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Arithmetic properties of periodic points of quadratic maps. II. (English) Zbl 1029.12002

From the introduction: In this second part [Part I, Acta Arith. 62, 343–372 (1992; Zbl 0767.11016)] the author proves that the quadratic map \(x\to f(x)=x^2+c\), for \(c\) in \(\mathbb Q\) and \(x\) in the complex field \(\mathbb C\), has no rational 4-cycles. The periodic points of \(f\) of minimal period 4 are roots of the 12th degree polynomial, \[ \Phi_4(x,c)= {f^4(x)-x \over f^2(x)-x}, \] where \(f^n\) denotes the \(n\)th iterate of \(f\). He shows that the curve \(\Phi_4(x,c)=0\) has no rational points by proving it is modular, being a model for \(X_1(16)\), the compactification of the upper half-plane modulo the action of \(\Gamma_1(16)= \left\{\left( \begin{smallmatrix} a & b\\ c & d\end{smallmatrix} \right)\in \text{SL}_2 (\mathbb Z)\mid c\equiv 0,\;a\equiv d\equiv 1\pmod{16} \right\}\), and then using results of L. Washington [Math. Comput. 57, 763–775 (1991; Zbl 0743.11058)] on the rational points of \(X_1(16)\).
There are no finite rational points on \(\Phi_4(x,c)=0\) even though the curve \(\Phi_4(x,c)=0\) has infinitely many points defined over each \(p\)-adic completion \(\mathbb Q_p\) and over \(\mathbb R\). The latter property is shared by all the curves \(\Phi_n(x,c)=0\), \(n\geq 1\), where \(\Phi_n(x,c)\) is the polynomial whose roots are the periodic points of \(f\) of minimal period \(n\): \[ \Phi_n(x,c)= \prod_{d\mid n}\bigl(f^d(x)-x\bigr)^{ \mu(n/d)}. \] Thus the 12th degree curve \(\Phi_4(x,c)=0\) provides an affine counterexample to the Hasse principle and may be the first in an infinite family of such examples. The results of E. V. Flynn, B. Poonen and E. Schaefer [Duke Math. J. 90, 435–463 (1997; Zbl 0958.11024)] show that \(f\) has no rational 5-cycles either, so that the 30th degree curve \(\Phi_5 (x,c)=0\) provides another such counterexample.
In Part I the author asked if the curves \(\Phi_n(x,c)=0\) with \(n\geq 5\) might also be modular, but in Flynn et al. (loc. cit.) it is shown that \(\Phi(x,c)=0\) is definitely not modular, and from the arguments of that paper it seems unlikely that \(\Phi_n(x,c)=0\) would be modular for \(n>5\).
This parametrization of the curve \(\Phi_4(x,c)=0\) by modular functions leads to the simple substitution \(c=-1/(4q^2)-3/4\), which is considered in Section 4. He shows in Proposition 5 that all the periodic points of \(f_q(x)= x^2-1/(4q^2) -3/4\) in the algebraic closure of the Laurent series field \(\mathbb Q((q))\) have \(q\)-expansions of the form \[ {\pm 1\over 2q}\pm {1\over 2}+ \sum^\infty_{k=1} a_kq^k, \] where the coefficients \(a_k\) are rational integers. In particular, when \(q\) is a \(p\)-adic integer divisible by \(p\), these expansions give convergent \(p\)-adic expressions for the roots of the polynomial \(\Phi_n (x,-1/(4q^2) -3/4)\) in \(\mathbb Q_p\). Furthermore, if \(\{\xi_0,\xi_1,\dots,\xi_{n-1}\}\) is an \(n\)-cycle of the map \(f_q\) in \(\mathbb Q((q))\), then \(\xi_i= {\varepsilon_i\over 2q}+\sum^\infty_{k=0} a_k^{(i)}q^k\), where \(\varepsilon_i= \pm 1\), the sequence \(\{\varepsilon_k\}_{k\geq 0}\) has minimal period \(n\), and the coefficients \(a_k^{(i)}\) satisfy the following system of recurrences: \[ \varepsilon_i a_{k+1}^{(i)} =a_k^{(i+1)}-\sum^k_{j=0} a_j^{(i)}a_{k-j}^{(i)} \quad\text{for }k\geq 1\text{ and }i=0, \dots,n-1, \] with initial conditions \(a_0^{(i)} =\varepsilon_i \varepsilon_{i+1}/2\), \(a_1^{(i)}=\varepsilon_i (a_0^{(i+1)} +1/2)\). The sequence \(\{a_k^{(0)}\}\) obtained in this manner for \(n=1\) is essentially the sequence of Catalan numbers, and for \(n=3\) and 4 the sum of the sequences \(\{a_k^{(i)}\}\) in a given orbit appears to be an analogue of the Catalan numbers. In Section 4, he gives an application of the above \(q\)-expansions to computing various polynomials related to the dynamical system \(x\to x^2+c\), including the polynomials whose roots are the multipliers (or traces) of the orbits of a given period \(n\).

MSC:

37P35 Arithmetic properties of periodic points
11R09 Polynomials (irreducibility, etc.)
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
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