KAWA 2015：动态模空间和椭圆曲线
KAWA 2015: Dynamical moduli spaces and elliptic curves
摘要：
在这些课堂讲稿中，我们提出了一个有理函数族$f_t: \mathbb{P}^1\to \mathbb{P}^1$的复动力学之间的联系，在黎曼曲面$X$中由$t$参数化，以及$f_t$在有理点$\mathbb{P}^1(k)$上的算术动力学，其中$k = \mathbb{C}(X)$或$\bar{\mathbb{Q}}(X)$。一个明确的稳定性和规范的高度之间的关系进行了解释，证明包含一块的Mordell-Weil定理的椭圆曲线的功能领域。我们的主要目标是提出一些关于这些家庭的问题和建议，由算术几何的“不太可能的交叉”原则指导，如[Zannier 2012]。我们还包括一个证明，双曲型后临界有限映射是Zagliki稠密的模空间的有理映射的任何给定的程度$d>1$。这些笔记基于KAWA 2015的四个讲座，在意大利比萨，专为专门从事复杂分析的观众设计，扩展了[Baker-DeMarco 2013，DeMarco 2016，DeMarco-Wang-Ye 2016]的主要结果。
英文摘要：
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].
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