9. Differential Embedding Problems over Complex Function Fields

  • Annette Bachmayr Mathematisches Institut der Universit\"at Bonn, 53115 Bonn, Germany
  • David Harbater Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
  • Julia Hartmann Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
  • Michael Wibmer Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395, USA
Keywords: Differential algebra, linear algebraic groups and torsors, patching, Picard-Vessiot theory, embedding problems, inverse differential Galois problem, Riemann surfaces


We introduce the notion of differential torsors, which allows the adaptation of constructions from algebraic geometry to differential Galois theory. Using these differential torsors, we set up a general framework for applying patching techniques in differential Galois theory over fields of characteristic zero. We show that patching holds over function fields over the complex numbers. As the main application, we prove the solvability of all differential embedding problems over complex function fields, thereby providing new insight on the structure of the absolute differential Galois group, i.e., the fundamental group of the underlying Tannakian category.


Katsutoshi Amano, Akira Masuoka, Mitsuhiro Takeuchi.
\newblock Hopf algebraic approach to Picard-Vessiot theory.
\newblock In: {\em Handbook of algebra}. Vol.~6, 127--171,
Elsevier/North-Holland, Amsterdam, 2009.
DOI 10.1016/S1570-7954(08)00204-0;
zbl 1218.12003;

Annette Bachmayr, David Harbater, and Julia Hartmann.
\newblock Differential {G}alois groups over {L}aurent series fields.
\newblock Proceedings of the London Mathematical Society,
\textbf{112}(3) (2016), 455--476.
DOI 10.1112/plms/pdv070;
zbl 1348.12003;
arxiv 1501.06884

Annette Bachmayr, David Harbater, Julia Hartmann, and Florian Pop.
\newblock Differential {G}alois theory over large fields.
\newblock Preprint.

Annette Bachmayr, David Harbater, and Julia Hartmann.
\newblock Differential embedding problems over Laurent series fields.
\newblock Preprint.

Annette Bachmayr and Michael Wibmer.
\newblock $\sigma$-parameterized differential Galois groups over $C(x)$.
\newblock Preprint.

Michel Demazure and Pierre Gabriel.
\newblock {\em Groupes Alg\'ebriques. {T}ome {I}: {G}\'eom\'etrie alg\'ebrique,
g\'en\'eralit\'es, groupes commutatifs}.
\newblock Masson \& Cie, \'Editeur, Paris; North-Holland Publishing Co.,
Amsterdam, 1970.

Tobias Dyckerhoff.
\newblock The inverse problem of differential Galois theory over the field
\newblock 2008 manuscript, available at arXiv:0802.2897.

Stefan Ernst.
\newblock Iterative differential embedding problems in positive characteristic.
\newblock J.\ Algebra \textbf{402} (2014), 544--564.
DOI 10.1016/j.jalgebra.2013.12.023;
zbl 1308.12006;
arxiv 1107.1962

Otto Forster.
\newblock {\em Lectures on Riemann Surfaces}. Translated from the German by Bruce Gilligan.
\newblock Graduate Texts in Mathematics, vol.~81.
\newblock Springer-Verlag, New York-Berlin, 1981.

Alexander Grothendieck.
\newblock {\em Rev\^etements \'etales et groupe fondamental.}
\newblock S{\'e}minaire de g{\'e}om{\'e}trie alg{\'e}brique du Bois Marie
1960--61 ({SGA} 1).
\newblock Lecture Notes in Math., vol.~224, Springer, Berlin, 1971.

David Harbater.
\newblock Abhyankar's conjecture on Galois groups over curves.
\newblock Invent.\ Math.\ \textbf{117} (1994), 1--25.
DOI 10.1007/BF01232232;
zbl 0805.14014;

David Harbater.
\newblock Fundamental groups and embedding problems in characteristic $p$.
\newblock In: Recent developments in the inverse Galois problem (Seattle, WA, 1993), 353--369, Contemp.\ Math., vol.~186, Amer.\ Math.\ Soc., Providence, RI, 1995.

