KAWA 2015: Dynamical moduli spaces and elliptic curves

TL;DR

This paper explores the interplay between complex and arithmetic dynamics of rational functions, establishing links between stability, canonical heights, and the density of special maps in moduli spaces, with conjectures inspired by unlikely intersections.

Contribution

It introduces explicit relations between stability and canonical height in families of rational maps, and proves the Zariski density of hyperbolic postcritically-finite maps in moduli spaces.

Findings

Hyperbolic postcritically-finite maps are Zariski dense in moduli space.

Explicit relation between stability and canonical height for rational functions.

Formulation of conjectures based on unlikely intersections in arithmetic geometry.

Abstract

In these lecture notes, we present a connection between the complex dynamics of a family of rational functions $f_t: \mathbb{P}^1\to \mathbb{P}^1$, parameterized by $t$ in a Riemann surface $X$, and the arithmetic dynamics of $f_t$ on rational points $\mathbb{P}^1(k)$ where $k = \mathbb{C}(X)$ or $\bar{\mathbb{Q}}(X)$. An explicit relation between stability and canonical height is explained, with a proof that contains a piece of the Mordell-Weil theorem for elliptic curves over function fields. Our main goal is to pose some questions and conjectures about these families, guided by the principle of "unlikely intersections" from arithmetic geometry, as in [Zannier 2012]. We also include a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space of rational maps of any given degree $d>1$. These notes are based on four lectures at KAWA 2015, in Pisa, Italy, designed for an audience specializing in complex analysis, expanding upon the main results of [Baker-DeMarco 2013, DeMarco 2016, DeMarco-Wang-Ye 2016].