Cubic points on dynamical modular curves

TL;DR

The paper analyzes cubic points on dynamical modular curves $X_1(\P)$ arising from quadratic polynomials $f_c(x)=x^2+c$, linking these points to preperiodic portraits realized by cubic fields. It defines and studies dynamical modular curves via dynatomic polynomials, establishes a framework of morphisms between such curves, and applies Castelnuovo–Severi, reduction techniques, and elliptic-subcover theory to classify which $X_1(\P)$ have infinitely many cubic points. A main result identifies a finite set $\Gamma(3)$ of portraits for which infinitely many cubic points occur, and shows that portraits realizing these cubic points correspond to infinite realizations over cubic fields via Hilbert irreducibility and good-reduction arguments. The work also refines the understanding of c-coordinates, distinguishing strictly versus weakly cubic points, and provides computational evidence and explicit models, extending prior results for $d=1,2$ to the cubic case with a detailed catalog of portraits and their cubic-point behavior.

Abstract

We consider the family of dynamical modular curves associated to quadratic polynomial maps and determine precisely which of these curves have infinitely many cubic points. We use this to prove a classification statement on preperiodic points for quadratic polynomials over cubic fields, extending previous work of Poonen, Faber, and the first author and Krumm.

Figures (11)

• Figure 1: A generic quadratic portrait $\mathcal{P}$
• Figure 2: A portrait $\mathcal{P}$
• Figure 3: Portraits with cycle structure $(1,1)$ and at most ten vertices
• Figure 4: Portraits with cycle structure $(2)$ and at most ten vertices
• Figure 5: Portraits with cycle structure $(3)$ and at most ten vertices

Theorems & Definitions (62)

• Conjecture 1.1: Morton-Silverman morton/silverman:1994
• Conjecture 1.2
• Theorem 1.3: doyle/krumm:2024
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Example 2.1
• Definition 2.2
• Remark 2.3