David Harbater.
\newblock Patching and Galois theory. In: {\em Galois groups and fundamental groups}, 313--424,
Math.\ Sci.\ Res.\ Inst.\ Publ., vol.~41, Cambridge Univ.\ Press, Cambridge, 2003.
zbl 1071.14029;

David Harbater and Julia Hartmann.
\newblock Patching over fields.
\newblock Israel J.\ Math.\ \textbf{176} (2010), 61--107.
DOI 10.1007/s11856-010-0021-1;
zbl 1213.14052;
arxiv 0710.1392

David Harbater, Julia Hartmann, and Daniel Krashen.
\newblock Local-global principles for torsors over arithmetic curves.
\newblock Amer.\ J.\ Math., \textbf{137}(6) (2015), 1559--1612.
DOI 10.1353/ajm.2015.0039;
zbl 1348.11036;
arxiv 1108.3323

Julia Hartmann.
\newblock On the inverse problem in differential Galois theory.
\newblock J.\ reine angew.\ Math.\ \textbf{586} (2005), 21--44
DOI 10.1515/crll.2005.2005.586.21;
zbl 1090.12003;

\bibitem[Hum75]{Humphreys:Linear algebraic groups}
James~E.\ Humphreys.
\newblock {\em Linear Algebraic Groups}.
\newblock Graduate Texts in Mathematics, No. 21,
Springer-Verlag, New York-Heidelberg, 1975.

Kenkichi Iwasawa.
\newblock On solvable extensions of algebraic number fields.
\newblock Ann.\ of Math.\ (2) \textbf{58} (1953), 548--572.
DOI 10.2307/1969754;
zbl 0051.26602;

Lourdes Juan.
\newblock Generic Picard-Vessiot extensions for connected-by-finite groups.
\newblock J. Algebra~\textbf{312} (2007), no.1, 194--206.
DOI 10.1016/j.jalgebra.2007.02.043;
zbl 1183.20054;

Lourdes Juan and Arne Ledet.
\newblock Equivariant vector fields on non-trivial $\operatorname{SO}_n$-torsors and differential Galois theory.
\newblock J. Algebra~\textbf{316} (2007), no.2, 735--745.
DOI 10.1016/j.jalgebra.2007.01.005;
zbl 1173.12001;

Lourdes Juan and Arne Ledet.
\newblock On generic differential $\operatorname{SO}_n$-extensions.
\newblock Proc. Amer. Math. Soc.~\textbf{136} (2008), 1145--1153,
DOI 10.1090/S0002-9939-07-09314-8;
zbl 1170.12001;

Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol.
\newblock {\em The Book of Involutions}, volume~44 of American
Mathematical Society Colloquium Publications.
\newblock American Mathematical Society, Providence, RI, 1998.

Ellis R.~Kolchin.
\newblock {\em Differential Algebra and Algebraic Groups},
Pure and Applied Mathematics, Vol.~54.
\newblock Academic Press, New York-London, 1973.

\newblock The inverse problem in the Galois theory of differential fields.
\newblock Ann.\ of Math., \textbf{89} (1969), 583--608.
DOI 10.2307/1970653;
zbl 0188.33801;

\newblock On the inverse problem in the Galois theory of differential fields.
\newblock Ann.\ of Math.\ (2), \textbf{93} (1971), 269--284.

Jerald~J.\ Kovacic.
\newblock The differential Galois theory of strongly normal extensions.
\newblock Trans.\ Amer.\ Math.\ Soc., \textbf{355}(11) (2003) (electronic), 4475--4522.
DOI 10.1090/S0002-9947-03-03306-3;
zbl 1036.12005;

Hideyuki Matsumura.
\newblock {\em Commutative Ring Theory}, volume~8 of {\em Cambridge Studies in
Advanced Mathematics}.
\newblock Cambridge University Press, second edition, 1989.

B. Heinrich Matzat and Marius van der Put.
\newblock Constructive differential Galois theory.
\newblock In: {\em Galois Groups and fundamental groups}, Math. Sci. Res. Inst. Publ., \textbf{41}, 425--467.
zbl 1070.12002;

Andreas Maurischat.
\newblock Galois theory for iterative connections and nonreduced {G}alois
\newblock Trans.\ Amer.\ Math.\ Soc., \textbf{362} (10) (2010), 5411--5453.
DOI 10.1090/S0002-9947-2010-04966-9;
zbl 1250.13009;
arxiv 0712.3748

James~S.\ Milne.
\newblock {\em \'Etale Cohomology}, Princeton University Press, 1980.

