---
abstract: 'Given a polynomial $\phi$ over a global function field $K/\mathbb{F}_q(t)$ and a wandering base point $b\in K$, we give a geometric condition on $\phi$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$. As an application, we prove that the Galois groups (over $K$) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of $\mathcal{O}_\phi(b)$.'
author:
- Wade Hindes
title: Prime divisors in polynomial orbits over function fields
---

[^1] [^2]

[Introduction]{} From elliptic divisibility sequences to the Fibonacci numbers, it is an important problem in number theory to prove the existence of “new" prime divisors of an arithmetically defined sequence. For example, such ideas have applications ranging from the undecidability of Hilbert’s $10$th problem [@Poonen], to the classification of certain families of subgroups of finite linear groups [@Feit; @primcycl; @Linear; @trans; @alg]. In this paper, we study the set of prime divisors of polynomial recurrence sequences defined by iteration over global function fields.

To wit, let $K/\mathbb{F}_q(t)$ be a finite extension, let $V_K$ be a complete set of valuations on $K$, and let $\mathcal{B}=\{b_n\}\subseteq K$ be any sequence. We say that $v\in V_K$ is a *primitive prime divisor* of $b_n$ if $$v(b_n)>0\;\;\;\text{and}\;\;\; v(b_m)=0\;\; \text{for all}\; 1\leq m\leq n-1.$$ Likewise, we define the *Zsigmondy set of* $\mathcal{B}$ to be $$\mathcal{Z}(\mathcal{B}):=\big\{n\geq1\;\big\vert\; b_n\;\text{has no primitive prime divisors}\big\}.$$ Over number fields, there are numerous results regarding the finiteness (and size) of $\mathcal{Z}(\mathcal{B})$; for example, see [@PrimDiv; @Tucker; @Silv-Ing; @Krieger; @Silv-Vojta; @SilvPrimDiv].

In this paper, we are interested in studying the finiteness of $\mathcal{Z}(\phi,b):=\mathcal{Z}(\mathcal{O}_\phi(b))$, where $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$ is the *orbit* of $b\in K$ for $\phi\in K[x]$; here the superscript $n$ denotes iteration (of $\phi$).

The key geometric notion, allowing us to use techniques in the theory of rational points on curves over $K$ to study $\mathcal{Z}(\phi,b)$, is the following:

Let $\phi\in K(x)$ and let $\ell\geq2$ be an integer. Then we say that $\phi$ is *dynamically $\ell$-power non-isotrivial* if there exists an integer $m\geq1$ such that, $${\label{Curve}} C_{m,\ell}(\phi):=\big\{(X,Y)\in \mathbb{A}^2(\bar{K})\;\big\vert\;Y^\ell=\phi^m(X)=(\underbracket{\phi\circ\phi\dots \circ\phi}_m)(X)\big\}$$ is a non-isotrivial curve [@isotrivial] of genus at least $2$.

Similarly, we have the following refined notions of primitive prime divisors and Zsigmondy sets:

Let $\phi\in K(x)$, let $b\in K$, and let $\ell$ be an integer. We say that a place $v\in V_K$ is an *$\ell$-primitive prime divisor* for $\phi^n(b)$ if all of the following conditions are satisfied:

1.  $v(\phi^n(b))>0$,

2.  $v(\phi^m(b))=0$ for all $1\leq m\leq n-1$ such that $\phi^m(b)\neq0$,

3.  $v(\phi^n(b))\not\equiv0{\ (\textup{mod}\ \ell)}$.

Moreover, we call $${\label{Zig}} \mathcal{Z}(\phi,b,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\big\}$$ the *$\ell$-th Zsigmondy set* for $\phi$ and $b$.

Note that $\mathcal{Z}(\phi,b)\subseteq\mathcal{Z}(\phi,b,\ell)$ for all $\ell$. Hence, it suffices to show that $\mathcal{Z}(\phi,b,\ell)$ is finite for a single $\ell$ to ensure that all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors. Moreover, we use the notions of height $h_K$ and canonical height $\hat{h}_\phi$ found in [@Baker].

