Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory

Details Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory Chebycheff and Belyi polynomials, dessins d'enfants, Beauville surfaces and group theory

2006-05-10

arxiv.org/abs/math/0605258v1

Mathematics

Algebraic Geometry

25 pages, to appear in the Special Volume of Mediterranean Journal of Mathematics, dedicated to the Almeria Mediterranean Cong

Scientific paper

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.

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