Claude Mitschi and Michael Singer.
\newblock Connected linear groups as differential Galois groups.
\newblock J.\ Algebra \ \textbf{184} (1996), 333--361.
DOI 10.1006/jabr.1996.0263;
zbl 0867.12004;

Claude Mitschi and Michael Singer.
\newblock Solvable-by-finite groups as differential Galois groups.
\newblock Ann. Fac. Sci. Toulouse Math.~(6)\ \textbf{11/3} (2002), 403--423.

J\"urgen Neukirch, Alexander Schmidt, and Kay Wingberg.
\newblock {\em Cohomology of number fields}. Second edition.
\newblock Grundlehren der Mathematischen Wissenschaften, vol.~323. Springer-Verlag, Berlin, 2008.
zbl 1136.11001;

Thomas Oberlies.
\newblock Einbettungsprobleme in der {D}ifferentialgaloistheorie.
\newblock Dissertation, Universit\"at Heidelberg, 2003. Available at

Josip Plemelj.
\newblock Riemannsche Funktionenscharen mit gegebener Monodromiegruppe.
\newblock Monatsh.\ Math.\ Phys.\ \textbf{19} (1908), 211--245.

Florian Pop.
\newblock \'Etale Galois covers of affine smooth curves. The geometric case of a conjecture of Shafarevich. On Abhyankar's conjecture.
\newblock Invent.\ Math.\ \textbf{120} (1995), no.~3, 555--578.
DOI 10.1007/BF01241142;
zbl 0842.14017;

Michel Raynaud.
\newblock Faisceaux amples sur les sch\'emas en groupes et les espaces homog\'enes.
\newblock Lecture Notes in Math., vol.~119, Springer, Berlin, 1970.
DOI 10.1007/BFb0059504;
zbl 0195.22701;

Michel Raynaud.
\newblock Rev\^etements de la droite affine en caract\'eristique $p>0$ et conjecture d'Abhyankar.
\newblock Invent.\ Math.\ \textbf{116} (1994), 425--462.

Jean-Pierre Serre.
\newblock G\'eom\'etrie alg\'ebrique et g\'eom\'etrie analytique.
\newblock Ann.\ Inst.\ Fourier, Grenoble, \textbf{6} (1955--1956), 1--42.

Jean-Pierre Serre.
\newblock Construction de rev\^etements
\'etales de la droite affine en caract\'eristique $p$.
\newblock C.\ R.\ Acad.\ Sci.\ Paris S\'er. I Math.\ \textbf{311} (1990), no.~6, 341--346.

Jean-Pierre Serre.
\newblock {\em Galois Cohomology}.
\newblock Translated from the French by Patrick Ion and revised by the author.
\newblock Springer-Verlag, Berlin, 1997.

Igor R.\ Shafarevich.
\newblock Construction of fields of algebraic numbers with given solvable Galois group. (Russian) Izv.\ Akad.\ Nauk SSSR. Ser.\ Mat.\ \textbf{18} (1954), 525--578.
English translation in Amer.\ Math.\ Soc.\ Transl.\ \textbf{4} (1956), 185--237.

Michael Singer.
\newblock Moduli of linear differential equations on the Riemann sphere with fixed Galois groups.
\newblock Pac.\ J.\ Math.\ \textbf{106}(2) (1993), 343--395.
DOI 10.2140/pjm.1993.160.343;
zbl 0778.12007;

T.~A. Springer.
\newblock {\em Linear Algebraic Groups}.
\newblock Birkh\"auser Boston Inc., second edition, 2009.

Carol Tretkoff and Marvin Tretkoff.
\newblock Solution of the inverse problem of differential {G}alois theory in
the classical case.
\newblock Amer.\ J.\ Math., \textbf{101}(6) (1979), 1327--1332.
DOI 10.2307/2374143;
zbl 0423.12021;

Marius van~der Put and Michael~F. Singer.
\newblock {\em Galois Theory of Linear Differential Equations}.
\newblock Springer, Berlin, 2003.
zbl 1036.12008;
Unassigned Articles