Frobenius modules and Galois representations
Annales de l'Institut Fourier, Volume 59 (2009) no. 7, p. 2805-2818

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for p-adic differential equations with (strong) Frobenius structure over p-adic differential fields with algebraically closed residue field.

Les modules de Frobenius sont des modules aux différences par rapport à un opérateur de Frobenius. Nous montrons ici que, sur des corps différentiels complets et non archimédiens, les modules de Frobenius définissent des modules différentiels ayant le même anneau de Picard-Vessiot, et quitte à étentre le corps des constantes, le même schéma en groupes de Galois. De plus, ces modules de Frobenius sont classifiés par des représentations galoisiennes non ramifiées sur le corps de base. Cela donne, entre autres, la solution du problème de Galois différentiel inverse pour les équations différentielles p-adiques avec une structure de Frobenius (forte), définies sur les corps différentiels p- adiques ayant un corps résiduel algébriquement clos.

Classification:  12H25,  12F12,  12H05,  12H10
Keywords: Frobenius modules, iterative differential modules, Galois representations, p-adic differential equations, inverse differential Galois theory
     author = {Matzat, B. Heinrich},
     title = {Frobenius modules and Galois representations},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {7},
     year = {2009},
     pages = {2805-2818},
     doi = {10.5802/aif.2508},
     mrnumber = {2649339},
     zbl = {1185.12004},
     language = {en},
     url = {}
Matzat, B. Heinrich. Frobenius modules and Galois representations. Annales de l'Institut Fourier, Volume 59 (2009) no. 7, pp. 2805-2818. doi : 10.5802/aif.2508.

[1] Eisenbud, David Commutative algebra, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 150 (1995) (With a view toward algebraic geometry) | MR 1322960 | Zbl 0819.13001

[2] Engler, Antonio J.; Prestel, Alexander Valued fields, Springer-Verlag, Berlin, Springer Monographs in Mathematics (2005) | MR 2183496 | Zbl 1128.12009

[3] Malle, Gunter; Matzat, B. H. Inverse Galois theory, Springer-Verlag, Berlin, Springer Monographs in Mathematics (1999) | MR 1711577 | Zbl 0940.12001

[4] Matzat, B. H. Differential Galois Theory in Positive Characteristic, IWR (2001) no. 35 (Preprint)

[5] Matzat, B. H. Frobenius modules and Galois groups, Galois theory and modular forms, Kluwer Acad. Publ., Boston, MA (Dev. Math.) Tome 11 (2004), pp. 233-267 | MR 2059766 | Zbl 1111.12002

[6] Matzat, B. H. Integral p-adic differential modules, Groupes de Galois arithmétiques et différentiels, Soc. Math. France, Paris (Sémin. Congr.) Tome 13 (2006), pp. 263-292 | MR 2316354 | Zbl 1158.13009

[7] Matzat, B. H. From Frobenius structures to differential equations, DART II Proceedings, World Scientific Publisher (2009)

[8] Matzat, B. H.; Van Der Put, Marius Iterative differential equations and the Abhyankar conjecture, J. Reine Angew. Math., Tome 557 (2003), pp. 1-52 | Article | MR 1978401 | Zbl 1040.12010

[9] Maurischat, A. Galois theory for iterative connections and nonreduced Galois groups, Trans. AMS (to appear)

[10] Papanikolas, Matthew A. Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Tome 171 (2008) no. 1, pp. 123-174 | Article | MR 2358057 | Zbl 1235.11074 | Zbl pre05236753

[11] Stalder, N. R. Algebraic Monodromy Groups of A-Motives, ETH Zürich (2007) (Ph. D. Thesis)

[12] Van Der Put, Marius Bounded p-adic differential equations, Circumspice, Various Papers in and around Mathematics in Honor of Arnoud van Rooij, Kath. Univ. Nijmegen (2001) | MR 1908143

[13] Van Der Put, Marius; Singer, Michael F. Galois theory of difference equations, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1666 (1997) | MR 1480919 | Zbl 0930.12006