On -adic Iterated Integrals V: Linear Independence, Properties of -adic Polylogarithms, -adic Sheaves

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


In a series of papers we have introduced and studied -adic polylogarithms and -adic iterated integrals which are analogues of the classical complex polylogarithms and iterated integrals in -adic Galois realizations. In this note we shall show that in the generic case -adic iterated integrals are linearly independent over \({\mathbb{Q}}_{\mathcal{l}}\). In particular they are non trivial. This result can be viewed as analogous of the statement that the classical iterated integrals from 0 to z of sequences of one forms \(\frac{dz} {z}\) and \(\frac{dz} {z-1}\) are linearly independent over \(\mathbb{Q}\). We also study ramification properties of -adic polylogarithms and the minimal quotient subgroup of the absolute Galois group G K of a number field K on which -adic polylogarithms are defined. In the final sections of the paper we study -adic sheaves and their relations with -adic polylogarithms. We show that if an -adic sheaf has the same monodromy representation as the classical complex polylogarithms then the action of G K in stalks is given by -adic polylogarithms.


Fundamental Group Algebraic Variety Galois Group Galois Representation Iterate Integral 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de Nice-Sophia Antipolis, Laboratoire Jean Alexandre DieudonnéNice Cedex 2France

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