Arithmetic properties of periodic points of quadratic maps

Abstract

Wellesley, Mass.) 1. Introduction.Iterating polynomial maps gives a convenient way of finding extensions of Q whose Galois groups are subgroups of special imprimitive groups known as wreath products, as has been shown by Odoni ([o1], [o2]). Subgroups of wreath products occur as Galois groups not only for the iterates studied by Odoni, but also for the algebraic number fields generated by the periodic points of a polynomial map (see [m1] and [vh]). In particular, studying periodic points of iterated maps over a number field leads naturally to some parametrized families of polynomials with special Galois groups. One of the purposes of this paper is to illustrate this by investigating the algebraic and number-theoretic properties of periodic points of order 3 of a quadratic map over an arbitrary field κ whose characteristic is different from 2. The investigation shows that the arithmetic properties of these periodic points are related to an interesting curve of genus 4. This is part of a larger project to study the Galois groups of periodic points of arbitrary polynomial maps. (See [m1] and [pa].) Thus, let σ(x) be a polynomial over a field κ, and denote by σ n (x) the n-fold iteration of σ with itself:

Arithmetic properties of periodic points of quadratic maps, II

“…This gives a generic factorization which is similar to the factorization of Φ 3 (x, −(s 2 + 7)/4) in [m1,Lemma 4]. Similarly, after replacing z by (w 2 − 1)/w and factoring over Q(w) we find that x is a root of the quadratic polynomial…”

“…It can also be shown that the series ξ i is convergent for complex q satisfying |q| −1 > M for large enough M , so that Φ n (x, c) = 0 also has infinitely many points defined over R. Alternatively, by [m1,Thm. 4] or [rw], the rational curve Φ 3 (x, c) = 0 has infinitely many real points (x, c), and for any such c, f (x) = x 2 + c has real periodic points of all periods, by Sharkovskiȋ's theorem [de, p. 60].…”

The classification of rational preperiodic points of quadratic polynomials over ${\Bbb Q}$ : a refined conjecture

“…Conjecture 2 has been verified for N = 4 and N = 5 (see [18] and [10], respectively), and [10] presents some evidence that it holds for N = 6 as well.…”

“…But X 1 (18) has only six rational points (these are all cusps), so the only rational points on C besides the two points at infinity (on the nonsingular model) are (−1, 1), (−1, −1), (0, 1), and (0, −1). These do not give rise to a valid pair (τ, ρ), since τ is not allowed to be 0 or −1 in Theorem 1.…”

On Poonen's conjecture concerning rational preperiodic points of quadratic maps

“…We will say that α ∈ P 1 (Q) is a periodic point with exact period n for ϕ c if ϕ n c (α) = α, while ϕ m c (α) = α for 0 < m < n. For example, the point at infinity is a point with exact period 1 for ϕ c , as are α = 1 2 ± √ 1 − 4c 2 in the case where 1 − 4c is a rational square. It is equally easy to construct infinite families of quadratic polynomials ϕ c with points with (exact) period 2 or 3, but there are no such polynomials with points of period 4, 5, or (assuming certain conjectures) 6 [1,6,10]. It is reasonable to ask for which N there exists a pair α, c ∈ Q such that α is a point with exact period N for ϕ c (or how many rational periodic points a quadratic map defined over Q may have, in total; the questions are explicitly related [8]).…”