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2016 | OriginalPaper | Buchkapitel

4. Galois Actions

verfasst von : Gareth A. Jones, Jürgen Wolfart

Erschienen in: Dessins d'Enfants on Riemann Surfaces

Verlag: Springer International Publishing

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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.

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Literatur
1.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
3.
Zurück zum Zitat 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)MathSciNet 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)MathSciNet
4.
Zurück zum Zitat 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) 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)
5.
Zurück zum Zitat 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)CrossRef 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)CrossRef
6.
Zurück zum Zitat Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, Berlin/Heidelberg/New York (1991)MATH Farkas, H.M., Kra, I.: Riemann Surfaces. Springer, Berlin/Heidelberg/New York (1991)MATH
7.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
9.
Zurück zum Zitat 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) 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)
10.
Zurück zum Zitat 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) 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)
11.
Zurück zum Zitat Guillot, P.: An elementary approach to dessins d’enfants and the Grothendieck-Teichmüller group. Enseign. Math. 60(3–4), 293–375 (2014)MathSciNetCrossRefMATH Guillot, P.: An elementary approach to dessins d’enfants and the Grothendieck-Teichmüller group. Enseign. Math. 60(3–4), 293–375 (2014)MathSciNetCrossRefMATH
12.
Zurück zum Zitat Guillot, P.: Some computations with the Grothendieck-Teichmüller group and equivariant dessins d’enfants (2014). arXiv:1407:3112 [math.GR] Guillot, P.: Some computations with the Grothendieck-Teichmüller group and equivariant dessins d’enfants (2014). arXiv:1407:3112 [math.GR]
13.
Zurück zum Zitat Hammer, H., Herrlich, F.: A remark on the moduli field of a curve. Arch. Math. (Basel) 81, 5–10 (2003) Hammer, H., Herrlich, F.: A remark on the moduli field of a curve. Arch. Math. (Basel) 81, 5–10 (2003)
14.
Zurück zum Zitat Herradón Cueto, M.: The field of moduli and fields of definition of dessins d’enfants (2014). arXiv:1409.7736 [math.AG]. Accessed 20 Jan 2015 Herradón Cueto, M.: The field of moduli and fields of definition of dessins d’enfants (2014). arXiv:1409.7736 [math.AG]. Accessed 20 Jan 2015
15.
Zurück zum Zitat 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) 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)
17.
Zurück zum Zitat 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)CrossRef 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)CrossRef
18.
Zurück zum Zitat 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)MathSciNetCrossRefMATH 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)MathSciNetCrossRefMATH
19.
Zurück zum Zitat Koeck, B.: Belyi’s theorem revisited. Beiträge Algebra Geom. 45, 253–275 (2004)MATH Koeck, B.: Belyi’s theorem revisited. Beiträge Algebra Geom. 45, 253–275 (2004)MATH
20.
Zurück zum Zitat 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) 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)
21.
Zurück zum Zitat Malle, G., Matzat, B.H.: Inverse Galois Theory. Springer, Berlin/Heidelberg/New York (1999)CrossRefMATH Malle, G., Matzat, B.H.: Inverse Galois Theory. Springer, Berlin/Heidelberg/New York (1999)CrossRefMATH
22.
Zurück zum Zitat Oesterlé, J.: Dessins d’enfants. Astérisque (Sém. Bourbaki 2001/02, Exp. 907) 290, 285–305 (2003) Oesterlé, J.: Dessins d’enfants. Astérisque (Sém. Bourbaki 2001/02, Exp. 907) 290, 285–305 (2003)
23.
Zurück zum Zitat 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)CrossRef 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)CrossRef
24.
Zurück zum Zitat Schneps, L. (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200. Cambridge University Press, Cambridge (1994) Schneps, L. (ed.): The Grothendieck Theory of Dessins d’Enfants. London Mathematical Society Lecture Note Series, vol. 200. Cambridge University Press, Cambridge (1994)
25.
Zurück zum Zitat 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)CrossRef 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)CrossRef
26.
Zurück zum Zitat Schneps, L., Lochak, P. (eds.): Geometric Galois Actions 1, 2. London Mathematical Society Lecture Note Series, vols. 242, 243. Cambridge University Press, Cambridge (1997) Schneps, L., Lochak, P. (eds.): Geometric Galois Actions 1, 2. London Mathematical Society Lecture Note Series, vols. 242, 243. Cambridge University Press, Cambridge (1997)
27.
Zurück zum Zitat Shimura, G.: On the field of rationality of an abelian variety. Nagoya Math. J. 45, 167–178 (1972)MathSciNetMATH Shimura, G.: On the field of rationality of an abelian variety. Nagoya Math. J. 45, 167–178 (1972)MathSciNetMATH
30.
Zurück zum Zitat 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) 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)
Metadaten
Titel
Galois Actions
verfasst von
Gareth A. Jones
Jürgen Wolfart
Copyright-Jahr
2016
DOI
https://doi.org/10.1007/978-3-319-24711-3_4