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Wilson’s map operations on regular dessins and cyclotomic fields of definition. (English) Zbl 1191.14034

Dessins d’enfants were introduced by Grothendieck in his “esquisse d’un programme” as a tool to understand the structure of the absolute Galois group \(\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\). The general idea is that a dessin \(D\) on a closed orientable surface \(S\) defines a unique (up to conformal equivalence) Riemann surface structure on it which can be defined by an irreducible non-singular projective algebraic variety defined over \(\overline{\mathbb Q}\) by Belyi’s theorem. The absolute Galois group acts at the level of those algebraic curves, so it also acts on the collection of dessins d’enfants. Unfortunately, such an action is still not completely understood. Some particular examples have been worked out by the authors and others, but only few general results are known. A dessin can be seen as a (bipartite) map on the surface and it has naturally associated a graph. Wilson’s operation on a map is given so that it keeps the graph invariant but changes the cyclic order of its edges at vertices. With such an operation the map may change but the underlying graph is still the same; in this way a new dessin is obtained.
In this paper the authors consider certain families of regular dessins, each family being invariant under Wilson’s operations and the absolute Galois group. The main result is the following.
A Galois invariant family \(\{ D_{j}\}\) of regular dessins forming an orbit under Wilson’s operations is defined over a cyclotomic field, and the Wilson operations are equivalent to the Galois conjugations. Conversely, let \(\{D_{j}\}\) be a family of regular dessins defined over a cyclotomic field and all of them Galois conjugate. If the Galois conjugation preserves adjacency between the vertices of the dessins, then the algebraic conjugations act as Wilson’s operations on the dessins.
The paper is nicely written with explicit examples so the non-experts will find no problem to follow the arguments and to grasp the beautiful theory behind.

MSC:

14H45 Special algebraic curves and curves of low genus
14H25 Arithmetic ground fields for curves
14H55 Riemann surfaces; Weierstrass points; gap sequences
05C10 Planar graphs; geometric and topological aspects of graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
30F10 Compact Riemann surfaces and uniformization
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