Iterated line integrals over Laurent series fields of characteristic p

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Inspired by Besser’s work on Coleman integration, we use ∇-modules to define iterated line integrals over Laurent series fields of characteristic p taking values in double cosets of unipotent n×n matrices with coefficients in the Robba ring divided out by unipotent n×n matrices with coefficients in the bounded Robba ring on the left and by unipotent n×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 ∇-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.