[\[PrimDivThm\]]{} Suppose that $\phi\in K[x]$, $b\in K$, and $\ell\geq2$ satisfy the following conditions:

1.  $\phi$ is dynamically $\ell$-power non-isotrivial,

2.  $b$ is wandering (i.e. $\hat{h}_\phi(b)>0$).

Then $\mathcal{Z}(\phi,b,\ell)$ and $\mathcal{Z}(\phi,b)$ are finite. In particular, all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors.

In addition to determining whether or not a sequence has primitive prime divisors, it is interesting to compute the “size" of its set of prime divisors (in terms of density) [@Hasse; @Lagarias], a problem which has applications to the dynamical Mordell-Lang conjecture [@Mordell-Lang] and to questions regarding the size of hyperbolic Mandelbrot sets [@RafeThesis].

To do this, let $\mathcal{O}_K$ be the integral closure of $\mathbb{F}_q[t]$ in $K$ and let $\mathfrak{q}\subseteq \mathcal{O}_K$ be a prime ideal, determining a valuation $v_\mathfrak{q}$ on $K$. For such $\mathfrak{q}$, define the *norm* of $\mathfrak{q}$ to be $N(\mathfrak{q}):=\#(\mathcal{O}_K/\mathfrak{q}\mathcal{O}_K)$, and let $\delta(\mathcal{P})$ be the *Dirchlet density* of a set of primes $\mathcal{P}$ of $K$: $$\delta(\mathcal{P}):=\lim_{s\rightarrow 1^+}\frac{\sum_{\mathfrak{q}\in\mathcal{P}}N(\mathfrak{q})^{-s}}{\sum_{\mathfrak{q}}N(\mathfrak{q})^{-s}}$$ We use Theorem \[PrimDivThm\] and ideas from the Galois theory of iterates to compute the density of $$\mathcal{P}_\phi(b):=\big\{\mathfrak{q}\in{\operatorname{Spec}}(\mathcal{O}_K)\;\big\vert\; v_\mathfrak{q}(\phi^n(b))>0\;\text{for some $n\geq0$}\big\},$$ the set of prime divisors of the orbit $\mathcal{O}_\phi(b)$. In particular, we establish a version of [@R.Jones Conj. 3.11]; see [@uniformity Theorem 1] for the corresponding statement in characteristic zero (with uniform bounds) and [@B-J; @R.Jones] for introductions to dynamical Galois theory.

[\[Galois\]]{} Let $K/\mathbb{F}_q(t)$ for some odd $q$ and let $\phi\in K[x]$ be a quadratic polynomial.\
Write $\phi(x)=(x-\gamma)^2+c$ and suppose that $\phi$ satisfies the following conditions:

1.  $\phi$ is not post-critically finite (i.e. $\gamma$ is wandering),

2.  the adjusted critical orbit $\widebar{\mathcal{O}}_\phi(\gamma)=\{-\phi(\gamma),\phi^n(\gamma)\}_{n\geq2}$ contains no squares in K,

3.  the $j$-invariant of the elliptic cure $E_\phi: Y^2=(X-c)\cdot\phi(X)$ is non-constant.

Then all of the following statements hold:

1.  $\mathcal{Z}(\phi,b,2)$ is finite for all wandering points $b\in K$,

2.  $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup,

3.  $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$.

One expects similar statements to hold for $\phi(x)=x^\ell+c$ and $\ell$ a prime. Namely, if one can show that $\phi$ is dynamically $\ell$-power non-isotrivial, then $\mathcal{Z}(\phi,0,\ell)$ is finite by Theorem \[PrimDivThm\] and $G_\infty(\phi)$ is a finite index subgroup of ${\operatorname{Aut}}(T(\phi))$ by [@Eventually Theorem 25].

