Arithmetic properties of periodic points of quadratic maps, II

Morton, Patrick

doi:10.4064/aa-87-2-89-102

Cited by 49 publications

(24 citation statements)

References 15 publications

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“…These results are in contrast with the case of quadratic polynomials, where it is known that there exists a one-parameter family of maps having rational points of period 3, and there are no Q-rational points of primitive period 4 (see [10]) or 5 (see [4]). Evidence suggests that a quadratic polynomial defined over Q cannot have a rational periodic point of primitive period N > 3.…”

Section: Introductionmentioning

confidence: 65%

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

Manes

2007

Proceedings of the London Mathematical Society

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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...

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Section: Introductionmentioning

confidence: 65%

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

Manes

2007

Proceedings of the London Mathematical Society

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“…It is known that all cusps on X dyn 1 (N ) and hence also on X dyn 0 (N ) are rational points, for all N . This follows from the Laurent series expansions for the periodic points in terms of q = (−4c − 3) −1/2 , which have rational coefficients; compare [16]. In fact, the map…”

Section: Rational Pointsmentioning

confidence: 90%

“…It is easy to see that fixed points and 2-cycles are each parameterized by a rational curve; the same is true for 3-cycles. Morton [16] has shown that 4-cycles are parameterized by the modular curve X 1 (16); he used this to show that there do not exist rational 4-cycles. Flynn, Poonen, and Schaefer [9] proved that there are no rational 5-cycles and make a preliminary study of 6-cycles.…”

Section: Introductionmentioning

confidence: 99%

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Rational 6-Cycles Under Iteration of Quadratic Polynomials

Stoll¹

2008

LMS J. Comput. Math.

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“…It has been conjectured that f c does not have rational periodic orbits of period greater than three, but so far this conjecture has been proved only for periods four and five. Morton [8] proved the impossibility of rational 4-cycles by using a result of Washington [11], and later Flynn, Poonen and Schaefer [2] settled the period-five case by using deep results of arithmetic and algebraic geometry.…”

Section: Introductionmentioning

confidence: 99%