We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not conjugate to a power map, Chebyshev polynomial, or Lattès map. The associated Galois action on an infinite ternary rooted tree has Hausdorff dimension bounded strictly between that of the infinite wreath product of cyclic groups and that of the infinite wreath product of symmetric groups. We deduce a zero-density result for prime divisors in an orbit under this polynomial. We also obtain a zero-density result for the set of places of convergence of Newton's method for a certain cubic polynomial, thus resolving the first nontrivial case of a conjecture of Faber and Voloch.
Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $ϕ_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $ϕ_f$ is surjective. We show that this conjecture is false.
We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of their Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. This allows us to study the behavior of the monodromy groups under specialization of the maps, and to derive applications to dynamical sequences.
Let~$E$ be a Hilbertian field of characteristic~$0$. R.W.K. Odoni conjectured that for every positive integer~$n$ there exists a polynomial~$f\in E[X]$ of degree~$n$ such that each iterate~$f^{\circ{k}}$ of~$f$ is irreducible and the Galois group of the splitting field of~$f^{\circ k}$ is isomorphic to the automorphism group of a regular,~$n$-branching tree of height~$k.$ We prove this conjecture when~$E$ is a number field.
Given a field $K$, a polynomial $f \in K[x]$, and a suitable element $t \in K$, the set of preimages of $t$ under the iterates $f^{\circ n}$ carries a natural structure of a $d$-ary tree. We study conditions under which the absolute Galois group of $K$ acts on the tree by the full group of automorphisms. When $K=\mathbb{Q}$ we exhibit examples of polynomials of every even degree with maximal Galois action on the preimage tree, partially affirming a conjecture of Odoni. We also study the case of $K=F(t)$ and $f \in F[x]$ in which the corresponding Galois groups are the monodromy groups of the r…
Let $ϕ$ be a quadratic, monic polynomial with coefficients in $\mathcal O_{F,D}[t]$, where $\mathcal O_{F,D}$ is a localization of a number ring $\mathcal O_F$. In this paper, we first prove that if $ϕ$ is non-square and non-isotrivial, then there exists an absolute, effective constant $N_ϕ$ with the following property: for all primes $\mathfrak p\subseteq\mathcal O_{F,D}$ such that the reduced polynomial $ϕ_\mathfrak p\in (\mathcal O_{F,D}/\mathfrak p)[t][x]$ is non-square and non-isotrivial, the squarefree Zsigmondy set of $ϕ_{\mathfrak p}$ is bounded by $N_ϕ$. Using this result, we prove th…
We give a complete description of the normal subgroups of arboreal Galois groups of Belyi maps. The normal groups form a unique chief series. We also carefully compute the discriminate of the iterate of a polynomial minus an algebraic number, which allows us to predict when a such discriminate is a perfect square in the base field or intermediate field for a postcritically finite polynomials (PCF). As a consequence we are able to find another PCF cubic polynomial that has the same arboreal Galois group as the one of Belyi maps.
Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $α\in E(F)$ of infinite order. Attached to the pair $(E,α)$ is the $\ell$-adic arboreal Galois representation $ω_{E,α,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $β_{n}$ so that $\ell^{n} β_{n} = α$. We give an explicit bound on the index of the image of $ω_{E,α,\ell^{\infty}}$ depending on how $\ell$-divisible the point $α$ is, and the image of the ordinary $\el…
In this paper, we prove several results on finitely generated dynamical Galois groups attached to quadratic polynomials. First we show that, over global fields, quadratic post-critically finite polynomials are precisely those having an arboreal representation whose image is topologically finitely generated. To obtain this result, we also prove the quadratic case of Hindes' conjecture on dynamical non-isotriviality. Next, we give two applications of this result. On the one hand, we prove that quadratic polynomials over global fields with abelian dynamical Galois group are necessarily post-criti…
The arboreal Galois group of a polynomial $f$ over a field $K$ encodes the action of Galois on the iterated preimages of a root point $x_0\in K$, analogous to the action of Galois on the $\ell$-power torsion of an abelian variety. We compute the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 - 1$ when the field $K$ and root point $x_0$ satisfy a simple condition. We call the resulting group the arithmetic basilica group because of its relation to the basilica group associated with the complex dynamics of $f$. For $K=\mathbb{Q}$, our condition holds for infinitely man…
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
In this paper, we consider minimal equicontinuous actions of discrete countably generated groups on Cantor sets, obtained from the arboreal representations of absolute Galois groups of fields. In particular, we study the asymptotic discriminant of these actions. The asymptotic discriminant is an invariant obtained by restricting the action to a sequence of nested clopen sets, and studying the isotropies of the enveloping group actions in such restricted systems. An enveloping (Ellis) group of such an action is a profinite group. A large class of actions of profinite groups on Cantor sets is gi…
We prove Odoni's conjecture in all prime degrees; namely, we prove that for every positive prime $p$, there exists a degree $p$ polynomial $φ\in\mathbb{Z}[x]$ with surjective arboreal Galois representation. We also show that Vojta's conjecture implies the existence of such a polynomial in many degrees $d$ which are not prime.
Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups of the extensions of $K$ generated by the solutions to $f^n(z)=a$. The behavior depends upon $v(c)$, and we find that it shifts dramatically as $v(c)$ crosses a certain value: $0$ in the case $p \nmid \ell$, and $-p/(p-1)$ in the case $p=\ell$.
We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and we prove surjectivity for some small degree examples, including a new family of quadratic polynomials. To do this, we use a combination of local techniques including the Chabauty-Coleman method and the Mordell-Weil sieve.
Arboreal Galois Groups of a PCF Map with Strictly Pre-periodic Critical Points
Koc University（科奇大学）；Adam Mickiewicz University（亚当·密茨凯维奇大学）；Bogazici University（博阿齐奇大学）；Johann Wolfgang Goethe-Universität（约翰·沃尔夫冈·歌德大学）；Academia Sinica, Institute of Mathematics（中央研究院数学研究所）
We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that the index is bounded, in most cases, independently of the polynomial.
The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a global field $K$ on a tree $T(ϕ, α)$ of iterated preimages of $α\in \mathbb{P}^1(K)$ under $ϕ\in K(x)$ with $\text{deg}(ϕ) \geq 2$ induces a homomorphism $ρ: \text{Gal}(K^{\text{ksep}}/K) \to \text{Aut}(T(ϕ, α))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(ϕ,α) := \text{im} ρ= \underset{\leftarrow n}\lim\text{Gal}(K(ϕ^{-n}(α))/K)$. Specifically, we consider two cases for the pair $(ϕ, α)$: (1) $ϕ$ is such th…
We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $\mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.
Given a set $S=\{x^2+c_1,\dots,x^2+c_s\}$ defined over a field and an infinite sequence $γ$ of elements of $S$, one can associate an arboreal representation to $γ$, generalizing the case of iterating a single polynomial. We study the probability that a random sequence $γ$ produces a ``large-image'' representation, meaning that infinitely many subquotients in the natural filtration are maximal. We prove that this probability is positive for most sets $S$ defined over $\mathbb{Z}[t]$, and we conjecture a similar positive-probability result for suitable sets over $\mathbb{Q}$. As an application o…
Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $β\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(β)=\text{Gal}(K(f^{-n}(β))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\text{Aut}(T_\infty):G_\infty]<\infty$. When $f$ is a cubic polynomial and $K$ is a function field of transcendence degree $1$ over an algebraic extension of $\mathbb{Q}$, we res…
