Variation of Canonical Height for Fatou points on $\mathbb{P}^1$

TL;DR

This paper establishes a relationship between the canonical heights of Fatou points in dynamical systems over function fields and Weil heights on the base curve, providing both global and local height comparisons under stability conditions.

Contribution

It introduces a stability condition for points and proves that the canonical height variation induces a Weil height on the base curve, extending known results to more general maps and points.

Findings

Existence of a divisor D on the base curve relating canonical and Weil heights.

Local canonical heights differ from Weil functions by a continuous function.

Characterization of stability condition via the geometry of the induced map.

Abstract

Let $f: \mathbb{P}^1\to \mathbb{P}^1$ be a map of degree $>1$ defined over a function field $k = K(X)$, where $K$ is a number field and $X$ is a projective curve over $K$. For each point $a \in \mathbb{P}^1(k)$ satisfying a dynamical stability condition, we prove that the Call-Silverman canonical height for specialization $f_t$ at point $a_t$, for $t \in X(\bar{\mathbb{Q}})$ outside a finite set, induces a Weil height on the curve $X$; i.e., we prove the existence of a $\mathbb{Q}$-divisor $D = D_{f,a}$ on $X$ so that the function $t\mapsto \hat{h}_{f_t}(a_t) - h_D(t)$ is bounded on $X(\bar{\mathbb{Q}})$ for any choice of Weil height associated to $D$. We also prove a local version, that the local canonical heights $t\mapsto \hat{\lambda}_{f_t, v}(a_t)$ differ from a Weil function for $D$ by a continuous function on $X(\mathbb{C}_v)$, at each place $v$ of the number field $K$. These results were known for polynomial maps $f$ and all points $a \in \mathbb{P}^1(k)$ without the stability hypothesis, and for maps $f$ that are quotients of endomorphisms of elliptic curves $E$ over $k$. Finally, we characterize our stability condition in terms of the geometry of the induced map $\tilde{f}: X\times \mathbb{P}^1 \rightarrow X\times \mathbb{P}^1$ over $K$; and we prove the existence of relative N\'eron models for the pair $(f,a)$, when $a$ is a Fatou point at a place $\gamma$ of $k$, where the local canonical height $\hat{\lambda}_{f,\gamma}(a)$ can be computed as an intersection number.