Cubic points on dynamical modular curves

TL;DR

This paper classifies which dynamical modular curves related to quadratic polynomials have infinitely many cubic points, and applies this to classify preperiodic points over cubic fields, extending prior research.

Contribution

It provides a complete classification of dynamical modular curves with infinitely many cubic points and applies this to preperiodic points over cubic fields, extending earlier work.

Findings

Identifies which dynamical modular curves have infinitely many cubic points.

Classifies preperiodic points for quadratic polynomials over cubic fields.

Extends previous classifications to cubic field cases.

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.