Overgroups of the arboreal representation of PCF polynomials
Wayne Peng
The 11th East Asia Number Theory Conference
<p class="fragment" data-fragment-index="0"> Arithmetic dynamics = number theory + dynamical systems</p> <p class="fragment" data-fragment-index="1"> A dynamical system here refers to the iteration of a rational function $f(x)$ over a number field $K$, and we denote $$f^n=f\circ f^{n-1},\quad f^0(x)=x$$ </p>
<h2>Terminologies</h2> <ul> <li class="fragment" data-fragment-index="1"><em>Orbit</em> of a point $p$ under $f$: $\{f^n(p)| n=0,\ 1,\ 2,\ \ldots \}$</li> <li class="fragment" data-fragment-index="2"><em>Preperiodic point</em> $p$: if $p$ has a finite orbit, i.e. $$f^n(p)=f^m(p)\quad\text{ for some }n< m.$$ </li> <li class="fragment" data-fragment-index="3"><em>Periodic point</em> $p$: if $n=0$.</li> <li class="fragment" data-fragment-index="4"><em>Wandering point</em> $p$: if $p$ has an infinite orbit</li> </ul>
<h2>PCF maps</h2> <p> Recall that a <em>critical point</em> $p$ of $f$ is a solution of $f'(x)=0$. </p> <p class="fragment" data-fragment-index="1"> A rational function $f$ is called <em>post-critically finite (PCF)</em> map if all critical points of $f$ are preperiodic points. </p> <div class="example fragment" data-fragment-index="2"> <p>The following are PCF maps: $x^n$, $x^2-1$, $x^2-2$ ....</p> </div>
<div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">(Benedetto, Ingram, Jones, and Levy)</span> </div> <p> For any fixed integers $d \geq 2$ and $B \geq 1$, there are only finitely many conjugacy classes of post-critically finite rational maps of degree $d$ which can be defined over a number field of degree at most $B$, except (when $d$ is a perfect square) for the infinite family of flexible Lattès maps. </p> </div> <p class="fragment" data-fragment-index="1"> Proposed by Silverman, it is natural to define $$h_{crit}(f)=\sum_{p\in\mathbb{P}^1(\bar{K})}(e_p(f)-1)h_f(p)$$ where $e_p(f)$ is the ramification index of $f$ at $p$ which measures the post-critical complexity of a rational map defined over $\bar{K}$ </p>
<p> It is immediate that $h_{crit}(f)=0$ if and only if $f$ is post-critically finite. </p> <p class="fragment" data-fragment-index="1"> We then show that $|h_{crit}(f)-h(f)|< C$. (This step is very subtle.) </p>
<p> List of all degree $2$ PCF rational maps </p> <div class="grid-container"> <div class="grid-item">\( z^{2} \)</div> <div class="grid-item">\( \frac{1}{z^{2}} \)</div> <div class="grid-item">\( z^{2} - 2 \)</div> <div class="grid-item">\( z^{2} - 1 \)</div> <div class="grid-item">\( \tfrac{1}{2}(z - 1)^{2} \)</div> <div class="grid-item">\( \frac{1}{(z - 1)^{2}} \)</div> <div class="grid-item">\( -\frac{1}{4}z^{2} - 4z \)</div> <div class="grid-item">\( -\frac{4}{9}z^{2} - 12z \)</div> <div class="grid-item">\( \frac{2}{(z - 1)^{2}} \)</div> <div class="grid-item">\( -\frac{2z}{2z^{2} - 4z + 1} \)</div> <div class="grid-item">\( \frac{3z^{2} - 4z + 1}{1 - 4z} \)</div> </div>
