Kawa 2015 – Dynamical Moduli Spaces and Elliptic Curves

In these notes, we present a connection between the complex dynamics of a rational function f : P → P, specifically when examined in algebraic families ft with t in a Riemann surface X, and the arithmetic dynamics on rational points P(k) where k = C(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. We also present a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space Md of rational maps of any given degree d > 1. Finally, we include some open questions and conjectures, guided by the principle of “unlikely intersections” from arithmetic geometry, as in [Za], and their dynamical counterparts. 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 [BD2, De3, DWY2]. Résumé. Dans ces notes, nous donnons un lien entre la dynamique complexe d’une fraction rationnelle f : P → P, plus spécifiquement quand on l’examine dans des familles algébriques ft paramétrée par une surface de Riemann X, et la dynamique arithmétique des points rationnels de P(k), où k = C(X). Une relation explicite entre stabilité et hauteur canonique est établie, avec une preuve qui contient une partie du théorème de Mordell-Weil pour les courbes elliptiques sur un corps de fonctions. Nous donnons aussi une preuve du fait que les applications hyperboliques postcritiquement-finies sont Zariski denses dans l’espace des modules Md des applications rationnelles de degré donné d > 1. Finalement nous incluons quelques questions ouvertes et des conjectures, guidés par le principe de “unlikely intersections” en géométrie arithmétique (cf. [Za]) et leurs homologues dynamiques. Ces notes sont basées sur un cours de 4 séances données à KAWA 2015 à Pise, Italie, destinées à une audience spécialisée en analyse complexe, et développent les principaux résultats de [BD2, De3, DWY2]. These notes are based on a series of four lectures at KAWA 2015, the sixth annual school and workshop in complex analysis, this year held at the Centro di Ricerca Matematica Ennio de Giorgi in Pisa, Italy. The goal of the lectures was to explain some of the background and context for my recent research, concentrating on the main results of [BD2, De3, DWY2], developing connections between one-dimensional complex dynamics and the arithmetic of elliptic curves. The text below is essentially a transcription of the lectures, with each section corresponding to one lecture, with a few added details and additional references. I have included an Appendix containing a proof that the hyperbolic postcritically finite maps are Zariski dense in the moduli space Md; this fact was mentioned during a lecture but has not previously appeared in print. Date: October 9, 2015.