependent, as integration along the Frobenius-invariant lift of a path in the dual graph. Vologodsky's path-independent integration theory which was previously described using a monodromy condition can now be identified as Berkovich--Coleman integration along a combinatorial canonical path arising from the theory of combinatorial iterated integration as developed by the first-named author and Cheng.