The arithmetic of curves defined by iteration.

*(English)*Zbl 1330.14032Given 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.

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.

Reviewer: Yu Yasufuku (Tokyo)

##### 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 |

##### Keywords:

quadratic polynomial; dynamical arboreal representation; Hall-Lang conjecture; rational points on curves##### References:

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