Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-th
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The categories of this paper are: patt-sol