Abelian-by-Central Galois Theory of Prime Divisors

  • Florian PopEmail author
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 2)


In this manuscript I show how to recover some of the inertia structure of (quasi) divisors of a function field K | k over an algebraically closed base field k from its maximal mod ℓabelian-by-central Galois theory of K, provided td(K | k) > 1. This is a first technical step in trying to extend Bogomolov’s birational anabelian program beyond the full pro- situation, which corresponds to the limit case mod .


Maximal Subgroup Galois Group Prime Divisor Canonical Projection Closed Subgroup 
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  1. AEJ87.
    J. K. Arason, R. Elman, and B. Jacob. Rigid elements, valuations, and realization of Witt rings. J. Algebra, 110:449–467, 1987.CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bog91.
    F. A. Bogomolov. On two conjectures in birational algebraic geometry. In A. Fujiki et al., editors, Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings. Springer Verlag, Tokyo, 1991.Google Scholar
  3. Bou64.
    N. Bourbaki. Algèbre commutative. Hermann Paris, 1964.Google Scholar
  4. BT02.
    F. A. Bogomolov and Yu. Tschinkel. Commuting elements in Galois groups of function fields. In F. A. Bogomolov and L. Katzarkov, editors, Motives, Polylogarithms and Hodge theory, pages 75–120. International Press, 2002.Google Scholar
  5. BT08.
    F. Bogomolov and Yu. Tschinkel. Reconstruction of function fields. Geometric And Functional Analysis, 18:400–462, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  6. EE77.
    O. Endler and A. J. Engler. Fields with Henselian valuation rings. Math. Z., 152:191–193, 1977.CrossRefzbMATHMathSciNetGoogle Scholar
  7. Gro83.
    A. Grothendieck. Brief an Faltings (27/06/1983). In L. Schneps and P. Lochak, editors, Geometric Galois Action 1, volume 242 of LMS Lecture Notes, pages 49–58. Cambridge, 1997.Google Scholar
  8. Gro84.
    A. Grothendieck. Esquisse d’un programme. In L. Schneps and P. Lochak, editors, Geometric Galois Action 1, volume 242 of LMS Lecture Notes, pages 5–48. Cambridge, 1997.Google Scholar
  9. Koe01.
    J. Koenigsmann. Solvable absolute Galois groups are metabelian. Inventiones Math., 144:1–22, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  10. MMS04.
    L. Mahé, J. Mináč, and T. L. Smith. Additive structure of multiplicative subgroups of fields and Galois theory. Doc. Math., 9:301–355, 2004.zbMATHMathSciNetGoogle Scholar
  11. Neu69.
    J. Neukirch. Kennzeichnung der p-adischen und endlichen algebraischen Zahlkörper. Inventiones Math., 6:269–314, 1969.CrossRefMathSciNetGoogle Scholar
  12. NSW08.
    J. Neukirch, A. Schmidt, and K. Wingberg. Cohomology of number fields, volume 323 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2nd edition, 2008.Google Scholar
  13. Pop94.
    F. Pop. On Grothendieck’s conjecture of birational anabelian geometry. Ann. of Math., 138:145–182, 1994.CrossRefGoogle Scholar
  14. Pop98.
    F. Pop. Glimpses of Grothendieck’s anabelian geometry. In L. Schneps and P. Lochak, editors, Geometric Galois Action 1, volume 242 of LMS Lecture Notes, pages 113–126. Cambridge, 1997.Google Scholar
  15. Pop06.
    F. Pop. Pro- Galois theory of Zariski prime divisors. In Débès and others, editor, Luminy Proceedings Conference, SMF No 13. Hérmann, Paris, 2006.Google Scholar
  16. Pop10.
    F. Pop. Pro- abelian-by-central Galois theory of Zariski prime divisors. Israel J. Math., 180:43–68, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Pop11.
    F. Pop. On the birational anabelian program initiated by Bogomolov I. Inventiones Math., pages 1–23, 2011.Google Scholar
  18. Sza04.
    T. Szamuely. Groupes de Galois de corps de type fini (d’après Pop). Astérisque, 294:403–431, 2004.MathSciNetGoogle Scholar
  19. Top11.
    A. Topaz.  ∕  commuting liftable pairs. Manuscript.Google Scholar
  20. Uch79.
    K. Uchida. Isomorphisms of Galois groups of solvably closed Galois extensions. Tôhoku Math. J., 31:359–362, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  21. War81.
    R. Ware. Valuation rings and rigid elements in fields. Can. J. Math., 33:1338–1355, 1981.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

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