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Table of contents for this volume  Previous article  Next article Ambrus Pál Iterated line integrals over Laurent series fields of characteristic $p$ Publications mathématiques de Besançon (2017), p. 109126, doi: 10.5802/pmb.17 Article PDF Class. Math.: 14K15, 14F30, 14F35 Keywords: $p$adic integration, Laurent series fields 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 nontrivial, we show that it includes the $p$adic formal logarithm as a special case. Bibliography [2] Kiran S. Kedlaya, $p$adic differential equations, Cambridge Studies in Advanced Mathematics 125, Cambridge University Press, 2010 [3] Minhyong Kim, “The unipotent Albanese map and Selmer varieties for curves”, Publ. Res. Inst. Math. Sci. 45 (2009) no. 1, p. 89133 [4] Christopher Lazda, “Relative fundamental groups and rational points”, Rend. Semin. Mat. Univ. Padova 134 (2015), p. 145 