Combinatorics of planar trees and arithmetics of hyperelliptic curves

The aim of this article is to propose a new method for the study of some questions related to the action of the absolute Galois group on the bet of plane trees, in the context of the Grotendieck theory of "Dessins.d'enfants". Starting with a plane tree with n edges, we construct a hyperelliptic curve with an n-division point. This enables us to establish a relation between the theory of "Dessins d'enfants" and the theory of torsion on hyperelliptic curves. In particular, using the results on torsion on elliptic curves we obtain the lower bounds on the degrees of the fields of moduli of some trees. On the other hand, we obtain the interesting series of examples of torsion on hyperelliptic curves.