Galois Action on Diameter Four Trees

TL;DR

This paper investigates the Galois action on diameter four trees, a class of dessins d'enfants, revealing how Galois groups act on these covers and analyzing wild ramification to gain arithmetic insights.

Contribution

It provides new descriptions of Galois orbits and the action of decomposition groups on diameter four trees, and reduces wild ramification to tame cases for detailed analysis.

Findings

Galois orbits can be distinguished in many cases.

Action of decomposition groups is explicitly described.

Wild ramification can be reduced to tame ramification for analysis.

Abstract

The object of this paper is the study of a class of dessins d'enfants, the so-called diameter four trees. These objects, first introduced by G. Shabat, can be considered as the simplest non trivial example of etale covers of the projective line minus three points. Their arithmetic properties are still mysterious, and their study can inspire the understanding of more general situations. Here, the main interest is devoted to the action of the absolute Galois group on these (isomorphism classes of) coverings. In particular, in many cases, we are able to distinguish Galois orbits and to describe the action of the decomposition groups. One other central result concerns the study of wild ramification, for which we show how to reduce to the tame case, and then deduce some detailed arithmetical informations.