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Real and \(p\)-adic Picard-Vessiot fields. (English) Zbl 1344.34096
The authors consider differential modules \(M\) over real and \(p\)-adic differential fields \(K\) (with fields of constants \(k\) real closed or \(p\)-adic closed). A Picard-Vessiot field (PVF) \(L\) for \(M/K\) is a field extension of \(K\) with the same field of constants \(k\), where \(L\) is equipped with a differentiation extending the one of \(K\) and there exists an invertible \(d\times d\)-matrix \(F\) (\(d=\)dim\(M\)) with entries in \(L\) satisfying \(F'=AF\), as a field \(L\) being generated over \(K\) by the entries of \(F\). Using results of J.-P. Serre and P. Deligne, the authors obtain a purely algebraic proof of the existence and unicity of PVFs. They treat the inverse problem for real forms of a semisimple group and they give examples illustrating the relations between differential modules, PVFs and real forms of a linear algebraic group.

MSC:
12H20 Abstract differential equations
12H25 \(p\)-adic differential equations
34M50 Inverse problems (Riemann-Hilbert, inverse differential Galois, etc.) for ordinary differential equations in the complex domain
11E10 Forms over real fields
11R34 Galois cohomology
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