an:06441794
Zbl 1330.14032
Hindes, Wade
The arithmetic of curves defined by iteration
EN
Acta Arith. 169, No. 1, 1-27 (2015).
00344667
2015
j
14G05 37P05 12F10 37P15 11G30 11G05 14H45 20E08 37P55 14H25
quadratic polynomial; dynamical arboreal representation; Hall-Lang conjecture; rational points on curves
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 \textit{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.
Yu Yasufuku (Tokyo)
Zbl 0758.11045