<p> List of all degree $3$ PCF polynomials. </p> <div class="grid-container"> <div class="grid-item">\( z^{3} \)</div> <div class="grid-item">\( -z^{3} + 1 \)</div> <div class="grid-item">\( -2z^{3} + 3z^{2} + \tfrac{1}{2} \)</div> <div class="grid-item">\( -2z^{3} + 3z^{2} \)</div> <div class="grid-item">\( -z^{3} + \tfrac{3}{2}z^{2} - 1 \)</div> <div class="grid-item">\( 2z^{3} - 3z^{2} + 1 \)</div> <div class="grid-item">\( 2z^{3} - 3z^{2} + \tfrac{1}{2} \)</div> <div class="grid-item">\( z^{3} - \tfrac{3}{2}z^{2} \)</div> <div class="grid-item">\( -3z^{3} + \tfrac{9}{2}z^{2} \)</div> <div class="grid-item">\( -4z^{3} + 6z^{2} - \tfrac{1}{2} \)</div> <div class="grid-item">\( 4z^{3} - 6z^{2} + \tfrac{3}{2} \)</div> <div class="grid-item">\( 3z^{3} - \tfrac{9}{2}z^{2} + 1 \)</div> <div class="grid-item">\( -z^{3} + \tfrac{3}{2}z^{2} - 1 \)</div> <div class="grid-item">\( -\tfrac{1}{4}z^{3} + \tfrac{3}{2}z + 2 \)</div> <div class="grid-item">\( -\tfrac{1}{28}z^{3} - \tfrac{3}{4}z + \tfrac{7}{2} \)</div> </div>
<h2>Arboreal representation</h2> <img src="/assets/img/slides/overgroup/tree.png" alt="Tree diagram" width="400">
<h3>Wreath Product</h3> <p class="fragment" data-fragment-index="1">Let $G$ and $H$ be two groups, and $G$ acts on the a index set $I=\{1,2,\ldots,d\}$.</p> <p class="fragment" data-fragment-index="2">The <em>wreath product</em> of $H$ by $G$ is a group extension of $G$ by $H^{|I|}$.</p> <p class="fragment" data-fragment-index="3">An element in the wreath product $H\wr_{I} G$ is expressed as $$((h_1,\ldots, h_d);g)$$ with $h_i\in H$ and $g\in G$.</p>
<p>The operation is defined as $$((h_i)_{i\in I}; g)((h'_i)_{i\in I} ; g')=((h_{g'(i)})_{i\in I}(h'_i)_{i\in I};gg').$$</p> <p class="fragment" data-fragment-index="1">The index set $I$ is often omitted when it is clear from the context.</p> <p class="fragment" data-fragment-index="2">We define $$[G]^2=G\wr G\quad\text{and}\quad [G]^n=[G]^{n-1}\wr G\text{ for }n> 2$$ </p>
<h3>Automorphisms of a $d$-ary Tree</h3> <p> Let $T_d^n$ be a <em>(regular) level-$n$ $d$-ary tree</em>. An automorphism of a tree (viewed as a graph) is a bijection on the vertices that preserves edge–vertex incidence. </p> <p class="fragment" data-fragment-index="1"> Let's define $\operatorname{Aut}(T_d^n)$ be the group of automorphisms. We have </p> <div class="proposition fragment"> <p> Let $\operatorname{Aut}(T_d^n)\cong [S_d]^n$ where $S_d$ is the symmetric group of $d$ elements. </p> </div>
<div class="example"> <ol> <li>$\operatorname{Aut}(T_d^1)\cong S_d$</li> <li class="fragment" data-fragment-index="1">$((1,(13),(132));(23))[1,1]=[1,1]$</li> <li class="fragment" data-fragment-index="2">$((1,(13),(132));(23))[1,2]=[3,3]$</li> <li class="fragment" data-fragment-index="3">$((1,(13),(132));(23))[1,3]=[3,2]$</li> </ol> <img src="/assets/img/slides/overgroup/3-ary-tree.png" alt="Tree diagram" width="600"> </div>
<h3>Dynamical Galois Group</h3> <ul> <li class="fragment" data-fragment-index="1">Let \(K\) be a number field.</li> <li class="fragment" data-fragment-index="2">Let \(f\) be a polynomial of degree \(d\) over \(K\).</li> <li class="fragment" data-fragment-index="3">\(K_n(\alpha) = K(f^n(X)-\alpha)\) for some \(\alpha\in K\).</li> <li class="fragment" data-fragment-index="4">\(\alpha\) is called <em>base point</em>.</li> <li class="fragment" data-fragment-index="5">\(G_n(\alpha)= \mathrm{Gal}(K_n/K)\); \(G_\infty = \varprojlim_n G_n\).</li> </ul> <p class="fragment" data-fragment-index="6">We may define a tree structure on $\alpha$ and its preimages under $f$ by creating an edge between two points $a$ and $b$ if $f(a)=b$. This tree is denoted as \( \mathbb{T}_f(\alpha) \)</p> <div class="proposition fragment" data-fragment-index="7"> <p> For $\sigma\in G_n(\alpha)$, $\sigma$ is a tree automorphism. </p> </div>
