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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.

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 $T_{n}^{d}$ formed by all nodes at distance at most $n$ from the root. The group of automorphisms of $T_{n}^{d}$ 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 $f\colon \mathbb {P} _{K}^{1}\to \mathbb {P} _{K}^{1}$ 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 first three levels of the tree of preimages of $0$ under the map $x^{2}+1$

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 $K=\mathbb {Q}$, and in an unpublished paper Specter layed out a proof of Odoni's conjecture for $K$ a number field and $n$ any positive integer.

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 $\gamma _{1},\gamma _{2}\in \mathbb {P} _{K}^{1}$ 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 $d\geq 2$ 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 $K=\mathbb {Q}$.