# Iterated line integrals over Laurent series fields of characteristic p

## TL;DR

This paper develops a new theory of iterated line integrals over Laurent series fields of characteristic p, extending classical concepts via $
abla$-modules and including the $p$-adic formal logarithm as a special case.

## Contribution

It introduces a novel $p$-adic iterated integral framework using $
abla$-modules, generalizing classical theory to characteristic p fields.

## Key findings

- Includes the $p$-adic formal logarithm as a special case
- Defines iterated line integrals over Laurent series fields of characteristic p
- Establishes a connection between classical and $p$-adic theories

## Abstract

Inspired by Besser's work on Coleman integration, we use $\nabla$-modules to define iterated line integrals over Laurent series fields of characteristic $p$ taking values in double cosets of unipotent $n\times n$ matrices with coefficients in the Robba ring divided out by unipotent $n\times n$ matrices with coefficients in the bounded Robba ring on the left and by unipotent $n\times n$ matrices with coefficients in the constant field on the right. We reach our definition by looking at the analogous theory for Laurent series fields of characteristic $0$ first, and reinterpreting the classical formal logarithm in terms of $\nabla$-modules on formal schemes. To illustrate that the new $p$-adic theory is non-trivial, we show that it includes the $p$-adic formal logarithm as a special case.