<h3>Arboreal Representation</h3> <p> This proposition yields an embedding $$\phi:G_\infty\to \operatorname{Aut}(T_d^\infty)$$ (by taking an inverse limit on $n$), and is refered to as as <em>arboreal representation.</em> </p>
<p> A motivation for understanding the dynamical Galois group for number theorists is that this group can be used to estimate $$ \dfrac{\# f^n(\mathbb{P}^1(\mathbb{F}_p))}{p+1}. $$ This implies, for a generic rational function $p$, $$ \limsup_{p\to\infty}\dfrac{\operatorname{Per}(f_p,\mathbb{P}^1(\mathbb{F}_p))}{p} = 0. $$ </p>
<h3>When Will $\phi$ be Surjective?</h3>
<h2>Serre's open image theorem</h2> <ul> <li class="fragment">Let \(E\) be an elliptic curve over \(K\).</li> <li class="fragment">For a prime integer \(p\), multiplication-by-\(p\) is a self-map of \(E\): \[ E \xrightarrow{[p]} E \] </li> <li class="fragment">Let \(E[p^n]\) be the \(p^n\)-torsion points in \(E(\overline{K})\). (We know \(E[p^n]\cong (\mathbb{Z}/p^n\mathbb{Z})^2\).)</li> </ul>
<p class="fragment">The inverse limit of the following chain is the Tate module:</p> <p class="fragment"> $$ \{1\}\xleftarrow{[p]}E[p]\xleftarrow{[p]}E[p^2]\xleftarrow{[p]}E[p^3]\xleftarrow{[p]}\cdots $$ </p> <p class="fragment"> $$ \{1\}\xleftarrow{\times p}(\mathbb{Z}/p\mathbb{Z})^2 \xleftarrow{\times p}(\mathbb{Z}/p^2\mathbb{Z})^2 \xleftarrow{\times p}(\mathbb{Z}/p^3\mathbb{Z})^2 \xleftarrow{\times p}\cdots $$ </p> <p class="fragment"> $$ T_p(E)\cong \mathbb{Z}_p^2 $$ </p>
<p class="fragment"> Taking the Absolute Galois group \(G_K = \operatorname{Gal}(\overline{K}/K)\) acting on \(T_p(E)\) induces an action of \(\operatorname{Gal}(\overline{K}/K)\) on a \(\mathbb{Z}_p\)-module, i.e. a representation $$ G_K \xrightarrow{\rho} \operatorname{GL}_2(\mathbb{Z}_p)\subseteq \operatorname{GL}_2(\mathbb{Q}_p). $$ </p> <div class="theorem fragment"> <div class="theorem-header"> <span class="theorem-authors">(Serre's Open Image Theorem)</span> </div> <p> If \(E\) is not a CM-curve, then \([\operatorname{GL}_2(\mathbb{Z}_p):\rho(G_K)]=1\) for almost all \(p\). </p> </div>
## A dynamic analog <p class="fragment"> Taking the absolute Galois group action on the root tree \( \mathbb{T}_f(\alpha) \), the Galois action on the tree gives an embedding \[ G_k \xrightarrow{\ \rho_{f,\alpha}\ } \operatorname{Aut}(T_d). \] </p> <p class="fragment"> We call \([ \operatorname{Aut}(T_d) : G_k ] = [ \operatorname{Aut}(T_d) : G_\infty ]\) <em>Odoni's index</em>, and ask: when the Odoni index is finite? </p>
## Finite Odoni's index <ul> <li class="fragment">(Juul, Odoni, and etc.) For generic rational function defined over a global field $K$, $\operatorname{Gal}(f^n(X)-t/K)\cong [S_d]^n$, i.e. the Odoni's index is $1$.</li> <li class="fragment">(Bridy, L.-C. Hsia, Jones, H.-C. Li, Tucker, and etc.) For certain families of rational functions, e.g. unit critical polynomial $x^n-c$, Odoni's index is finite. </li> </ul>
## Infinite Odoni's index <p class="fragment"> We note that if the discriminant of a polynomial $f$, denoted by $\operatorname{disc}(f)$, is a perfect square over the defining field of $f$, then the Galois action on the roots of $f$ induces an even permutation. <span class="fragment">That is $$[S_d:\operatorname{Gal}(f)]\leq [S_d:A_d]=2$$ </span> </p> <p class="fragment"> How