Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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Ambrus Pál
Iterated line integrals over Laurent series fields of characteristic $p$
Publications mathématiques de Besançon (2017), p. 109-126, doi: 10.5802/pmb.17
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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 non-trivial, we show that it includes the $p$-adic formal logarithm as a special case.

Bibliography

[1] Amnon Besser, “Coleman integration using the Tannakian formalism”, Math. Ann. 322 (2002) no. 1, p. 19-48
[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. 89-133
[4] Christopher Lazda, “Relative fundamental groups and rational points”, Rend. Semin. Mat. Univ. Padova 134 (2015), p. 1-45

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