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Galois Actions

  • Gareth A. Jones
  • Jürgen Wolfart
Chapter
  • 1.2k Downloads
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

This chapter first collects basic material about Galois theory for finite and infinite field extensions, with examples chosen from number fields and function fields. The latter examples provide a link between Galois groups and covering groups for regular coverings. Another important example is the absolute Galois group \(\mathbb{G}\), the automorphism group of the field of all algebraic numbers: as the projective limit of the (finite) Galois groups of the Galois extensions of the rationals, this is a profinite group, with a natural topology, the Krull topology, making it a topological group. Belyĭ’s Theorem implies that \(\mathbb{G}\) has a natural action on dessins, through its action on the algebraic numbers defining them. As observed by Grothendieck, this action is faithful, so it gives a useful insight into the Galois theory of algebraic number fields. In the second section, moduli fields of algebraic curves are defined, and we discuss their relation to fields of definition. Weil’s cocycle condition is explained. We sketch two proofs of the other direction of Belyĭ’s theorem, that a curve can be defined over an algebraic number field if it admits a Belyĭ function. We list some Galois invariants and non-invariants of dessins, which are useful in determining orbits of \(\mathbb{G}\), and we give a proof due to Lenstra and Schneps that \(\mathbb{G}\) acts faithfully on the set of dessins formed from trees in the plane.

Keywords

Absolute Galois group Algebraic number Field extension Field of definition Galois group Galois invariant Grothendieck-Teichmüller tower Krull topology Moduli field Number field Profinite completion Projective limit 

