Dessin D'enfant - The Absolute Galois Group and Its Invariants

The Absolute Galois Group and Its Invariants

The polynomial

may be made into a Shabat polynomial by choosing

The two choices of a lead to two Belyi functions ƒ₁ and ƒ₂. These functions, though closely related to each other, are not equivalent, as they are described by the two nonisomorphic trees shown in the figure.

However, as these polynomials are defined over the algebraic number field, they may be transformed by the action of the absolute Galois group Γ of the rational numbers. An element of Γ that transforms √21 to −√21 will transform ƒ₁ into ƒ₂ and vice versa, and thus can also be said to transform each of the two trees shown in the figure into the other tree. More generally, due to the fact that the critical values of any Belyi function are the pure rationals 0, 1, and ∞, these critical values are unchanged by the Galois action, so this action takes Belyi pairs to other Belyi pairs. One may define an action of Γ on any dessin d'enfant by the corresponding action on Belyi pairs; this action, for instance, permutes the two trees shown in the figure.

Due to Belyi's theorem, the action of Γ on dessins is faithful (that is, every two elements of Γ define different permutations on the set of dessins), so the study of dessins d'enfants can tell us much about Γ itself. In this light, it is of great interest to understand which dessins may be transformed into each other by the action of Γ and which may not. For instance, one may observe that the two trees shown have the same degree sequences for their black nodes and white nodes: both have a black node with degree three, two black nodes with degree two, two white nodes with degree two, and three white nodes with degree one. This equality is not a coincidence: whenever Γ transforms one dessin into another, both will have the same degree sequence. The degree sequence is one known invariant of the Galois action, but not the only invariant.

The stabilizer of a dessin is the subgroup of Γ consisting of group elements that leave the dessin unchanged. Due to the Galois correspondence between subgroups of Γ and algebraic number fields, the stabilizer corresponds to a field, the field of moduli of the dessin. An orbit of a dessin is the set of all other dessins into which it may be transformed; due to the degree invariant, orbits are necessarily finite and stabilizers are of finite index. One may similarly define the stabilizer of an orbit (the subgroup that fixes all elements of the orbit) and the corresponding field of moduli of the orbit, another invariant of the dessin. The stabilizer of the orbit is the maximal normal subgroup of Γ contained in the stabilizer of the dessin, and the field of moduli of the orbit corresponds to the smallest normal extension of Q that contains the field of moduli of the dessin. For instance, for the two conjugate dessins considered in this section, the field of moduli of the orbit is . The two Belyi functions ƒ₁ and ƒ₂ of this example are defined over the field of moduli, but there exist dessins for which the field of definition of the Belyi function must be larger than the field of moduli.