ℚ-rational cycles for degree-2 rational maps having an automorphism

Abstract

Let φ : P 1 → P 1 be a rational map of degree d = 2 defined over Q and assume that f −1 • φ • f = φ for exactly one nontrivial f ∈ PGL2 Q . We describe families of such maps that have Q-rational periodic points of period 1, 2, and 4, and we prove that no such map has a Q-rational periodic point of exact period 3. We give a complete description of the Q-rational preperiodic points with period at most 4 and show in particular that there are at most 12 such points.Theorem 2. Let φ : P 1 → P 1 be a morphism of degree 2 defined over Q, and suppose that Aut(φ) ∼ = C 2 . Then #{P ∈ Q | P is preperiodic and lands on a cycle of length at most 4} 12.The complete list of possible directed graphs can be found in Figure 1. This work is inspired by Poonen's paper [14], in which he provides a complete analysis of directed graphs that occur as PrePer(f, Q) for points of primitive period N 3 and f ∈ Q[x] a quadratic polynomial. Specifically, he shows that if a quadratic polynomial f has no rational points of primitive period greater than 3, then (counting the fixed point at infinity) # PrePer(f, Q) 9.As Poonen explains in [14], these directed graphs can be thought of as the analogs of possible torsion subgroups for elliptic curves over Q as classified in Mazur's theorem [6]. Let φ(z) be a rational map of degree 2 with Aut(φ) ∼ = C 2 (up to linear conjugacy), and let G be a specific graph of rational preperiodic points. Pairs (φ(z), G) are parametrized by points on an algebraic curve, just as elliptic curves with given level structure correspond to points on modular curves. Deciding whether a given graph is possible, then, reduces to finding rational points on these algebraic curves.The particular curves with rational points that we need to determine have genus 0, 1, or 3. The genus 0 and 1 curves have rational points at infinity, so they are, respectively, P 1 and elliptic curves. The elliptic curves all have small conductor and rank 0, so we are able to list their rational points completely. In the case of the genus 3 curve, it covers an elliptic curve, which unfortunately has rank 1, so this does not allow us to list completely the (necessarily finitely many) rational points on the curve. However, it also covers a genus 2 curve. We find all of the rational points on the genus 2 curve and use that result to find all of the rational points on the original genus 3 curve.Northcott proved in [13] that for a fixed morphism φ, there are at most finitely many preperiodic points in P 1 (Q). Lying deeper is the uniform boundedness conjecture of Morton and Silverman (see [11]). Conjecture 2. Let K/Q be a number field of degree D, and let φ : P N → P N be a morphism of degree d 2 defined over K. There is a constant κ (D, N, d) such that # PrePer(φ, K) κ (D, N, d).This conjecture implies, for example, uniform boundedness for torsion points on abelian varieties over number fields (see [2]). Even the special case n = 1 and d = 4 is enough to imply Merel's uniform boundedness of torsion points on elliptic curves proved in [7]. Torsion points o...

Current trends and open problems in arithmetic dynamics

“…Manes, 2008,[157]). A Q-rational periodic point of a map of the form bx + cx −1 with b, c ∈ Q * has period 1, 2, or 4.…”

A census of quadratic post-critically finite rational functions defined over

“…In [9], Milnor described the symmetry locus for quadratic rational maps; the second author investigated the arithmetic of these maps in [6,7]. Jones and the second author found a height bound on PCF maps with nontrivial stabilizer and used that bound to show that over Q, the only maps meeting these criteria must be conjugate to either…”

“…(1) The map φ b always has a rational fixed point at infinity and a rational point of type 1 In this case, there are no finite rational fixed points [6,Proposition 6] and no rational points of period 2 [6, Proposition 9]. (6) The map φ b cannot have rational points of type 1 n for n 3 [6, Propositions 7 and 8].…”

“…In [12], Poonen undertakes this task for quadratic polynomials defined over Q, subject to the condition that no rational point is on a cycle of length greater than 3. In [6], the second author gives a classification for quadratic rational maps with nontrivial PGL 2 stabilizer, subject to a similar condition. Given the comprehensive list in Theorem 1, we are able to describe all possible rational preperiodic structures for quadratic PCF maps defined over Q with no additional hypotheses.…”

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

“…In [8], Poonen assumes that M (1) = 3 and shows that there can be at most 9 Q-rational preperiodic points, which is achieved by c = − 29 16 . Manes [3] considered a certain family of degree-two rational maps on P 1 , and showed that if the largest possible exact period of a Q-rational periodic point for such a map is 4, then there are at most 12 Q-rational preperiodic points. In the quadratic field case for the quadratic polynomials ϕ c , there can be at least 15 K-rational preperiodic points achieved for c = − 29 16 and the field K = Q( √ 17).…”