In particular, we apply Theorem \[PrimDivThm\] and Corollary \[Galois\] to the explicit family $$\phi_{(f,g)}(X)=\big(X-f(t)\cdot g(t)^d\big)^2+g(t)\;\;\;\;\text{for}\;\;\; f\in\mathbb{F}_q(t),-g\notin(\mathbb{F}_q(t))^2,\;\text{and}\;d\geq1.$$ Note that by letting $f=0$ and $g=t$, we recover the main result of [@RafeThesis].

[\[eg\]]{} Let $K=\mathbb{F}_q(t)$ and let $\phi:=\phi_{(f,g)}$ be such that the $j$-invariant of the elliptic curve $$E_{\phi_{(f,g)}}: Y^2=(X-g(t))\cdot \phi_{(f,g)}(X)$$ is non-constant. Then $\mathcal{Z}(\phi,b,2)$ is finite for all wandering base points, $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup, and $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$.

[Primitive Prime Divisors and Superelliptic Curves]{} To prove Theorem \[PrimDivThm\], we build on our techniques from the characteristic zero setting [@uniformity]. There, among other things, we prove the uniform bound $$\#\mathcal{Z}(\phi,\gamma,2)\leq17$$ for all $\phi(x)=(x-\gamma(t))^2+c(t)$ satisfying $\deg(\gamma)\neq \deg(c)$; here $\gamma$ and $c$ are polynomials with coefficients in any field of characteristic zero. Additionally, we adapt ideas from [@Tucker] and use linear height bounds for rational points on non-isotrivial curves [@Kim] to prove our results.

Throughout the proof, it will be useful to consider only $\ell$-primitive prime divisors avoiding some finite subset $S\subseteq V_K$. Therefore, we make the following convention: $$\mathcal{Z}(\phi,b,S,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\; v\in V_K{\mathbin{\fgebackslash}}S\big\}.$$ Clearly $\mathcal{Z}(\phi,b,\ell)\subseteq\mathcal{Z}(\phi,b,\ell,S)$ for all $S$, and so it suffices to show that $\mathcal{Z}(\phi,b,\ell,S)$ is finite for some $S\subseteq V_K$. Note that there is no harm in enlarging $S$. In particular, we may assume that $$\text{(a)}.\;\;\;b\in\mathcal{O}_{K,S}\;\;\;\;\;\; \text{(b)}.\;\;\;\phi\in\mathcal{O}_{K,S}[x]\;\;\;\;\;\;\text{(c)}.\;\;\; v(a_d)=0\;\;\text{for all}\; v\notin S\;\;\;\;\;\; \text{(d)}.\;\;\; \mathcal{O}_{K,S}\;\text{is a UFD,}$$ where $a_d$ is the leading term of $\phi$. Note that condition (d) is made possible by [@Rosen Prop. 14.2 ]. Similarly, we see that $$\mathcal{Z}(\phi,b,S,\ell)\subseteq\mathcal{Z}(\phi,\phi^n(b),S,\ell)\cup\{t\in\mathbb{Z}\;\vert\; 1\leq t\leq n\}\;\;\,\text{for all}\;n\geq0.$$ Therefore, after replacing $b$ with $\phi^n(b)$ for some $n$, we may assume that $0\notin\mathcal{O}_\phi(b)$. By the assumptions on $S$ above, we see that $\phi^n(b)\in\mathcal{O}_{K,S}$ for all $n$, permitting us to write $${\label{decomp}} \phi^n(b)=u_n\cdot d_n\cdot y_n^\ell,\;\;\text{for some}\;\;\; d_n,y_n\in\mathcal{O}_{K,S}\;,\;u_n\in\mathcal{O}_{K,S}^*.$$ However, since $\mathcal{O}_{K,S}^*$ is a finitely generated group [@Rosen Prop. 14.2 ], we may write $u_n=\textbf{u}_1^{r_1}\cdot \textbf{u}_2^{r_2}\dots \textbf{u}_t^{r_t}$ for some basis $\{\textbf{u}_i\}$ of $\mathcal{O}_{K,S}^*$ and some integers $0\leq r_i\leq\ell-1$. In particular, the height $h_K(u_n)$ is bounded independently of $n\geq0$. Similarly, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\notin S$. To see this, we use the correspondence $V_K{\mathbin{\fgebackslash}}S\longleftrightarrow{\operatorname{Spec}}(\mathcal{O}_{K,S})$ discussed in [@Rosen Ch. 14] and the fact that $\mathcal{O}_{K,S}$ is a UFD to write $$d_n=p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big(p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)^\ell,\;\;\;\;\; p_i\in{\operatorname{Spec}}(\mathcal{O}_{K,S})$$ for some integers $e_i, q_i$ satisfying $v_{p_i}(d_n)=q_i\cdot\ell+e_i$ and $0\leq e_i<\ell$. In particular, by replacing $d_n$ with $\big(p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big)$ and $y_n$ with $\big(y_n\cdot p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)$, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\in V_K{\mathbin{\fgebackslash}}S$ as claimed.