can we construct a sequence where $\operatorname{disc}(f^n-\alpha)$ is always square? </p>
<div class="proposition fragment"> <p> Let $f$ be a polynomial over a field $K$, let $a_f$ be its leading coefficient, let $d$ be its degree, let $\alpha\in K$, and let $n\geq 2$ be an integer. The discriminant of the $n$-th iteration of $f$, with multiple roots considered to be different elements, is given by: $$\operatorname{disc}(f^n(x)-\alpha)=(-1)^{A(d,n)}a_f^{B(d,n)}d^{d^n}\operatorname{disc}(f^{n-1}(x)-\alpha)^{d}\prod_{c\in \mathcal{C}_f}(f^n(c)-\alpha),$$ where $$ \begin{align*} A(d,n) & =d^n\left(\dfrac{d^n-1}{2}+\dfrac{d^{n-1}-1}{2}\right), \\ B(d,n) & =d^{2n-1}-1. \end{align*} $$ In particular, $(-1)^{A(d,n)}=\left(\frac{d}{4}\right)$ is equal to $1$ if $d$ is even or if $d \equiv 1 \mod 4$, and equal to $-1$ otherwise. </p> </div>
<h3>What's Special About PCF Maps for Arboreal Representation?</h3> <p><center>PCF maps have infinite Odoni's index! They are the dynamical analogue of CM-curves.</center></p>
<h3>Geometric viewpoint (by Jones)</h3> <ul> <li class="fragment"> We refer to $G_n=\operatorname{Gal}(f^n(X)-t/K)$ as the <em>arithmetic Galois group</em>. </li> <li class="fragment"> We refer to $\overline{G_n}=\operatorname{Gal}(f^n(X)-t/\overline{K})$ as the <em>geometric Galois group</em>. </li> <li class="fragment"> Let's consider the <em>strictly post-critical orbit</em> $P= \displaystyle \bigcup_{n\geq 1}f^n(C_f)$. </li> </ul>
<ul> <li class="fragment"> If $\#P<\infty$, then the geometric Galois group is isomorphic to the image of the monodromy representation $$\rho:\pi_1(\mathbb{P}^1_{\overline{K}}\setminus f^{-1}(P))\to \operatorname{Aut}(T_d)$$ where the kernel is isomorphic to $$\cap_{n\in \mathbb{N}}\ker F_n$$ where $F_n:\pi_1(\operatorname{P}^1_{\overline{K}}\setminus P)\to \operatorname{Sym}f^{-n}(t)$. The quotient group is called <em>iteration monodromy group</em>, denoted by $\widetilde{G_n}$. </li> </ul>
We have the following exact sequence $$1\to \overline{G_\infty}\cong\widetilde{G_\infty}\to G_\infty\to \operatorname{Gal}(K'/K)\to 1$$ where $K'=\varinjlim_{n}K(f^n(x)-t)\cap \overline{K}$. <ul> <li class="fragment"> $G_\infty$ is finitely generated. </li> <li class="fragment"> $[S_d]^\infty$ is infinitely generated </li> </ul> <p class="fragment">Therefore, Odoni's index is infinite.</p>
<h3>What is the overgroup of the arboreal representation for PCF polynomials?</h3>
### A key lemma follows from the computation of discriminants <div class="lemma"> <p> Let $f$ be a PCF polynomial over $K$, and $\alpha\in K$. Let $L$ be the minimal integer such that $f^L(\mathcal{C}_f)$ is a periodic set, and let $O$ be the minimal integer such that $f^{L+O}(\mathcal{C}_f)=f^L(\mathcal{C}_f)$. Then, we have the following: <ol> <li>Suppose that all critical points are periodic, and the degree of $f$ is odd. Then $\operatorname{disc}(f^{2O}-\alpha)$ is a square in $K$.</li> <li>Suppose that all critical points are periodic, and the degree of $f$ is even. Then, $\operatorname{disc}(f^{O+1}-\alpha)$ is a square in $K_f^1(\alpha)$.</li> <li>Suppose that a critical point is strictly preperiodic, and the degree of $f$ is odd. Then, $\operatorname{disc}(f^{L+2O-1}-\alpha)$ is a square in $K_f^{L-1}(\alpha)$. In particular, $K_f^{L-1}(\alpha)= K$ if $L=1$.