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Wade Hindes

Publications and source records attributed to Wade Hindes.

34 records · Page 2Linked to original sources
arXiv2018

Finite orbit points for sets of quadratic polynomials

Let $S=\{x^2+c_1, x^2+c_2,\dots, x^2+c_s\}$ be a set of quadratic polynomials with rational coefficients, and let $P$ be a rational basepoint. We classify the pairs $(S,P)$ for which $P$ has finite orbit for $S$, assuming that the maximum period length for each individual polynomial is at most three (conjectured by Poonen). In particular, under these hypotheses we prove that if $s\geq4$, then there are no points $P$ with finite orbit for $S$. Moreover, we use this perspective to formulate an analog of the Morton-Silverman Conjecture for sets of rational functions.

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arXiv2018

Stochastic Canonical Heights

We construct height functions defined stochastically on projective varieties equipped with endomorphisms, and we prove that these functions satisfy analogs of the usual properties of canonical heights. Moreover, we give a dynamical interpretation of the kernel of these stochastic height functions, and in the case of the projective line, we relate the size of this kernel to the Julia sets of the original maps. Finally, as an application, we establish the finiteness of some generalized Zsigmondy sets over global fields.

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arXiv2017

Riccati equations and polynomial dynamics over function fields

Given a function field $K$ and $ϕ\in K[x]$, we study two finiteness questions related to iteration of $ϕ$: whether all but finitely many terms of an orbit of $ϕ$ must possess a primitive prime divisor, and whether the Galois groups of iterates of $ϕ$ must have finite index in their natural overgroup $\text{Aut}(T_d)$, where $T_d$ is the infinite tree of iterated preimages of $0$ under $ϕ$. We focus particularly on the case where $K$ has characteristic $p$, where far less is known. We resolve the first question in the affirmative under relatively weak hypotheses; interestingly, the main step in our proof is to rule out "Riccati differential equations" in backwards orbits. We then apply our result on primitive prime divisors and adapt a method of Looper to produce a family of polynomials for which the second question has an affirmative answer; these are the first non-isotrivial examples of such polynomials. We also prove that almost all quadratic polynomials over $\mathbb{Q}(t)$ have iterates whose Galois group is all of $\text{Aut}(T_d)$.

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arXiv2017

The average number of integral points in orbits

Over a number field $K$, a celebrated result of Silverman states that if $φ(z)\in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit $\text{Orb}_φ(P)=\{φ^n(P)\}_{n\geq0}$ is finite for all $P\in \mathbb{P}^1(K)$. In this paper, we show that if we vary $φ$ and $P$ in a suitable family, the number of $S$-integral points in $\text{Orb}_φ(P)$ is absolutely bounded. In particular, if we fix $φ$ and vary the basepoint $P\in \mathbb{P}^1(K)$, then we show that $\#(\text{Orb}_φ(P)\cap\mathcal{O}_{K,S})$ is zero on average. Finally, we prove a zero-average result in general, assuming a standard height uniformity conjecture in arithmetic geometry.

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arXiv2017

Classifying Galois groups of small iterates via rational points

We establish several surjectivity theorems regarding the Galois groups of small iterates of $ϕ_c(x)=x^2+c$ for $c\in\mathbb{Q}$. To do this, we use explicit techniques from the theory of rational points on curves, including the method of Chabauty-Coleman and the Mordell-Weil sieve. For example, we succeed in finding all rational points on a hyperelliptic curve of genus $7$, with rank $5$ Jacobian, whose points parametrize quadratic polynomials with a "newly small" Galois group at the fifth stage of iteration.

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arXiv2017

Galois groups of some iterated polynomials over cyclotomic extensions

Let $φ_p(z)=(z-1)^p+2-ζ_p$, where $ζ_p\in\bar{\mathbb{Q}}$ is a primitive $p$-th root of unity for some odd prime $p$. Building on previous work, we show that the $n$-th iterate $φ_p^n(z)$ has Galois group $[C_p]^n$, an iterated wreath product of cyclic groups, whenever $p$ is not a Wieferich prime.

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arXiv2016

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 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.

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arXiv2016

Integral points of bounded degree on the projective line and in dynamical orbits

Let $D$ be a non-empty effective divisor on $\mathbb{P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb{P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $φ(z)\in \overline{\mathbb{Q}}(z)$ be a rational function of degree at least two whose second iterate $φ^2(z)$ is not a polynomial. We show that as we vary over points $P\in\mathbb{P}^1(\overline{\mathbb{Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.

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arXiv2016

Average Zsigmondy sets, dynamical Galois groups, and the Kodaira-Spencer map

Let $K$ be a global function field and let $ϕ\in K[x]$. For all wandering basepoints $b\in K$, we show that there is a bound on the size of the elements of the dynamical Zsigmondy set $\mathcal{Z}(ϕ,b)$ that depends only on $ϕ$, the poles of the $b$, and $K$. Moreover, when we order $b\in\mathcal{O}_{K,S}$ by height, we show that $\mathcal{Z}(ϕ,b)$ is empty on average. As an application, we prove that the inverse limit of the Galois groups of iterates of $ϕ(x)=x^d+f$ is a finite index subgroup of an iterated wreath product of cyclic groups. Finally, we establish similar results on Zsigmondy sets when $K$ is the field of rational numbers or $K$ is a quadratic imaginary field subject to an added stipulation: either zero has finite orbit under iteration of $ϕ$ or the Vojta conjecture for algebraic points on curves holds.

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arXiv2015

Prime divisors in polynomial orbits over function fields

Given a polynomial $ϕ$ over a global function field $K$ and a wandering base point $b\in K$, we give a geometric condition on $ϕ$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_ϕ(b):=\{ϕ^n(b)\}_{n\geq0}$

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arXiv2014

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.

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arXiv2014

Galois uniformity in quadratic dynamics over rational function fields

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.

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arXiv2014

Rational points on certain families of symmetric equations

We generalize the work of Dem'janenko and Silverman for the Fermat quartics, effectively determining the rational points on the curves $x^{2m}+ax^m+ay^m+y^{2m}=b$ whenever the ranks of some companion hyperelliptic Jacobians are at most one. As an application, we explicitly describe $X_d(\mathbb{Q})$ for certain $d\geq3$, where $X_d: T_d(x)+T_d(y)=1$ and $T_d$ is the monic Chebychev polynomial of degree $d$. Moreover, we show how this later problem relates to orbit intersection problems in dynamics. Finally, we construct a new family of genus $3$ curves which break the Hasse principle, assuming the parity conjecture, by specifying our results to quadratic twists of $x^4-4x^2-4y^2+y^4=-6$.

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arXiv2014

The Arithmetic of Curves Defined by Iteration

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.

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arXiv2012

Points on Elliptic Curves Parametrizing Dynamical Galois Groups

We show how rational points on certain varieties parametrize phenomena arising in the Galois theory of iterates of quadratic polynomials. As an example, we characterize completely the set of quadratic polynomials $x^2+c$ whose third iterate has a "small" Galois group by determining the rational points on some elliptic curves. It follows as a corollary that the only such integer value with this property is $c=3$, answering a question of Rafe Jones. Furthermore, using a result of Granville's on the rational points on quadratic twists of a hyperelliptic curve, we indicate how the ABC conjecture implies a finite index result, suggesting a geometric interpretation of this problem.

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