Now suppose that $n\in Z(\phi,b,S,\ell)$. It is our goal to show that $n$ is bounded. To do this, first note that conditions (b) and (c) imply that $\phi$ has good reduction (see [@Silv-Dyn Thm. 2.15]) modulo the primes in ${\operatorname{Spec}}(\mathcal{O}_{K,S})$. In particular, if $p$ is such that $v_{p}(d_n)>0$ and $n\in Z(\phi,b,S,\ell)$, then $v_p(\phi^m(b))>0$ for some $1\leq m\leq n-1$. Moreover, $$\phi^{n-m}(0)\equiv\phi^{n-m}(\phi^{m}(b))\equiv\phi^n(b)\equiv0{\ (\textup{mod}\ p)};$$ see [@Silv-Dyn Thm. 2.18]. Therefore, we have the refinement $${\label{refinement}}  d_n=\prod p_i^{e_i},\;\;\text{where}\;\; p_i\big\vert\phi^{t_i}(b)\;\text{or}\; p_i\big\vert\phi^{t_i}(0)\;\; \text{for some}\; 1\leq t_i\leq\Big\lfloor \frac{n}{2}\Big\rfloor.$$ Moreover, as noted above, we may assume that $0\leq e_i\leq\ell-1$. Hence $${\label{htestimate}} \boxed{h_K(d_n)\leq (\ell-1)\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)}$$ Now, choose (and fix) an integer $m\geq1$ such that the curve $$C_{m,\ell}(\phi): Y^\ell=\phi^m(X)$$ is nonsingular and of genus at least two - possible, since $\phi$ is dynamically $\ell$-power non-isotrivial. If $n\leq m$ for all $n\in\mathcal{Z}(\phi,b,\ell,S)$, then we are done. Otherwise, we may assume that $n>m$ so that (\[decomp\]) implies that $$P_n:=\big(\phi^{n-m}(b)\;,\;y_n\cdot\sqrt[\ell]{u_n\cdot d_n}\, \big)\in C_m(\phi)\big(K\big(\sqrt[\ell]{u_n\cdot d_n}\big)\big)\;\;.$$ It follows from any of the bounds (suitable to positive characteristic) discussed in the introductions of [@Kim] or [@htineq] that there exist constants $A_1, A_2>0$ such that $${\label{Szpiro}}
h_{\kappa(\phi,m)}(P_n)\leq A_1\cdot d(P_n)+A_2,$$ where $\kappa(\phi,m)$ is the canonical divisor of $C_{m,\ell}(\phi)$, $$d(P_n):=\frac{2\cdot {\operatorname{genus}}\big(K_n\big)-2}{\big[K_n:K\big]},\;\;\;\;\text{and}\;\;\;K_n:=K\big(\sqrt[\ell]{u_n\cdot d_n}\,\big).$$ Crucially, the bounds $A_i=A_i(\phi,\ell,m)$ are independent of both $b$ and $n$. We note that the bounds on (\[Szpiro\]) have been improved by Kim [@Kim Theorem 1], although we do not need them here.

