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The arithmetic of curves defined by iteration. (English) Zbl 1330.14032
Given an irreducible quadratic polynomial $$f(x) = f_c(x) = x^2 + c \in \mathbb{Q}[x]$$ whose $$n$$-th iterate $$f^n$$ has distinct roots, let $$\mathbb {T}_n$$ be $$\{0\} \coprod f^{-1}(0) \coprod \cdots \coprod f^{-n}(0)$$. This becomes a $$2$$-ary rooted tree when we draw an edge between $$\alpha$$ and $$\beta$$ whenever $$f(\alpha) = \beta$$. The Galois group $$G_n$$ of $$f^n$$ acts on $$\mathbb {T}_n$$, and we can ask for which $$c$$ finiteness of $$[\mathrm{Aut}(\mathbb {T}_n): G_n]$$ holds (and in the limit $$n= \infty$$). This is a dynamical analog of Serre’s open image conjecture.
This paper focuses on two aspects of this problem. First, in Theorem 1.1, he studies $$c$$’s for which $$n=4$$ is the first non-maximality, i.e. $$G_3 = \mathrm{Aut}(\mathbb {T}_3)$$ but $$G_4 \neq \mathrm{Aut}(\mathbb {T}_4)$$. In particular, no such $$c$$ exists for $$c\in \mathbb Z$$, and only such $$c\in \mathbb Q$$ is $$\frac 23$$ and $$-\frac 67$$ as long as a certain curve has no rational points above a certain height. Secondly, the author shows in Theorem 1.2 that the Hall–Lang conjecture implies finiteness of $$[\mathrm{Aut}(\mathbb {T}_\infty):G_\infty]$$ for integers $$c$$ which are not negatives of squares, and shows that this index is $$2$$ when $$c = 3$$.
To prove these results, the author considers curves $$C_{c,n}: y^2 = f_c^n(x)$$ and $$B_{c,n}: y^2 = (x-c)f_c^n(x)$$, as well as their twists. By using M. Stoll’s criterion [Arch. Math. 59, No. 3, 239–244 (1992; Zbl 0758.11045)], the author relates the non-maximality to rational points on certain curves. More specifically, for Theorem 1.2, he constructs rational points on twists of $$C_{c,n}$$ and $$B_{c,1}$$. For Theorem 1.1, he shows that $$\sqrt{f_c^4(0)}$$ must be fixed by one of the $$7$$ distinct index-$$2$$ subgroups of $$G_3$$ if $$n=4$$ is the first non-maximality, resulting in a rational point on the corresponding hyperelliptic curves. Then standard techniques such as Chabauty and Runge’s method are used to find rational points.
In addition to these results, the author provides a detailed analysis of $$B_{-2,n}$$ and their Jacobians. In this Chebyshev case, he constructs characteristic polynomial of Frobenius for primes $$\equiv \pm 3 \pmod 8$$ and determines $$B_{-2,n}(\mathbb Q)$$. This leads to the decomposition of $$J(C_{c,n})$$ into simple factors when $$f_c \equiv x^2-2$$ modulo such primes.

##### MSC:
 14G05 Rational points 37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps 12F10 Separable extensions, Galois theory 37P15 Dynamical systems over global ground fields 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 11G05 Elliptic curves over global fields 14H45 Special algebraic curves and curves of low genus 20E08 Groups acting on trees 37P55 Arithmetic dynamics on general algebraic varieties 14H25 Arithmetic ground fields for curves
Magma; SageMath
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