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

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 $\tilde{f}:X\times {\mathbb{P}}^1\to X\times {\mathbb{P}}^1$ over $K$; and we prove the existence of relative Néron 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.

added a conjecture and made minor edits