On the other hand, it follows from [@ffields Prop. III.7.3 and Remark III.7.5] that $d(P_n)\leq B_1\cdot h_K(u_n\cdot d_n)+B_2$ for some positive constants $B_i=B_i(K)$. Likewise, $h_K(u_n\cdot d_n)\leq h_K(d_n)+B_3(K,S,\ell)$, since the height of $u_n$ is absolutely bounded. In particular, after combining these bounds with those on (\[Szpiro\]), we see that $${\label{combin1}}
h_{\kappa(\phi,m)}(P_n)\leq D_1\cdot h_K(d_n)+D_2, \;\;\;\;\;\;\; D_i=D_i(K,\phi,S,\ell,m)>0.$$ However, if $\mathcal{D}_1$ is an ample divisor on $C_{m,\ell}(\phi)$ and $\mathcal{D}_2$ is an arbitrary divisor, then $${\label{Divisor}} \lim_{h_{\mathcal{D}_1}(P)\rightarrow\infty}\frac{h_{\mathcal{D}_2}(P)}{h_{\mathcal{D}_1}(P)}=\frac{\deg\mathcal{D}_2}{\deg{\mathcal{D}_1}},\;\;\;\;\;\;P\in C_{m}(\phi);$$ see [@SilvA Thm III.10.2]. In particular, if $\pi:C_{m,\ell}(\phi)\rightarrow\mathbb{P}^1$ is the covering $\pi(X,Y)=X$, then a degree one divisor on $\mathbb{P}^1$ (giving the usual height $h_K$ on projective space) pulls back to a $\deg(\pi)$ divisor $\mathcal{D}_2$ on $C$ satisfying $h_{\mathcal{D}_2}(P)=h_K(\pi(P))$.

We deduce from (\[Divisor\]) that there exist constants $\beta$ and $E=E(\phi,m,\ell)$ such that $${\label{limit}} h_{\kappa(\phi,m)}(P)>\beta\;\;\;\;\;\text{implies}\;\;\;\;\; h_K(\pi(P))\leq E\cdot h_{\kappa(\phi,m)}(P)+1$$ for all $P\in C_m(\phi)(\bar{K})$. However, note that $${\label{finiteness}} 
T:=\big\{P_n\;\big\vert\; h_{\kappa(\phi,m)}(P_n)\leq\beta\}\subseteq\big\{P\in C_m(\phi)(\bar{K})\;\big\vert\;\; h_{\kappa(\phi,m)}(P)\leq\beta\;\;\,\text{and}\,\;\; [K(P):K]\leq\ell\big\},$$ and the latter set is finite, since $\kappa(\phi,m)$ is ample; see [@SilvA Thm. 10.3]. Hence, (\[htestimate\]),(\[combin1\]), (\[limit\]), and (\[finiteness\]) together imply that $${\label{combin2}} 
h_K(\phi^{n-m}(b))\leq F_1\cdot \bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)+F_2$$ for all but finitely many $n\in Z(\phi,b,S,\ell)$ and some positive constants $F_i=F_i(K,\phi,S,\ell,m)$.

