Preperiodic points for quadratic polynomials with small cycles over quadratic fields

Abstract

Given a number field K and a polynomial ϕ(z) ∈ K[z] of degree at least 2, one can construct a finite directed graph G(ϕ, K) whose vertices are the K-rational preperiodic points for f , with an edge α → β if and only if ϕ(α) = β. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field K, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over Q, while recent and ongoing work of the author, Faber, Krumm, and Wetherell has provided a detailed study of this question for all quadratic extensions of Q. In this article, we give a conjecturally complete classification like Poonen's, but over the cyclotomic quadratic fields Q( √ −1) and Q( √ −3). The main tools we use are dynamical modular curves and results concerning quadratic points on curves.Proof of Theorem 4.6. For (A), we note that 3 is a prime of good reduction for J 1 (4), and thatMoreover, since 5 is a prime of good reduction and #J 1 (4)(F 5 2 ) = 2 7 · 5, J 1 (4)(K) cannot have 3-torsion. Hence J 1 (4)(K) tors → (Z/2Z) 3 ⊕ Z/10Z.Since J 1 (4)(Q) tors ∼ = Z/2Z ⊕ Z/10Z, the only way for J 1 (4)(K) tors to be strictly larger than J 1 (4)(Q) tors is for J 1 (4) to gain a 2-torsion point upon base change from Q to K. By Lemma 3.1,

Dynamical modular curves for quadratic polynomial maps

“…Some progress toward Conjecture 1.3 has also been made in the case n = 2. Computational evidence in [10,16] suggests the following: Conjecture 1.5. Let K be a quadratic field, and let c ∈ K. Then G(f c , K) is isomorphic to one of the 46 directed graphs appearing in [10,App.…”

“…In addition to the computational evidence referenced above, there has been considerable progress made in the direction of Conjecture 1.5 in recent [7,10] and ongoing [11] work.…”

“…We specify these conditions in §2.3, and we call a graph satisfying these conditions admissible. Given an admissible graph G, we define a curve X 1 (G) whose points parametrize maps f c together with a collection of preperiodic points for f c which generate a directed graph isomorphic to G. Though several such curves have been explicitly constructed and used in work toward Conjecture 1.3 (e.g., in [6,7,10,11,30]), a formal treatment of these curves has not appeared in the literature. The purpose of this article is to give such a treatment.…”

A local–global principle in the dynamics of quadratic polynomials

“…Indeed, one can show that the Jacobian of the nonsingular projective model of X has a 2-dimensional isogeny factor J for which the group J(Q) has rank 1; this suggests that an application of Chabauty-Coleman techniques may succeed in determining all rational points on X . The details of this approach will appear in the article [7], which deals with several problems of this type.Based on the results of [8] and [14], Poonen [18] conjectured that if f ∈ Q[x] is a quadratic polynomial and n > 3, then f does not have a rational point of period n. In view of Theorems 1.3 and 1.4, which extend the results of [8] and [14], we propose the following stronger statement.Conjecture 1.5. Let f ∈ Q[x] be a quadratic polynomial and n > 3 an integer.…”

“…Indeed, one can show that the Jacobian of the nonsingular projective model of X has a 2-dimensional isogeny factor J for which the group J(Q) has rank 1; this suggests that an application of Chabauty-Coleman techniques may succeed in determining all rational points on X . The details of this approach will appear in the article [7], which deals with several problems of this type.…”