[Paper Review] Tree representations of Galois groups

Nigel Boston|ArXiv.org|Sep 18, 2000

Algebraic Geometry and Number Theory4 references4 citations

TL;DR

This paper proposes that infinite, finitely generated pro-p Galois groups—particularly those unramified at p—admit representations with large image in the automorphism group of a rooted p-ary tree, measured by positive Hausdorff dimension. It conjectures that such groups are branch groups, offering a new approach to the unramified Fontaine-Mazur conjecture via tree representations, which may reveal structural properties inaccessible to standard p-adic representations.

ABSTRACT

Much work has gone into matrix representations of Galois groups, but there is a whole new class of naturally occurring representations that have as yet gone almost unnoticed. In fact, it is well-known in various areas of mathematics that the main sources of totally disconnected groups are matrix groups over local fields AND automorphism groups of locally finite trees. It is perhaps surprising then that representations of Galois groups into the latter have been almost ignored, while at the same time Galois representations into the former have been enormously effective in resolving long-standing problems in number theory. These ``tree'' representations are important as regards topics such as the unramified Fontaine-Mazur conjecture. This conjecture states that any p-adic representation of the Galois group of an extension unramified at p (and ramified at only finitely many primes) should have finite image. In other words, p-adic representations say little about such Galois groups. This paper proposes the conjecture that these Galois groups should, on the other hand, have representations with large image (measured by Hausdorff dimension) in the automorphism group of a rooted tree. Thus, they might allow us to investigate the structure of the Galois group of infinite pro-p extensions (such as Hilbert p-class towers), something unapproachable by standard p-adic representation methods.

Motivation & Objective

• To explore representations of Galois groups into automorphism groups of rooted trees as an alternative to standard p-adic representations.
• To address the limitations of p-adic representations in studying infinite pro-p extensions like Hilbert p-class towers.
• To investigate whether just-infinite pro-p quotients of Galois groups GK,S are branch groups via Hausdorff dimension.
• To strengthen the unramified Fontaine-Mazur conjecture by linking it to the structure of automorphism groups of trees.
• To propose a new framework for understanding Galois group structure through geometric and group-theoretic methods.

Proposed method

• Define Hausdorff dimension of a closed subgroup G of the automorphism group W of a p-ary rooted tree as lim infₙ→∞ log|Gₙ| / log|Wₙ|, where Gₙ is the image in the finite quotient Wₙ.
• Use the pro-2 automorphism group W = lim←Wₙ of the infinite binary tree as the target for Galois representations.
• Apply the notion of 'large image' not in terms of finite index, but via positive Hausdorff dimension, to capture structural richness.
• Leverage known results on Grigorchuk-type (branch) groups and their Hausdorff dimensions, such as the first example with dimension 5/8.
• Use the embedding of finitely generated pro-p groups into W to define the Hausdorff dimension of a pro-p group as the supremum over all such embeddings.
• Draw on the Odoni-Stoll result that the Galois group of the iterated polynomial x²+1 surjects onto W, providing a concrete example of a large-image tree representation.

Experimental results

Research questions

• RQ1 Can Galois groups of number fields unramified outside a finite set S be represented with large image in the automorphism group of a rooted tree, as measured by positive Hausdorff dimension?
• RQ2 Are all just-infinite pro-p quotients of GK,S (for K a number field and S not containing primes above p) branch groups, i.e., subgroups of the automorphism group of a p-ary tree with nontrivial Hausdorff dimension?
• RQ3 Does the existence of such tree representations provide a viable alternative to p-adic representations for studying infinite pro-p extensions?
• RQ4 Can algebraic geometry be used to systematically construct such tree representations via iterated quadratic covers?
• RQ5 Do Frobenius elements in these tree representations carry arithmetic information analogous to traces in modular forms?

Key findings

• The paper proposes that every just-infinite pro-p group has Hausdorff dimension zero if and only if it is not branch, leading to Conjecture 1: a just-infinite pro-p group is branch if and only if its Hausdorff dimension is nonzero.
• Conjecture 2 posits that all just-infinite pro-p quotients of GK,S (for S unramified at p) are branch groups, which would imply the unramified Fontaine-Mazur conjecture.
• The Odoni-Stoll result provides a concrete example: the Galois group of the iterated polynomial x²+1 surjects onto the pro-2 automorphism group W of the infinite binary tree.
• The Hausdorff dimension of the first Grigorchuk group is 5/8, illustrating a nontrivial example of a branch group with positive dimension.
• The paper shows that linear representations over Zp and Fp[[T]] cannot describe all just-infinite pro-p groups, suggesting a need for alternative representations such as those into automorphism groups of trees.
• The conjecture implies that Galois groups with infinite pro-p quotients should admit embeddings into W with positive Hausdorff dimension, offering a new geometric invariant to study their structure.