On the other hand, it is well known that the canonical height $\hat{h}_\phi$ satisfies: $$\text{(a).}\;\;\;\hat{h}_\phi=h_K+O(1)\;\;\;\;\;\;\;\;\;\;\;\; \text{(b).}\;\;\;\hat{h}_\phi(\phi^s(\alpha))=d^s\cdot \hat{h}_\phi(\alpha)$$ for all $\alpha\in K$ and all integers $s\geq0$; see [@Silv-Dyn Thm. 3.20]. In particular, (\[combin2\]) implies that $${\label{combin3}} 
d^{n-m}\cdot\hat{h}_\phi(b)\leq G_1\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}d^i\cdot \hat{h}_\phi(b)+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}d^j\cdot\hat{h}_\phi(0) \bigg)+G_2\cdot n+G_3$$ for almost all $n\in Z(\phi,b,S,\ell)$ (those $n$ such that $P_n\notin T$) and some constants $G_i=G_i(K,\phi,S,\ell,m)$. In particular, for almost all $n\in Z(\phi,b,S,\ell)$, $$d^{n-m}\leq G\cdot \big(d^{\lfloor\frac{n}{2}\rfloor+1}+n+1\big),\;\;\;\;\text{where}\;\;G:=\max\bigg\{G_1,\frac{G_1\cdot \hat{h}_\phi(0)}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)}\bigg\}.$$ However, since $m$ is fixed, such $n$ are bounded.

[Dynamical Galois Groups]{} Our main interest in proving the finiteness of $\mathcal{Z}(\phi,b,\ell)$ comes from the Galois theory of iterates. In particular, if $\phi(x)=(x-\gamma)^2+c$ is a quadratic polynomial and $\mathcal{Z}(\phi,\gamma,2)$ is finite, then the Galois groups of $\phi^n$ are large [@RafeThesis Theorem 3.3], enabling us to compute the density of prime divisors $\mathcal{P}_\phi(b)$ (for all $b\in K$) via a suitable Chebotarev density theorem [@Eventually; @Jones].

To define the relevant dynamical Galois groups, let $\phi$ be a polynomial and assume that $\phi^n$ is separable for all $n\geq1$; hence, the set $T_n(\phi)$ of roots of $\phi, \phi^2,\dots ,\phi^n$ together with $0$, carries a natural $\deg(\phi)$-ary rooted tree structure: $\alpha,\beta\in T_n(\phi)$ share an edge if and only if $\phi(\alpha) =\beta$ or $\phi(\beta)=\alpha$. Furthermore, let $K_n:=K(T_n(\phi))$ and $G_n(\phi):={\operatorname{Gal}}(K_n/K)$. Finally, we set $${\label{Arboreal}}
T_\infty(\phi):=\bigcup _{n \geq 0} T_n(\phi)\;\;\text{and}\;\; G_\infty(\phi)=\varprojlim G_n(\phi).$$ Since $\phi$ is a polynomial with coefficients in $K$, it follows that $G_n(\phi)$ acts via graph automorphisms on $T_n(\phi)$. Hence, we have injections $G_n(\phi) \hookrightarrow {\operatorname{Aut}}(T_n(\phi))$ and $G_\infty(\phi) \hookrightarrow {\operatorname{Aut}}(T_\infty(\phi))$ called the *arboreal representations* associated to $\phi$.

A major problem in dynamical Galois theory, especially over global fields, is to understand the size of $G_\infty(\phi)$ in ${\operatorname{Aut}}(T_\infty(\phi))$; see [@B-J; @Me; @R.Jones; @Odoni; @Stoll]. We now use Theorem \[PrimDivThm\] to prove a finite index result for many quadratic polynomials (c.f. [@uniformity Theorem 1]), including the family defined in Corollary \[eg\], and provide an outline for further examples.

