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Real and p-adic Picard–Vessiot fields

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Abstract

We consider differential modules over real and p-adic differential fields K such that its field of constants k is real closed (resp., p-adically closed). Using P. Deligne’s work on Tannakian categories and a result of J.-P. Serre on Galois cohomology, a purely algebraic proof of the existence and unicity of real (resp., p-adic) Picard–Vessiot fields is obtained. The inverse problem for real forms of a semi-simple group is treated. Some examples illustrate the relations between differential modules, Picard–Vessiot fields and real forms of a linear algebraic group.

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Authors and Affiliations

  1. Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes, 08007, Barcelona, Spain

    Teresa Crespo

  2. Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Prof. S. Łojasiewicza 6, 30-348, Kraków, Poland

    Zbigniew Hajto

  3. Department of Mathematics, University of Groningen, P.O. Box 407, 9700AK, Groningen, The Netherlands

    Marius van der Put

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  1. Teresa Crespo
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  2. Zbigniew Hajto
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  3. Marius van der Put
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Correspondence to Teresa Crespo.

Additional information

M. van der Put thanks the Barcelona Knowledge Campus for the PIE-BKC grant financing his stay at the Institute of Mathematics of the University of Barcelona in July 2013. T. Crespo and Z. Hajto acknowledge support of grant MTM2012-33830, Spanish Science Ministry.

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Crespo, T., Hajto, Z. & van der Put, M. Real and p-adic Picard–Vessiot fields. Math. Ann. 365, 93–103 (2016). https://doi.org/10.1007/s00208-015-1272-2

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