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**Coleman integration using the Tannakian formalism.**
*(English)*
Zbl 1013.11028

In the 1980’s Coleman – partly in collaboration with de Shalit – developed a theory of \(p\)-adic integration for closed \(1\)-forms on rigid analytic spaces. A different approach to \(p\)-adic integration is due to Colmez and – independently – Zarhin and does not use rigid geometry. An excellent discussion of these approaches and the relevant literature is given by C. Breuil in his Bourbaki talk [Sémin. Bourbaki, Vol. 1998/99, Exposé 860, Astérisque 266, 319-350 (2000; Zbl 1013.11028)].

In the paper under review the author considers Coleman iterated integrals as solutions of unipotent systems of differential equations and provides the general theoretical background for the unique analytic continuation of the local solutions. This was previously done by Coleman only for curves.

The idea is to use the fact that the collection of all paths between a point \(x\) and another point \(y\), along which one can do analytic continuation, is a principal homogeneous space for a certain Tannakian fundamental group. The author uses work of B. Chiarellotto [Ann. Sci. Éc. Norm. Supér. (4) 31, 683-718 (1998; Zbl 0933.14008)] to show that there is a unique Frobenius invariant path on the space of paths, and therefore achieves canonical analytic continuation “along Frobenius”.

In the paper under review the author considers Coleman iterated integrals as solutions of unipotent systems of differential equations and provides the general theoretical background for the unique analytic continuation of the local solutions. This was previously done by Coleman only for curves.

The idea is to use the fact that the collection of all paths between a point \(x\) and another point \(y\), along which one can do analytic continuation, is a principal homogeneous space for a certain Tannakian fundamental group. The author uses work of B. Chiarellotto [Ann. Sci. Éc. Norm. Supér. (4) 31, 683-718 (1998; Zbl 0933.14008)] to show that there is a unique Frobenius invariant path on the space of paths, and therefore achieves canonical analytic continuation “along Frobenius”.

Reviewer: Manfred Kolster (Hamilton/Ontario)