Arboreal Galois Representation Research Articles

Odoni’s conjecture on arboreal Galois representations is false

Suppose f ∈ K [ x ] f \in K[x] is a polynomial. The absolute Galois group of K K acts on the preimage tree T \mathrm {T} of 0 0 under f f . The resulting homomorphism ϕ f : Gal K → Aut ⁡ T \phi _f\colon \operatorname {Gal}_K \to \operatorname {Aut} \mathrm {T} is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K K there exists a polynomial f f for which ϕ f \phi _f is surjective. We show that this conjecture is false.

An equivariant isomorphism theorem for mod $\mathfrak {p}$ reductions of arboreal Galois representations

An equivariant isomorphism theorem for mod $\mathfrak {p}$ reductions of arboreal Galois representations

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^{\infty}} : {\rm Gal}(\overline{F}/F) \to \mathbb{Z}_{\ell}^{2} \rtimes {\rm GL}_{2}(\mathbb{Z}_{\ell})$ describing the action of ${\rm Gal}(\overline{F}/F)$ on points $\beta_{n}$ so that $\ell^{n} \beta_{n} = \alpha$. We give an explicit bound on the index of the image of $\omega_{E,\alpha,\ell^{\infty}}$ depending on how $\ell$-divisible the point $\alpha$ is, and the image of the ordinary $\ell$-adic Galois representation. The image of $\omega_{E,\alpha,\ell^{\infty}}$ is connected with the density of primes $\mathfrak{p}$ for which $\alpha \in E(\mathbb{F}_{\mathfrak{p}})$ has order coprime to $\ell$.

The 𝐴𝐵𝐶-Conjecture implies uniform bounds on dynamical Zsigmondy sets

We prove that the a b c abc -Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the a b c abc -Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.

Arboreal Galois representation for a certain type of quadratic polynomials

We give a description of the arboreal Galois representation of a quadratic polynomial $$f(x)=x^2-(4k+1)$$, where $$k\in {\mathbb {N}}$$ and $$4k+1$$ is not a square in $${\mathbb {Z}}$$.

On iterated extensions of number fields arising from quadratic polynomial maps

A post-critically finite rational map ϕ of prime degree p and a base point β yield a tower of finitely ramified iterated extensions of number fields, and sometimes provide an arboreal Galois representation with a p-adic Lie image. In this paper, we take ϕ to be the monic Chebyshev polynomial x2−2, and we examine the size of the 2-part of the ideal class group of extensions in the resulting tower. In some cases, we prove an analogue of Greenberg's conjecture from Iwasawa theory. A key tool is a general theorem on p-indivisibility of class numbers of relative cyclic extensions of degree p2.

Density of odd order reductions for elliptic curves with a rational point of order 2

Suppose that [Formula: see text] is an elliptic curve with a rational point [Formula: see text] of order [Formula: see text] and [Formula: see text] is a point of infinite order. We consider the problem of determining the density of primes [Formula: see text] for which [Formula: see text] has odd order. This density is determined by the image of the arboreal Galois representation [Formula: see text]. Assuming that [Formula: see text] is primitive (that is neither [Formula: see text] nor [Formula: see text] is twice a point over [Formula: see text]) and that the image of the ordinary [Formula: see text]-adic Galois representation is as large as possible (subject to [Formula: see text] having a rational point of order [Formula: see text]), we determine that there are [Formula: see text] possibilities for the image of [Formula: see text]. As a consequence, the density of primes [Formula: see text] for which the order of [Formula: see text] is odd is between 1/14 and [Formula: see text].

Dynamical Galois groups of trinomials and Odoni's conjecture

We prove that for every prime p, there exists a degree p polynomial whose arboreal Galois representation is surjective, that is, whose iterates have Galois groups over Q that are as large as possible subject to a natural constraint coming from iteration. This resolves in the case of prime degree a conjecture of Odoni from 1985. We also show that a standard height uniformity conjecture in arithmetic geometry implies the existence of such a polynomial in many degrees d which are not prime.

A large arboreal Galois representation for a cubic postcritically finite polynomial

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.

Galois groups of iterates of some unicritical polynomials

We prove that the arboreal Galois representations attached to certain unicritical polynomials have finite index in an infinite wreath product of cyclic groups, and that this index is 1 in some special cases, including a new family of quadratic polynomials

The Vojta conjecture implies Galois rigidity in dynamical families

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.

On arboreal Galois representations of rational functions

The action of the absolute Galois group Gal(Ksep/K) of a global field K on a tree T(ϕ,α) of iterated preimages of α∈P1(K) under ϕ∈K(x) with deg⁡(ϕ)≥2 induces a homomorphism ρ:Gal(Ksep/K)→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 [5,6] about the size of the group G(ϕ,α):=imρ=lim←n⁡Gal(K(ϕ−n(α))/K). Specifically, we consider two cases for the pair (ϕ,α): (1) ϕ is such that the sequence {an} defined by a0=α and an=ϕ(an−1) is periodic, and (2) ϕ commutes with a nontrivial Möbius transformation that fixes α.In the first case, we resolve a question posed by Jones [5] about the size of G(ϕ,α), and taking K=Q, we describe the Galois groups of iterates of polynomials ϕ∈Z[x] that have the form ϕ(x)=x2+kx or ϕ(x)=x2−(k+1)x+k. When K=Q and ϕ∈Z[x], arboreal Galois representations are a useful tool for studying the arithmetic dynamics of ϕ. In the case of ϕ(x)=x2+kx for k∈Z, we employ a result of Jones [4] regarding the size of the group G(ψ,0), where ψ(x)=x2−kx+k, to obtain a zero-density result for primes dividing terms of the sequence {an} defined by a0∈Z and an=ϕ(an−1).In the second case, we resolve a conjecture of Jones [5] about the size of a certain subgroup C(ϕ,α)⊂Aut(T(ϕ,α)) that contains G(ϕ,α), and we present progress toward the proof of a conjecture of Jones and Manes [6] concerning the size of G(ϕ,α) as a subgroup of C(ϕ,α).

Galois uniformity in quadratic dynamics over [formula omitted

TextWe prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions k(t) in characteristic zero has finite index in the full automorphism group of the associated preimage tree. Moreover, we show that in most cases, the index is bounded independently of the polynomial. VideoFor a video summary of this paper, please visit http://youtu.be/loO7qn4ncWs.

Arboreal Galois representations and uniformization of polynomial dynamics

Given a polynomial f defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z↦ f(z) to the multiplicative action z↦ zdeg(f). The relation between this construction and the Böttcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplicative Kummer theory. As a consequence, we obtain some corollaries regarding Galois irreducibility of preimages of certain points under certain polynomials, as well as the rationality of preimages in one-parameter families.

The Image of an Arboreal Galois Representation

The Image of an Arboreal Galois Representation