[Paper Review] Galois Action on Diameter Four Trees

Leonardo Zapponi|ArXiv.org|Aug 5, 2001

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper investigates the action of the absolute Galois group on diameter four trees—special dessins d'enfants corresponding to étale covers of the projective line minus three points. Using normalized polynomial models and reduction modulo primes, it establishes conditions under which Galois orbits are trivial, proves that wild ramification can be reduced to tame cases via Kummer models, and derives precise bounds on ramification indices in number fields, particularly showing that certain trees form Galois orbits when the prime is regular at infinity.

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.

Motivation & Objective

• To understand the action of the absolute Galois group on isomorphism classes of diameter four trees.
• To determine when such trees form Galois orbits and compute their fields of moduli.
• To analyze wild ramification in covers of the projective line and reduce it to tame cases.
• To describe the structure of decomposition groups and their action on normalized models.
• To establish conditions under which a diameter four tree has good reduction and lifts uniquely from positive characteristic.

Proposed method

• Models of diameter four trees are represented as polynomials in K[X], with normalization fixing the ramified point above 1 at 0 and a root at 1.
• Standard and normalized models are used to define Galois-invariant invariants, enabling the study of fields of moduli and Galois orbits.
• The paper uses Hilbert's Theorem 90 to construct models defined over the field of moduli.
• It applies p-congruence and reduction modulo p to study good reduction and lift Kummer models from characteristic p to characteristic zero.
• The system of equations χ₁ = ⋯ = χₙ₋₁ = 0 is used to parametrize solutions in positive characteristic, linking them to Kummer models.
• The action of decomposition groups is analyzed via the induced action on coefficients of normalized models, with σ(β(X)) = β(aσX + bσ).

Experimental results

Research questions

• RQ1 When does a diameter four tree form a single Galois orbit under the absolute Galois group?
• RQ2 How can wild ramification in diameter four trees be reduced to tame cases?
• RQ3 What is the structure of the decomposition group acting on normalized models of such trees?
• RQ4 Under what conditions does a tree have good reduction modulo a prime p, and how does this relate to its Galois orbit?
• RQ5 What bounds can be placed on the ramification index of primes in the field of moduli of a diameter four tree?

Key findings

• For generic type (a₁,…,aₙ) with a₁ < ⋯ < aₙ, there are exactly (n−1)! diameter four trees over ḡQ.
• The field of moduli of a diameter four tree is the intersection of the fields of definition of all its normalized models.
• If p > n is regular at infinity and p divides a₁+⋯+aₙ = pʰm with (p,m)=1, then the ramification index eₚ of any prime above p in Q(𝒯) satisfies eₚ ≤ (n−1)/(n−1,h).
• When h=1, the bound becomes eₚ = n−1, and if this equals the degree [Q(𝒯):Q], then the set of trees forms a single Galois orbit.
• For type (1,…,1,a,b), if p divides a+b+n−2 and does not divide u(n,a,b), then p is regular at infinity and the Galois orbit is trivial.
• In the example with n=5, a=2, b=77, primes 7 and 11 are 77-regular, so IV₁,₁,₁,₂,₇₇(ḡQ) is a Galois orbit.