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Frobenius modules and Galois representations

B. Heinrich Matzat[1]

  • [1] University of Heidelberg Interdisciplinary Center for Scientific Computing Im Neuheimer Feld 368 69120 Heidelberg (Germany)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 7, page 2805-2818
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Matzat, B. Heinrich. "Frobenius modules and Galois representations." Annales de l’institut Fourier 59.7 (2009): 2805-2818. <http://eudml.org/doc/10472>.

@article{Matzat2009,
abstract = {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.},
affiliation = {University of Heidelberg Interdisciplinary Center for Scientific Computing Im Neuheimer Feld 368 69120 Heidelberg (Germany)},
author = {Matzat, B. Heinrich},
journal = {Annales de l’institut Fourier},
keywords = {Frobenius modules; iterative differential modules; Galois representations; $p$-adic differential equations; inverse differential Galois theory; -adic differential equations},
language = {eng},
number = {7},
pages = {2805-2818},
publisher = {Association des Annales de l’institut Fourier},
title = {Frobenius modules and Galois representations},
url = {http://eudml.org/doc/10472},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Matzat, B. Heinrich
TI - Frobenius modules and Galois representations
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 7
SP - 2805
EP - 2818
AB - 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.
LA - eng
KW - Frobenius modules; iterative differential modules; Galois representations; $p$-adic differential equations; inverse differential Galois theory; -adic differential equations
UR - http://eudml.org/doc/10472
ER -

References

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  1. David Eisenbud, Commutative algebra, 150 (1995), Springer-Verlag, New York Zbl0819.13001MR1322960
  2. Antonio J. Engler, Alexander Prestel, Valued fields, (2005), Springer-Verlag, Berlin Zbl1128.12009MR2183496
  3. Gunter Malle, B. H. Matzat, Inverse Galois theory, (1999), Springer-Verlag, Berlin Zbl0940.12001MR1711577
  4. B. H. Matzat, Differential Galois Theory in Positive Characteristic, (2001) 
  5. B. H. Matzat, Frobenius modules and Galois groups, Galois theory and modular forms 11 (2004), 233-267, Kluwer Acad. Publ., Boston, MA Zbl1111.12002MR2059766
  6. B. H. Matzat, Integral p -adic differential modules, Groupes de Galois arithmétiques et différentiels 13 (2006), 263-292, Soc. Math. France, Paris Zbl1158.13009MR2316354
  7. B. H. Matzat, From Frobenius structures to differential equations, DART II Proceedings (2009), World Scientific Publisher Zbl1185.12004
  8. B. H. Matzat, Marius van der Put, Iterative differential equations and the Abhyankar conjecture, J. Reine Angew. Math. 557 (2003), 1-52 Zbl1040.12010MR1978401
  9. A. Maurischat, Galois theory for iterative connections and nonreduced Galois groups, Trans. AMS Zbl1250.13009
  10. Matthew A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), 123-174 Zbl1235.11074MR2358057
  11. N. R. Stalder, Algebraic Monodromy Groups of A-Motives, (2007) 
  12. Marius van der Put, Bounded p -adic differential equations, Circumspice, Various Papers in and around Mathematics in Honor of Arnoud van Rooij (2001), Kath. Univ. Nijmegen Zbl0239.46046MR1908143
  13. Marius van der Put, Michael F. Singer, Galois theory of difference equations, 1666 (1997), Springer-Verlag, Berlin Zbl0930.12006MR1480919

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