</li> <li>Suppose that a critical point is strictly preperiodic, and the degree of $f$ is even. Then, $\operatorname{disc}(f^{L+O}-\alpha)$ is a square in $K_f^{L}$.</li> </ol> </p> </div>
### Construct sign functions <ul> <li>Recall that $\operatorname{sgn}:S_d\to\{\pm 1\}.$</li> <li class="fragment"> Given $\sigma\in \operatorname{Aut}(T^n_d)\cong\operatorname{Aut}(T^{n-m}_d)\wr \operatorname{Aut}(T^m_d)$, we can express $\sigma$ as $$\sigma = ((b_i)_{i\in I(T^m_d)};a)$$ with $b_i\in\operatorname{Aut}(T^{n-m}_d)$ and $a\in \operatorname{Aut}(T^m_d)$. </li> <li class="fragment">Define $\operatorname{res}_m(\sigma)=a$.</li> <li class="fragment">Consider the natural sign s$\operatorname{sgn}:\operatorname{Aut}(T^n_d)\subseteq S_{d^n}\to\{\pm 1\}$ for any $n$</li> <li class="fragment">Define $\operatorname{sgn}_m=\operatorname{sgn}\circ\operatorname{res}_m$.</li> </ul>
<ul> <li class="fragment"> Define $\operatorname{sgn}^m(((b_i)_{i\in I};a))=\displaystyle\prod_{i\in I}\operatorname{sgn}(b_i)$ </li> <li class="fragment"> Finally, we define, for $n>m_1>m_2$, $$\operatorname{sgn}^{(m_1,m_2)}_1=\operatorname{sgn}^{m_2}\circ\operatorname{res}_{m_1}\cdot \operatorname{sgn}_{m_1}$$ and $$\operatorname{sgn}^{(m_1,m_2)}_2=\operatorname{sgn}^{m_2}\circ\operatorname{res}_{m_1}$$ </li> </ul><table class="fragment" style="margin:auto; border-collapse: collapse; text-align: center;"> <tr> <th>\(\operatorname{sgn} \backslash d\)</th> <th>even</th> <th>odd</th> </tr> <tr> <td>\(\operatorname{sgn}_1\)</td> <td></td> <td>natural</td> </tr> <tr> <td>\(\operatorname{sgn}_2\)</td> <td>natural</td> <td></td> </tr> </table>
<p> Now we can define the target overgroups. We let $$ E_n^{(m_1,m_2)}(d)= \begin{cases} \operatorname{Aut}(T^n_d)\text{, if }n < m\\ E_{n-1}^m(d)\wr \operatorname{Aut}(T^1_d)\cap \ker(\operatorname{sgn}_1^{(m_1,m_2)})\text{, if }n\geq m_1 \end{cases} $$ and $$ F_n^{(m_1,m_2)}(d)= \begin{cases} \operatorname{Aut}(T^n_d)\text{, if }n < m\\ F_{n-1}^{(m_1,m_2)}(d)\wr \operatorname{Aut}(T^1_d)\cap \ker(\operatorname{sgn}_2^{(m_1,m_2)})\text{, if }n\geq m_1. \end{cases}. $$ </p>
<div class="theorem"> <div class="theorem-header"><span class="theorem-authors">(P.)</span></div> <p> Let $f$ be a degree $d$ PCF polynomial over a number field $K$, and let $\alpha\in K$ is not periodic under $f$. Let $a_f$ be the leading coefficient of $f$, let $\mathcal{C}_f$ be the set of all critical points of $f$, let $L$ be the minimal integer such that $f^L(\mathcal{C}_f)$ is a periodic set, and let $O$ be the minimal positive integer such that $f^{L+O}(\mathcal{C}_f)=f^{L}(\mathcal{C}_f)$. <ol> <li>If the degree of $f$ is odd and $L\leq 1$, then $\operatorname{Gal}_f^n(\alpha)$ is isomorphic to a subgroup of $E_n^{2O}(d)$.</li> <li>If the degree of $f$ is odd and $L > 1$, then $\operatorname{Gal}_f^n(\alpha)$ is isomorphic to a subgroup of $F_n^{(L+2O-1,L-1)}(d)$.</li> <li>If the degree of $f$ is even, then $\operatorname{Gal}_f^n(\alpha)$ is isomorphic to a subgroup of $E_n^{(m,m')}(d)$ with $(m,m')$ as follows:</li> <ol> <li>If $L=0$, then $(m,m') = (O+1,1)$.</li> <li>$L>1$, then $(m,m') = (L+O,L)$.</li> </ol> </ol> </p> </div>
Discussion: <ul> <li> Can we find a PCF map whose dynamical Galois group is isomorphic to $F_n^{(L+2O-1,L-1)}(d)$ </li> <li> We also show that $E_\infty^2$ has a unique chief series. Can we identify the normal series structure of other normal subgroups? </li> </ul>