Picard–Vessiot Extensions of Real Differential Fields

Teresa Crespo ; Zbigniew Hajto

Symmetry, integrability and geometry: methods and applications, Tome 15 (2019)

Résumé

For a linear differential equation defined over a formally real differential field $K$ with real closed field of constants $k$, Crespo, Hajto and van der Put proved that there exists a unique formally real Picard–Vessiot extension up to $K$-differential automorphism. However such an equation may have Picard–Vessiot extensions which are not formally real fields. The differential Galois group of a Picard–Vessiot extension for this equation has the structure of a linear algebraic group defined over $k$ and is a $k$-form of the differential Galois group $H$ of the equation over the differential field $K\big(\sqrt{-1}\big)$. These facts lead us to consider two issues: determining the number of $K$-differential isomorphism classes of Picard–Vessiot extensions and describing the variation of the differential Galois group in the set of $k$-forms of $H$. We address these two issues in the cases when $H$ is a special linear, a special orthogonal, or a symplectic linear algebraic group and conclude that there is no general behaviour.

Détail

Keywords: real Picard–Vessiot theory, linear algebraic groups, group cohomology, real forms of algebraic groups.

Bibliographie

[1] Amano K., Masuoka A., “Picard–Vessiot extensions of Artinian simple module algebras”, J. Algebra, 285 (2005), 743–767 | DOI | MR | Zbl

[2] André Y., “Différentielles non commutatives et théorie de {G}alois différentielle ou aux différences”, Ann. Sci. École Norm. Sup. (4), 34 (2001), 685–739, arXiv: math.GM/0203274 | DOI | MR | Zbl

[3] Audin M., “Exemples de hamiltoniens non intégrables en mécanique analytique réelle”, Ann. Fac. Sci. Toulouse Math. (6), 12 (2003), 1–23 | DOI | MR | Zbl

[4] Bachmayr A., Harbater D., Hartmann J., “Differential Galois groups over Laurent series fields”, Proc. Lond. Math. Soc., 112 (2016), 455–476 | DOI | MR | Zbl

[5] Bayer-Fluckiger E., Parimala R., “Classical groups and the {H}asse principle”, Ann. of Math., 147 (1998), 651––693 ; Correction, Ann. of Math., 163 (2006), 381 | DOI | MR | Zbl | DOI | MR

[6] Bochnak J., Coste M., Roy M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 36, Springer-Verlag, Berlin, 1998 | DOI | MR | Zbl

[7] Borel A., “Lie groups and linear algebraic groups. I Complex and real groups”, Lie Groups and Automorphic Forms, AMS/IP Stud. Adv. Math., 37, Amer. Math. Soc., Providence, RI, 2006, 1–49 | DOI | MR | Zbl

[8] Crespo T., Hajto Z., Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, 122, Amer. Math. Soc., Providence, RI, 2011 | DOI | MR | Zbl

[9] Crespo T., Hajto Z., “Real Liouville extensions”, Comm. Algebra, 43 (2015), 2089–2093, arXiv: 1206.2283 | DOI | MR | Zbl

[10] Crespo T., Hajto Z., Sowa E., “Picard–Vessiot theory for real fields”, Israel J. Math., 198 (2013), 75–89 | DOI | MR | Zbl

[11] Crespo T., Hajto Z., van der Put M., “Real and $p$-adic Picard–Vessiot fields”, Math. Ann., 365 (2016), 93–103, arXiv: 1307.2388 | DOI | MR | Zbl

[12] Deligne P., Milne J. S., “Tannakian categories”, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Math., 900, eds. P. Deligne, J.S. Milne, A. Ogus, K.-Y. Shih, Springer-Verlag, Berlin – New York, 1982, 101––228 | DOI | MR

[13] Dyckerhoff T., The inverse problem of differential Galois theory over the field ${\mathbb R}(z)$, arXiv: 0802.2897

[14] Gel'fond O. A., Hovanskii A. G., “Real Liouville functions”, Funct. Anal. Appl., 14 (1980), 122–123 | DOI | MR | Zbl

[15] Kamensky M., Pillay A., “Interpretations and differential Galois extensions”, Int. Math. Res. Not., 2016 (2016), 7390–7413 | DOI | MR | Zbl

[16] Kneser M., Lectures on Galois cohomology of classical groups, Tata Institute of Fundamental Research Lectures on Mathematics, 47, Tata Institute of Fundamental Research, Bombay, 1969 | MR | Zbl

[17] Knus M.-A., Merkurjev A., Rost M., Tignol J.-P., The book of involutions, American Mathematical Society Colloquium Publications, 44, Amer. Math. Soc., Providence, RI, 1998 | DOI | MR | Zbl

[18] Kolchin E. R., “Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations”, Ann. of Math., 49 (1948), 1–42 | DOI | MR | Zbl

[19] León Sánchez O., Pillay A., “Some definable Galois theory and examples”, Bull. Symb. Log., 23 (2017), 145–159, arXiv: 1511.05541 | DOI | MR

[20] Malle G., Testerman D., Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, 133, Cambridge University Press, Cambridge, 2011 | DOI | MR | Zbl

[21] Prestel A., Lectures on formally real fields, Lecture Notes in Math., 1093, Springer-Verlag, Berlin, 1984 | DOI | MR | Zbl

[22] Serre J.-P., Local fields, Graduate Texts in Mathematics, 67, Springer-Verlag, New York–Berlin, 1979 | DOI | MR | Zbl

[23] Serre J.-P., Cohomologie galoisienne, Lecture Notes in Math., 5, 5th ed., Springer-Verlag, Berlin, 1994 | DOI | MR | Zbl

[24] Springer T. A., Linear algebraic groups, Progress in Mathematics, 9, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 1998 | DOI | MR | Zbl

[25] Steinberg R., Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968 | MR | Zbl

[26] Tits J., “Classification of algebraic semisimple groups”, Algebraic Groups and Discontinuous Subgroups (Boulder, Colo., 1965), Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, R.I., 1966, 33–62 | DOI | MR

[27] van der Put M., Singer M. F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, 328, Springer-Verlag, Berlin, 2003 | DOI | MR | Zbl