Let $\phi(x)=(x-\gamma)^2+c$ and let $m\geq2$. Then we have a map $${\label{map}} \Phi_m:C_{2,m}(\phi)\rightarrow E_\phi,\;\;\;\; \Phi(x,y)=\big(\phi^{m-1}(x),\;y\cdot(\phi^{m-2}(x)-\gamma)\big).$$ It follows from Proposition \[prop\] below that $C_{2,m}$ is non-isotrivial. On the other hand, since $\widebar{\mathcal{O}}_\phi(\gamma)$ contains no squares in $K$, [@Jones Proposition 4.2] implies that $\phi^n$ is an irreducible polynomial over $K$ for all $n$; hence, $C_{2,m}(\phi)$ and $E_\phi$ are non-singular; see [@Jones Lemma 2.6]. Therefore, we may choose $m$ so that $C_{2,m}(\phi)$ is a non-isotrivial curve of genus at least $2$. In particular, $\phi$ is dynamically 2-power non-isotrivial and Theorem \[PrimDivThm\] implies that $\mathcal{Z}(\phi,b,2)$ is finite for all wandering $b\in K$.

For the second claim, we apply this fact to $b=\gamma$ and use [@RafeThesis Theorem 3.3] to deduce that $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup. Finally, [@R.Jones Theorem 4.2] an [@RafeThesis Theorem 1.3] imply that the density of $\mathcal{P}_\phi(b)$ is zero for all $b\in K$.

We now apply Corollary \[Galois\] to the family $\phi_{(f,g)}$ defined in Corollary \[eg\].

Let $K=\mathbb{F}_q(t)$ and let $\phi(x)=\phi_{(f,g)}=(x-f(t)\cdot g(t)^d)^2+g(t)$. It suffices to check conditions (a) and (b) of Corollary \[Galois\] hold to prove Corollary \[eg\]. We first show that the adjusted critical orbit of $\phi$, the set $\{-\phi(f\cdot g^d),\phi^2(f\cdot g^d),\phi^3(f\cdot g^d),\dots\}$, contains no perfect squares in $K$; in particular, $\phi^n$ is an irreducible polynomial over $K$ for all $n\geq1$; see [@Jones Proposition 4.2].

Note that $-\phi(f\cdot g^d)=-g$ is not a square in $K$ by assumption. On the other hand, we let $h:=g-f\cdot g^d$ and suppose that $${\label{iterate}} j^2=\phi^n(f\cdot g^d)=((((h^2+h)^2+h)^2+h)^2+\dots+h)^2+g$$ for some polynomial $j\in \mathbb{F}_q[t]$ and some $n\geq2$. Hence, $j^2=g^2\cdot k^2+g=g\cdot(g\cdot k^2+1)$ for some $k\in \mathbb{F}_q[t]$, since $g\vert h$. However, because $\mathbb{F}_q[t]$ is a UFD and $g$ and $g\cdot k^2+1$ are coprime, it follows that $g=l^2$ and $g\cdot h^2+1=m^2$ for some $l,m\in \mathbb{F}_q[t]$. In particular, $1=(m+l\cdot h)(m-l\cdot h)$ and both factors are constant. Hence, $2m=(m+l\cdot h)+(m-l\cdot h)$ is also constant. Finally, since $m^2-1=g\cdot h^2$ and $h=g-f\cdot g^d$, it follows that $g, h$, and $f$ are all constant. In particular, the $j$-invariant of $E_\phi$ is constant, a contradiction.

On the other hand, the right hand side of (\[iterate\]) implies that $$\deg(\phi^n(f\cdot g^d))\leq\max\big\{2^{n-1}\cdot\deg(h),\deg(g)\big\}=\max\big\{2^{n-1}\cdot[\deg(g)+\deg(1-f\cdot g^{d-1})],\deg(g)\big\},$$ with equality if the terms are unequal. In particular, if $\deg(h)\neq 0$, then we may choose $n$ large enough so that $\deg(\phi^n(f\cdot g^d))=2^{n-1}\cdot\deg(h)\geq 2^{n-1}$, and $\phi$ is post-critically infinite. Otherwise, we may assume that $h$ is constant. Consequently, since $h=g\cdot(1-f\cdot g^{d-1})$, we deduce that either $g$ is constant or $1-f\cdot g^{d-1}=0$. However, $g$ and $h$ constant implies that $f$ is constant, a contradiction. We deduce that $1-f\cdot g^{d-1}=0$. In particular, $g=f\cdot g^d$ and $\phi(x)=(x-g)^2+g$. In this case the elliptic curve $$E_\phi:\;Y^2=(X-g)\cdot\phi(X)=(X-g)\cdot\big((X-g)^2+g)$$ has $j$-invariant $1728$, contradicting our assumption that $E_\phi$ have non-constant $j$-invariant.

