[Paper Review] The inverse problem for arboreal Galois representations of index two

Andrea Ferraguti, Carlo Pagano+1 more|arXiv (Cornell University)|Jul 19, 2019

Algebraic Geometry and Number Theory22 references4 citations

TL;DR

This paper solves the inverse problem for arboreal Galois representations of index two by providing explicit criteria—based on linear dependence relations modulo squares in the post-critical orbit—for when the image of a quadratic polynomial's Galois representation equals a given index two subgroup of the infinite automorphism group of the regular binary tree. It proves that exactly five such subgroups arise infinitely often over ℚ, and establishes their topological non-isomorphism via combinatorial invariants, yielding five distinct solutions to the infinite inverse Galois problem.

ABSTRACT

This paper introduces a systematic approach towards the inverse problem for arboreal Galois representations of finite index attached to quadratic polynomials. Let $F$ be a field of characteristic $ eq 2$, $f\in F[x]$ be monic and quadratic and $ρ_f$ be the arboreal Galois representation associated to $f$, taking values in the group $Ω_{\infty}$ of automorphisms of the infinite binary tree. We give a complete description of the maximal closed subgroups of each closed subgroup of index at most two of $Ω_{\infty}$ in terms of linear relations modulo squares among certain universal functions evaluated in elements of the critical orbit of $f$. We use such description in order to derive necessary and sufficient criteria for the image of $ρ_f$ to be a given subgroup of index two of $Ω_\infty$. These depend exclusively on the arithmetic of the critical orbit of $f$. Afterwards, we prove that if $ϕ=x^2+t\in\mathbb Q(t)[x]$, then there exist exactly five distinct subgroups of index two of $Ω_{\infty}$ that can appear as images of $ρ_{ϕ_{t_0}}$ for infinitely many $t_0\in\mathbb Q$, where $ϕ_{t_0}$ is the specialized polynomial. We show that two of them appear infinitely often, and if Vojta's conjecture over $\mathbb Q$ holds true, then so do the remaining ones. Finally, we give an explicit description of the derived series of each subgroup of index two. Using this, we introduce a sequence of combinatorial invariants for subgroups of index two of $Ω_\infty$. With a delicate use of these invariants we are able to establish that such subgroups are pairwise non-isomorphic as topological groups, a result of independent interest. This implies, in particular, that the five aforementioned groups are pairwise distinct topological groups, and therefore yield five genuinely different instances of the infinite inverse Galois problem over $\mathbb Q$.

Motivation & Objective

• To classify all closed subgroups of index two in the automorphism group of the infinite rooted binary tree, Ω∞.
• To determine which such subgroups can arise as images of arboreal Galois representations attached to quadratic polynomials over fields of characteristic ≠ 2.
• To provide necessary and sufficient criteria for a given index two subgroup to be realized as the image of a quadratic polynomial’s representation.
• To establish that the five index two subgroups of Ω∞ are pairwise non-isomorphic as topological groups, implying five distinct instances of the infinite inverse Galois problem over ℚ.

Proposed method

• Classify maximal closed subgroups of each index two subgroup of Ω∞ using parametrization by non-zero vectors in 𝔽₂^(ℤ≥0), indexed by the post-critical orbit.
• Translate subgroup structure into linear dependence relations modulo squares among universal functions evaluated at elements of the post-critical orbit of f.
• Use the derived series of index two subgroups to define a sequence of combinatorial invariants that distinguish them topologically.
• Prove that if Vojta’s conjecture holds over ℚ, then three of the five index two subgroups appear as images for infinitely many specializations of x² + t.
• Apply results from Ihara’s theory and profinite group theory to analyze the fibred quotients and higher derived series of these subgroups.

Experimental results

Research questions

• RQ1 Which closed subgroups of index two in Ω∞ can arise as images of arboreal Galois representations attached to monic quadratic polynomials over fields of characteristic ≠ 2?
• RQ2 What arithmetic conditions on the post-critical orbit of a quadratic polynomial ensure that its image is a specific index two subgroup of Ω∞?
• RQ3 How can the derived series and combinatorial invariants of index two subgroups be used to distinguish them as topological groups?
• RQ4 How many distinct index two subgroups of Ω∞ can arise as images of specialized polynomials x² + t₀ over ℚ, and which appear infinitely often?
• RQ5 Under what number-theoretic assumptions (e.g., Vojta’s conjecture) do all five index two subgroups appear as images for infinitely many t₀ ∈ ℚ?

Key findings

• There exist exactly five distinct index two subgroups of Ω∞ that can appear as images of ρϕₜ₀ for infinitely many t₀ ∈ ℚ, where ϕ = x² + t.
• Two of these five subgroups appear infinitely often, as demonstrated by explicit constructions.
• If Vojta’s conjecture over ℚ holds, then the remaining three subgroups also appear infinitely often as images.
• The five index two subgroups of Ω∞ are pairwise non-isomorphic as topological groups, proven via a sequence of combinatorial invariants derived from their derived series.
• The image of ρf is a given index two subgroup of Ω∞ if and only if certain linear dependence relations modulo squares hold among universal functions evaluated at elements of the post-critical orbit of f.
• The paper establishes that monic quadratic polynomials over fields with finite F×/(F×)² or post-critically finite polynomials have arboreal representations of infinite index, extending prior results.