Quadratic points on dynamical modular curves

TL;DR

This paper classifies quadratic points on dynamical modular curves linked to quadratic polynomial maps, advancing understanding of preperiodic points over quadratic fields and extending prior research in the area.

Contribution

It provides a complete classification of which dynamical modular curves have infinitely many quadratic points, a significant extension of previous work.

Findings

Identifies curves with infinitely many quadratic points.

Classifies preperiodic points for quadratic polynomials over quadratic fields.

Extends previous classifications in dynamical systems.

Abstract

Among all the dynamical modular curves associated to quadratic polynomial maps, we determine which curves have infinitely many quadratic points. This yields a classification statement on preperiodic points for quadratic polynomials over quadratic fields, extending previous work of Poonen, Faber, and the authors.