We expect that most $\phi\in K[x]$ are dynamically $\ell$-power non-isotrivial for some $\ell$, and we state a conjecture along these lines:

Suppose that $\phi\in K[x]{\mathbin{\fgebackslash}}\widebar{\mathbb{F}}_q[x]$ satisfies the following conditions:

1.  $\deg(\phi)\geq2$,

2.  $\phi$ is not conjugate to $x^d$ (as a function on $\mathbb{P}^1$)

3.  $\gcd(\deg(\phi),q)=1$,

4.  $0\notin \mathcal{O}_\phi(\gamma)$ for all critical points $\gamma\in\bar{K}$ of $\phi$.

Then there exists $\ell\geq2$ such that $\phi$ is dynamically $\ell$-power non-isotrivial and $\gcd(\ell,q)=1$.

Conditions (3) and (4) imply that the discriminant of $\phi^n$ is non-zero for all $n$; see [@Jones Lemma 2.6]. In particular, the curves $C_{m,\ell}(\phi)$ are non-singular and eventually of large genus, since $\deg(\phi^m)$ grows exponentially by condition (1).

We finish by proving an auxiliary result used in the proof of Corollary \[eg\]. The argument below was suggested by Bjorn Poonen.

[\[prop\]]{} Let $\Phi: C_1\rightarrow C_2$ be a non-constant morphism. If $C_1$ is isotrivial, then $C_2$ is isotrivial.

Suppose not. That is, suppose that $C_1$ is isotrivial but $C_2$ is not. By replacing $K$ by a finite extension, we may assume that $C_1$ is constant. We may also assume that $\Phi$ is separable (if not, then it factors as a power of Frobenius composed with a separable morphism, say $C_1 \rightarrow C_1' \rightarrow C_2$, and then (maybe after a finite extension) $C_1'$ is isomorphic to the curve obtained by taking the $p^n$-roots of all the coefficients of $C_1$, so $C_1'$ is constant too, and is separable over $C_2$; rename $C_1'$ as $C_1$).

We fix some notation. Let $g_i$ be the genus of $C_i$, and let $J_i$ be the Jacobian of $C_i$. Since $C_2$ is non-isotrivial, $g_2>0$.

Let $\Omega$ be an uncountable algebraically closed field containing $F_q$. Specializing (by choosing an $F_q$-homomorphism $\sigma: K \rightarrow \Omega$) gives separable maps over $\Omega$ from the same $C_1$ to uncountably many non-isomorphic curves $C_2^\sigma$. Each isogeny class of an elliptic curve over $\Omega$ consists of at most countably many elliptic curves, so if $g_2=1$, this would imply that in the decomposition of $J_i$ up to isogeny, uncountably many isogeny factors occur, which is impossible. If $g_2>1$, then the infinitely many maps from $C_1$ to various curves $C_2^\sigma$ contradict [@Samuel Theorem 2].

With some additional hypothesis, Proposition \[prop\] holds for projective varieties of arbitrary dimension [@isotrivial]. Such a result is necessary if one hopes to generalize Theorem \[PrimDivThm\] as in [@Silv-Vojta]

**Acknowledgements:** It is a pleasure to thank Joseph Silverman, Bjorn Poonen, Felipe Voloch, and Rafe Jones for the discussions related to the work in this paper.

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[^1]: 2010 *Mathematics Subject Classification*: Primary: 14G05, 37P55. Secondary: 11R32.

[^2]: *Key words and phrases*: Arithmetic Dynamics, Rational Points on Curves, Galois Theory.
