Arboreal Galois representation

In arithmetic dynamics, an arboreal Galois representation is a continuous group homomorphism between the absolute Galois group of a field and the automorphism group of an infinite, regular, rooted tree.

The study of arboreal Galois representations of goes back to the works of Odoni in 1980s.

Definition
Let $$K$$ be a field and $$K^{sep}$$ be its separable closure. The Galois group $$G_K$$ of the extension $$K^{sep}/K$$ is called the absolute Galois group of $$K$$. This is a profinite group and it is therefore endowed with its natural Krull topology.

For a positive integer $$d$$, let $$T^d$$ be the infinite regular rooted tree of degree $$d$$. This is an infinite tree where one node is labeled as the root of the tree and every node has exactly $$d$$ descendants. An automorphism of $$T^d$$ is a bijection of the set of nodes that preserves vertex-edge connectivity. The group $$Aut(T^d)$$ of all automorphisms of $$T^d$$ is a profinite group as well, as it can be seen as the inverse limit of the automorphism groups of the finite sub-trees $$T^d_n$$ formed by all nodes at distance at most $$n$$ from the root. The group of automorphisms of $$T^d_n$$ is isomorphic to $$S_d\wr S_d\wr \ldots \wr S_d$$, the iterated wreath product of $$n$$ copies of the symmetric group of degree $$d$$.

An arboreal Galois representation is a continuous group homomorphism $$G_K \to Aut(T^d)$$.

Arboreal Galois representations attached to rational functions
The most natural source of arboreal Galois representations is the theory of iterations of self-rational functions on the projective line. Let $$K$$ be a field and $$f \colon \mathbb P^1_K\to \mathbb P^1_K$$ a rational function of degree $$d$$. For every $$n\geq 1$$ let $$f^n=f\circ f\circ \ldots \circ f$$ be the $$n$$-fold composition of the map $$f$$ with itself. Let $$\alpha\in K$$ and suppose that for every $$n\geq 1$$ the set $$(f^n)^{-1}(\alpha)$$ contains $$d^n$$ elements of the algebraic closure $$\overline{K}$$. Then one can construct an infinite, regular, rooted $$d$$-ary tree $$T(f)$$ in the following way: the root of the tree is $$\alpha$$, and the nodes at distance $$n$$ from $$\alpha$$ are the elements of $$(f^n)^{-1}(\alpha)$$. A node $$\beta$$ at distance $$n$$ from $$\alpha$$ is connected with an edge to a node $$\gamma$$ at distance $$n+1$$ from $$\alpha$$ if and only if $$f(\beta)=\gamma$$.



The absolute Galois group $$G_K$$ acts on $$T(f)$$ via automorphisms, and the induced homorphism $$\rho_{f,\alpha}\colon G_K\to Aut(T(f))$$ is continuous, and therefore is called the arboreal Galois representation attached to $$f$$ with basepoint $$\alpha$$.

Arboreal representations attached to rational functions can be seen as a wide generalization of Galois representations on Tate modules of abelian varieties.

Arboreal Galois representations attached to quadratic polynomials
The simplest non-trivial case is that of monic quadratic polynomials. Let $$K$$ be a field of characteristic not 2, let $$f=(x-a)^2+b\in K[x]$$ and set the basepoint $$\alpha=0$$. The adjusted post-critical orbit of $$f$$ is the sequence defined by $$c_1=-f(a)$$ and $$c_n= f^n(a)$$ for every $$n\geq 2$$. A resultant argument shows that $$(f^n)^{-1}(0)$$ has $$d^n$$ elements for ever $$n$$ if and only if $$c_n\neq 0$$ for every $$n$$. In 1992, Stoll proved the following theorem:


 * Theorem: the arboreal representation $$\rho_{f,0}$$ is surjective if and only if the span of $$\{c_1,\ldots,c_n\}$$ in the $$\mathbb F_2$$-vector space $$K^*/(K^*)^2$$ is $$n$$-dimensional for every $$n\geq 1$$.

The following are examples of polynomials that satisfy the conditions of Stoll's Theorem, and that therefore have surjective arboreal representations.


 * For $$K=\mathbb Q$$, $$f=x^2+a$$, where $$a\in \mathbb Z$$ is such that either $$a>0$$ and $$a\equiv 1,2\bmod 4$$ or $$a<0$$, $$a\equiv 0\bmod 4$$ and $$-a$$ is not a square.


