Mediterranean Journal of Mathematics

, Volume 3, Issue 2, pp 121–146

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Original Paper

DOI: 10.1007/s00009-006-0069-7

Cite this article as:
Bauer, I., Catanese, F. & Grunewald, F. MedJM (2006) 3: 121. doi:10.1007/s00009-006-0069-7

Abstract.

We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck’s program of ‘Dessins d’ enfants’, aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

Mathematics Subject Classification (2000).

11S05 12D99 11R32 14J10 14J29 14M99 20D99 26C99 30F99 

Keywords.

Polynomials Riemann existence theorem monodromy Galois group dessins d’enfants Belyi and Chebycheff difference polynomials algebraic surfaces moduli spaces Beauville surfaces simple groups 
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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  • Ingrid Bauer
    • 1
  • Fabrizio Catanese
    • 2
  • Fritz Grunewald
    • 3
  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Lehrstuhl Mathematik VIIIUniversität BayreuthBayreuthGermany
  3. 3.Mathematisches Institut derHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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