Galois Action on Diameter Four Trees
Abstract
The object of this paper is the study of a class of dessins d'enfants, the socalled 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 August 2001
 arXiv:
 arXiv:math/0108031
 Bibcode:
 2001math......8031Z
 Keywords:

 Algebraic Geometry;
 Number Theory
 EPrint:
 27 pages, 3 figures