Dynamical modular curves for quadratic polynomial maps

TL;DR

This paper constructs dynamical analogues of classical modular curves to study preperiodic points of quadratic polynomials, providing new tools for understanding their structure and Galois properties in number theory.

Contribution

It introduces a formal construction of curves parametrizing quadratic polynomials with specified preperiodic point structures, extending classical modular curve concepts to a dynamical setting.

Findings

Curves $X_1(G)$ are irreducible in characteristic zero.

Application of irreducibility results to number fields.

Discussion of Galois representations related to preperiodic points.

Abstract

Motivated by the dynamical uniform boundedness conjecture of Morton and Silverman, specifically in the case of quadratic polynomials, we give a formal construction of a certain class of dynamical analogues of classical modular curves. The preperiodic points for a quadratic polynomial map may be endowed with the structure of a directed graph satisfying certain strict conditions; we call such a graph admissible. Given an admissible graph $G$, we construct a curve $X_1(G)$ whose points parametrize quadratic polynomial maps -- which, up to equivalence, form a one-parameter family -- together with a collection of marked preperiodic points that form a graph isomorphic to $G$. Building on work of Bousch and Morton, we show that these curves are irreducible in characteristic zero, and we give an application of irreducibility in the setting of number fields. We end with a discussion of the Galois theory associated to the preperiodic points of quadratic polynomials, including a certain Galois representation that arises naturally in this setting.