References

  1. 1.
    Bauer, I., Catanese, F., Grunewald, F.: Faithful actions of the absolute Galois group on connected components of moduli spaces. Invent. Math. 199, 859–888 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dèbes, P., Emsalem, M.: On fields of moduli of curves. J. Algebra 211, 42–56 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Drinfel’d, V.G.: On quasitriangular quasi-Hopf algebras and a group closely connected with \(\mathrm{Gal}\,(\overline{<Emphasis Type="Bold">\text{Q}</Emphasis>}/<Emphasis Type="Bold">\text{Q}</Emphasis>)\). Leningrad Math. J. 2, 829–860 (1991)MathSciNetGoogle Scholar
  4. 4.
    Earle, C.J.: On the moduli of closed Riemann surfaces with symmetries. In: Ahlfors, L.V., et al. (eds.) Advances in the Theory of Riemann Surfaces. Annals of Mathematics Studies, vol. 6, pp. 119–130. Princeton University Press, Princeton (1971)Google Scholar
  5. 5.
    Ensalem, M., Lochak, P.: Appendix: the action of the absolute Galois group on the moduli spaces of spheres with four marked points. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, pp. 307–321. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  6. 6.
    Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, Berlin/Heidelberg/New York (1991)zbMATHGoogle Scholar
  7. 7.
    Girondo, E., González-Diez, G.: A note on the action of the absolute Galois group on dessins. Bull. Lond. Math. Soc. 39, 721–723 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    González-Diez, G.: Variations on Belyi’s theorem. Q. J. Math. 57, 339–354 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    González-Diez, G., Jaikin-Zapirain, A.: The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces. Proc. Lond. Math. Soc. (3) 111(4), 775–796 (2015)Google Scholar
  10. 10.
    Grothendieck, A.: Esquisse d’un Programme. In: Schneps, L., Lochak, P. (eds.) Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme, London Mathematical Society Lecture Note Series, vol. 242, pp. 5–48. Cambridge University Press, Cambridge (1997)Google Scholar
  11. 11.
    Guillot, P.: An elementary approach to dessins d’enfants and the Grothendieck-Teichmüller group. Enseign. Math. 60(3–4), 293–375 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guillot, P.: Some computations with the Grothendieck-Teichmüller group and equivariant dessins d’enfants (2014). arXiv:1407:3112 [math.GR]Google Scholar
  13. 13.
    Hammer, H., Herrlich, F.: A remark on the moduli field of a curve. Arch. Math. (Basel) 81, 5–10 (2003)Google Scholar
  14. 14.
    Herradón Cueto, M.: The field of moduli and fields of definition of dessins d’enfants (2014). arXiv:1409.7736 [math.AG]. Accessed 20 Jan 2015Google Scholar
  15. 15.
    Ihara, Y.: On the embedding of \(\mathrm{Gal}\,(\overline{<Emphasis Type="Bold">\text{Q}</Emphasis>}/<Emphasis Type="Bold">\text{Q}</Emphasis>)\) into \(\widehat{GT}\). In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, pp. 289–305. Cambridge University Press, Cambridge (1994)Google Scholar
  16. 16.
    Jarden, M.: Normal automorphisms of free profinite groups. J. Algebra 62, 118–123 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jones, G.A., Streit, M.: Galois groups, monodromy groups and cartographic groups. In: Schneps, L., Lochak, P. (eds.) Geometric Galois Actions 2. The Inverse Galois Problem, Moduli Spaces and Mapping Class Groups. London Mathematical Society Lecture Note Series, vol. 243, pp. 25–65. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  18. 18.
    Jones, G.A., Streit, M., Wolfart, J.: Wilson’s map operations on regular dessins and cyclotomic fields of definition. Proc. Lond. Math. Soc. 100, 510–532 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Koeck, B.: Belyi’s theorem revisited. Beiträge Algebra Geom. 45, 253–275 (2004)zbMATHGoogle Scholar
  20. 20.
    Lochak, P., Schneps, L. The Grothendieck-Teichmüller group and automorphisms of braid groups. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, pp. 323–358. Cambridge University Press, Cambridge (1994)Google Scholar
  21. 21.
    Malle, G., Matzat, B.H.: Inverse Galois Theory. Springer, Berlin/Heidelberg/New York (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Oesterlé, J.: Dessins d’enfants. Astérisque (Sém. Bourbaki 2001/02, Exp. 907) 290, 285–305 (2003)Google Scholar
  23. 23.
    Schneps, L.: Dessins d’enfants on the Riemann sphere. In: Schneps, L. (ed.) The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200, pp. 47–77. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
  24. 24.
    Schneps, L. (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200. Cambridge University Press, Cambridge (1994)Google Scholar
  25. 25.
    Schneps, L.: The Grothendieck-Teichmüller group \(\widehat{GT}\): a survey. In: Schneps, L., Lochak, P. (eds.) Geometric Galois Actions 1. London Mathematical Society Lecture Note Series, vol. 242, pp. 183–203. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  26. 26.
    Schneps, L., Lochak, P. (eds.): Geometric Galois Actions 1, 2. London Mathematical Society Lecture Note Series, vols. 242, 243. Cambridge University Press, Cambridge (1997)Google Scholar
  27. 27.
    Shimura, G.: On the field of rationality of an abelian variety. Nagoya Math. J. 45, 167–178 (1972)MathSciNetzbMATHGoogle Scholar
  28. 28.
    The GAP Group: GAP – Groups, Algorithms, and Programming. Version 4.7.6 (2014). http://www.gap-system.org. Accessed 20 Jan 2015
  29. 29.
    Weil, A.: The field of definition of a variety. Am. J. Math. 78, 509–524 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wolfart, J.: The ‘Obvious’ part of Belyi’s Theorem and Riemann surfaces with many automorphisms. In: Schneps, L., Lochak, P. (eds.) Geometric Galois Actions 1. Around Grothendieck’s Esquisse d’un Programme. London Mathematical Society Lecture Note Series, vol. 242, pp. 97–112. Cambridge University Press, Cambridge (1997)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gareth A. Jones
    • 1
  • Jürgen Wolfart
    • 2
  1. 1.School of MathematicsUniversity of SouthamptonSouthamptonUK
  2. 2.Johann Wolfgang Goethe-UniversitätFrankfurt am MainGermany

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