 * Let $$k$$ be a field of characteristic not $$2$$ and $$K=k(t)$$ be the rational function field over $$k$$. Then $$f=x^2+t\in K[x]$$ has surjective arboreal representation.

Higher degrees and Odoni's conjecture
In 1985 Odoni formulated the following conjecture.


 * Conjecture: Let $$K$$ be a Hilbertian field of characteristic $$0$$, and let $$n$$ be a positive integer. Then there exists a polynomial $$f\in K[x]$$ of degree $$n$$ such that $$\rho_{f,0}$$ is surjective.

Although in this very general form the conjecture has been shown to be false by Dittmann and Kadets, there are several results when $$K$$ is a number field. Benedetto and Juul proved Odoni's conjecture for $$K$$ a number field and $$n$$ even, and also when both $$[K:\mathbb Q]$$ and $$n$$ are odd, Looper independently proved Odoni's conjecture for $$n$$ prime and $$K=\mathbb Q$$.

Finite index conjecture
When $$K$$ is a global field and $$f\in K(x)$$ is a rational function of degree 2, the image of $$\rho_{f,0}$$ is expected to be "large" in most cases. The following conjecture quantifies the previous statement, and it was formulated by Jones in 2013.


 * Conjecture Let $$K$$ be a global field and $$f\in K(x)$$ a rational function of degree 2. Let $$\gamma_1,\gamma_2\in \mathbb P^1_K$$ be the critical points of $$f$$. Then $$[Aut(T(f)):Im(\rho_{f,0})]=\infty$$ if and only if at least one of the following conditions hold:

(1) The map $$f$$ is post-critically finite, namely the orbits of $$\gamma_1,\gamma_2$$ are both finite.

(2) There exists $$n\geq 1$$ such that $$f^n(\gamma_1)=f^n(\gamma_2)$$.

(3) $$0$$ is a periodic point for $$f$$.

(4) There exist a Möbius transformation $$m=\frac{ax+b}{cx+d}\in PGL_2(K)$$ that fixes $$0$$ and is such that $$m\circ f \circ m^{-1}=f$$.

Jones' conjecture is considered to be a dynamical analogue of Serre's open image theorem.

One direction of Jones' conjecture is known to be true: if $$f$$ satisfies one of the above conditions, then $$[Aut(T(f)):Im(\rho_{f,0})]=\infty$$. In particular, when $$f$$ is post-critically finite then $$Im(\rho_{f,\alpha})$$ is a topologically finitely generated closed subgroup of $$Aut(T(f))$$ for every $$\alpha\in K$$.

In the other direction, Juul et al. proved that if the abc conjecture holds for number fields, $$K$$ is a number field and $$f\in K[x]$$ is a quadratic polynomial, then $$[Aut(T(f)):Im(\rho_{f,0})]=\infty$$ if and only if $$f$$ is post-critically finite or not eventually stable. When $$f\in K[x]$$ is a quadratic polynomial, conditions (2) and (4) in Jones' conjecture are never satisfied. Moreover, Jones and Levy conjectured that $$f$$ is eventually stable if and only if $$0$$ is not periodic for $$f$$.

Abelian arboreal representations
In 2020, Andrews and Petsche formulated the following conjecture.


 * Conjecture Let $$K$$ be a number field, let $$f \in K[x]$$ be a polynomial of degree $$d\ge 2$$ and let $$\alpha\in K$$. Then $$Im(\rho_{f,\alpha})$$ is abelian if and only if there exists a root of unity $$\zeta$$ such that the pair $$(f,\alpha)$$ is conjugate over the maximal abelian extension $$K^{ab}$$ to $$(x^d,\zeta)$$ or to $$(\pm T_d,\zeta+\zeta^{-1})$$, where $$T_d$$ is the Chebyshev polynomial of the first kind of degree $$d$$.

Two pairs $$(f,\alpha),(g,\beta)$$, where $$f,g\in K(x)$$ and $$\alpha,\beta\in K$$ are conjugate over a field extension $$L/K$$ if there exists a Möbius transformation $$m=\frac{ax+b}{cx+d}\in PGL_2(L)$$ such that $$m\circ f \circ m^{-1}=g$$ and $$m(\alpha)=\beta$$. Conjugacy is an equivalence relation. The Chebyshev polynomials the conjecture refers to are a normalized version, conjugate by the Möbius transformation $$2x$$ to make them monic.

It has been proven that Andrews and Petsche's conjecture holds true when $$K=\mathbb Q$$.