Dynamical moduli spaces and elliptic curves

Laura De Marco

doi:10.5802/afst.1573

Laura De Marco ¹

¹ Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 2, pp. 389-420.

Abstract
Abstract

In these notes, we present a connection between the complex dynamics of a family of rational functions $f_{t} : ℙ^{1} \to ℙ^{1}$ , parameterized by $t$ in a Riemann surface $X$ , and the arithmetic dynamics of $f_{t}$ on rational points $ℙ^{1} (k)$ where $k = ℂ (X)$ or $\bar{ℚ} (X)$ . 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 [53]. We also include a proof that the hyperbolic postcritically-finite maps are Zariski dense in the moduli space $𝕄_{d}$ 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 [6, 17, 14].

Dans ces notes, nous donnons un lien entre la dynamique complexe d’une famille de fractions rationnelles $f_{t} : ℙ^{1} \to ℙ^{1}$ , paramétrée par une surface de Riemann $X$ , et la dynamique arithmétique de $f_{t}$ sur les points rationnels de $ℙ^{1} (k)$ , 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. Notre but principal est de poser quelques questions et conjectures, guidés par le principe des « unlikely intersections » en géométrie arithmétique (cf. [53]). Nous incluons aussi une preuve du fait que les applications hyperboliques postcritiquement-finies sont Zariski denses dans l’espace des modules $𝕄_{d}$ des applications rationnelles de degré donné $d > 1$ . 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 [6, 17, 14].

Published online: 2018-06-17

MR Zbl

DOI: 10.5802/afst.1573

Author's affiliations:

Laura De Marco ¹

¹ Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Laura De Marco},
     title = {Dynamical moduli spaces and elliptic curves},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {389--420},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {2},
     year = {2018},
     doi = {10.5802/afst.1573},
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Laura De Marco. Dynamical moduli spaces and elliptic curves. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 2, pp. 389-420. doi : 10.5802/afst.1573. https://afst.centre-mersenne.org/articles/10.5802/afst.1573/

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