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The Arithmetic of Fundamental Groups pp 3–52Cite as

Heidelberg Lectures on Coleman Integration

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Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 2))

Abstract

Coleman integration is a way of associating with a closed one-form on a p-adic space a certain locally analytic function, defined up to a constant, whose differential gives back the form. This theory, initially developed by Robert Coleman in the 1980s and later extended by various people including the author, has now found various applications in arithmetic geometry, most notably in the spectacular work of Kim on rational points. In this text we discuss two approaches to Coleman integration, the first is a semi-linear version of Coleman’s original approach, which is better suited for computations. The second is the author’s approach via unipotent isocrystals, with a simplified and essentially self-contained presentation. We also survey many applications of Coleman integration and describe a new theory of integration in families.

Part of the research described in these lectures was conducted with the support of the Israel Science Foundation, grant number: 1129/08, whose support I would like to acknowledge.

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  1. Department of Mathematics, Ben-Gurion, University of the Negev, Be’er-Sheva, Israel

    Amnon Besser

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  1. Amnon Besser
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Correspondence to Amnon Besser .

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  1. MATCH - Mathematics Center Heidelberg, Dept. of Mathematics, Heidelberg University, Im Neuenheimer Feld 288, Heidelberg, 69120, Germany

    Jakob Stix

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Besser, A. (2012). Heidelberg Lectures on Coleman Integration. In: Stix, J. (eds) The Arithmetic of Fundamental Groups. Contributions in Mathematical and Computational Sciences, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23905-2_1

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