zbMATH — the first resource for mathematics

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.

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
Full Text: DOI arXiv
[1] Buzzard, K.: Forms of reductive algebraic groups, 30 Sept 2013. http://www2.imperial.ac.uk/ buzzard/maths/research/notes/
[2] Cassidy, Ph.J., Singer, M.F.: Galois theory of parametrized differential equations. In: Bertrand, D., et al. (eds.) Differential Equations and Quantum Groups. IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, pp. 113-156. EMS, Zürich (2007)
[3] Deligne, P.: Catégories Tannakiennes. In: The Grothendieck Festschrift, vol. 2. Progress in Mathematics, vol. 87, pp. 111-195. Birkhäuser, Boston (1990) · Zbl 0867.12004
[4] Deligne, P., Milne, J.S.: Tannakian Categories. In: Deligne, P., et al. (eds.) Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900, pp. 101-228. Springer, Berlin (1982)
[5] Gille, Ph., Moret-Bailly, L.: Actions algébriques de groupes arithmétiques. In: Torsors, Étale Homotopy and Applications to Rational Points. London Mathematical Society Lecture Note Series, vol. 405, pp. 231-249 (2013) · Zbl 1317.14101
[6] Lam, TY, An introduction to real algebra, Rocky Mt. J. Math., 14, 767-814, (1984) · Zbl 0577.14016
[7] Mitschi, C; Singer, MF, Connected linear algebraic groups as differential Galois groups, J. Algebra, 184, 333-361, (1996) · Zbl 0867.12004
[8] Prestel, A., Roquette, P.: Formally p-Adic Fields. Lecture Notes in Mathematics, vol. 1050. Springer, Berlin (1984) · Zbl 0523.12016
[9] van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations. Grundlehren, vol. 328. Springer, Berlin (2003) · Zbl 1036.12008
[10] Serre, J.-P.: Cohomologie Galoisienne, Cinquième édition, révisée et complétée. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1994)
[11] Seidenberg, A, Contributions to the Picard-Vessiot theory of homogeneous linear differential equations, Am. J. Math., 78, 808-817, (1956) · Zbl 0072.26502
[12] Springer, T.A.: Linear Algebraic Groups, 2nd edn. Progress in Mathematics, vol. 9. Birkhäuser, Boston (1998) · Zbl 0927.20024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.