Arboreal Galois representation

Last updated October 31, 2024

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
{\displaystyle 0}
under the map
x
2
+
1
{\displaystyle x^{2}+1} PreimageTree.png — 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^[1] 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:^[2]

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. ^[2]

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.^[3]

Higher degrees and Odoni's conjecture

In 1985 Odoni formulated the following conjecture.^[4]

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,^[5] 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,^[6] Looper independently proved Odoni's conjecture for $n$ prime and $K=\mathbb {Q}$ .^[7]

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.^[8]

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:

The map $f$ is post-critically finite, namely the orbits of $\gamma _{1},\gamma _{2}$ are both finite.
There exists $n\geq 1$ such that $f^{n}(\gamma _{1})=f^{n}(\gamma _{2})$ .
$0$ is a periodic point for $f$ .
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 $f$ .^[9]

Abelian arboreal representations

In 2020, Andrews and Petsche formulated the following conjecture.^[10]

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 $K=\mathbb {Q}$ .^[11]

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References

↑ Jones, Rafe (2008). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv: math/0612415 . doi:10.1112/jlms/jdn034. S2CID 15310955.
1 2 Stoll, Michael (1992). "Galois groups over $cQ$ of some iterated polynomials". Archiv der Mathematik. 59 (3): 239–244. doi:10.1007/BF01197321. S2CID 122514918.
↑ Ferraguti, Andrea; Micheli, Giacomo (2020). "An equivariant isomorphism theorem for mod ${\mathfrak {p}}$ reductions of arboreal Galois representations". Trans. Amer. Math. Soc. 373 (12): 8525–8542. arXiv: 1905.00506 . doi:10.1090/tran/8247.
↑ Odoni, R.W.K. (1985). "The Galois theory of iterates and composites of polynomials". Proceedings of the London Mathematical Society. 51 (3): 385–414. doi:10.1112/plms/s3-51.3.385.
↑ Dittmann, Philip; Kadets, Borys (2022). "Odoni's conjecture on arboreal Galois representations is false". Proc. Amer. Math. Soc. 150 (8): 3335–3343. arXiv: 2012.03076 . doi:10.1090/proc/15920.
↑ Benedetto, Robert; Juul, Jamie (2019). "Odoni's conjecture for number fields". Bulletin of the London Mathematical Society. 51 (2): 237–250. arXiv: 1803.01987 . doi:10.1112/blms.12225. S2CID 53400216.
↑ Looper, Nicole (2019). "Dynamical Galois groups of trinomials and Odoni's conjecture". Bulletin of the London Mathematical Society. 51 (2): 278–292. arXiv: 1609.03398 . doi:10.1112/blms.12227.
↑ Jones, Rafe (2013). Galois representations from pre-image trees: an arboreal survey. Actes de la Conférence Théorie des Nombres et Applications. pp. 107–136.
↑ Jones, Rafe; Levy, Alon (2017). "Eventually stable rational functions". Int. J. Number Theory. 13 (9): 2299–2318. arXiv: 1603.00673 . doi:10.1142/S1793042117501263. S2CID 119704204.
↑ Andrews, Jesse; Petsche, Clayton (2020). "Abelian extensions in dynamical Galois theory". Algebra Number Theory. 14 (7): 1981–1999. arXiv: 2001.00659 . doi:10.2140/ant.2020.14.1981. S2CID 209832399.
↑ Ferraguti, Andrea; Ostafe, Alina; Zannier, Umberto (2024). "Cyclotomic and abelian points in backward orbits of rational functions". Advances in Mathematics. 438. arXiv: 2203.10034 . doi:10.1016/j.aim.2023.109463. S2CID 247594240.