Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Arboreal Galois Representations of Rational Functions   The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}/K)$ of a
global field $K$ on a tree $T(\phi, \alpha)$ of iterated preimages of $\alpha
\in \mathbb{P}^1(K)$ under $\phi \in K(x)$ with $\text{deg}(\phi) \geq 2$
induces a homomorphism $\rho: \text{Gal}(K^{\text{ksep}}/K) \to
\text{Aut}(T(\phi, \alpha))$, 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 $G(\phi,\alpha) := \text{im} \rho =
\underset{\leftarrow n}\lim\text{Gal}(K(\phi^{-n}(\alpha))/K)$. Specifically,
we consider two cases for the pair $(\phi, \alpha)$: (1) $\phi$ is such that
the sequence $\{a_n\}$ defined by $a_0 = \alpha$ and $a_n = \phi(a_{n-1})$ is
periodic, and (2) $\phi$ commutes with a nontrivial Mobius transformation that
fixes $\alpha$.
  In the first case, we resolve a question posed by Jones about the size of
$G(\phi, \alpha)$, and taking $K = \mathbb{Q}$, we describe the Galois groups
of iterates of polynomials $\phi \in \mathbb{Z}[x]$ that have the form $\phi(x)
= x^2 + kx$ or $\phi(x) = x^2 - (k+1)x + k$. When $K = \mathbb{Q}$ and $\phi
\in \mathbb{Z}[x]$, arboreal Galois representations are a useful tool for
studying the arithmetic dynamics of $\phi$. In the case of $\phi(x) = x^2 + kx$
for $k \in \mathbb{Z}$, we employ a result of Jones regarding the size of the
group $G(\psi, 0)$, where $\psi(x) = x^2 - kx + k$, to obtain a zero-density
result for primes dividing terms of the sequence $\{a_n\}$ defined by $a_0 \in
\mathbb{Z}$ and $a_n = \phi(a_{n-1})$.
  In the second case, we resolve a conjecture of Jones about the size of a
certain subgroup $C(\phi, \alpha) \subset \text{Aut}(T(\phi, \alpha))$ that
contains $G(\phi, \alpha)$, and we present progress toward the proof of a
conjecture of Jones and Manes concerning the size of $G(\phi, \alpha)$ as a
subgroup of $C(\phi, \alpha)$.

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The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an