Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

Abstract

Let $K$ be a number field, let $S$ be a finite set of places of $K$, and let $R_S$ be the ring of $S$-integers of $K$. A $K$-morphism $f:{\mathbb{P}}_K^1\to {\mathbb{P}}_K^1$ has simple good reduction outside $S$ if it extends to an $R_S$-morphism ${\mathbb{P}}_{R_S}^1\to {\mathbb{P}}_{R_S}^1$. A finite Galois invariant subset $X\subset {\mathbb{P}}_K^1(\bar{K})$ has good reduction outside $S$ if its closure in ${\mathbb{P}}_{R_S}^1$ is étale over $R_S$. We study triples $(f,Y,X)$ with $X=Y\cup f(Y)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many ${\text{PGL}}_2(R_S)$-equivalence classes of triples with $\text{deg}(f)=d$ and ${\sum }_{P\in Y}{e}_f(P)\ge 2d+1$ and $X$ having good reduction outside $S$. We consider refined questions in which the weighted directed graph structure on $f:Y\to X$ is specified, and we give an exhaustive analysis for degree $2$ maps on ${\mathbb{P}}^1$ when $Y=X$.

47 pages - the original version of this paper is being split into two pieces. This piece contains the material on dynamical Shafarevich theorems. The second part, which will be posted separately, will contain material on moduli spaces for dynamical systems with portraits

References in corpus (2)

• Portraits of preperiodic points for rational maps
• Critically separable rational maps in families

Cited by in corpus (1)

• Moduli Spaces for Dynamical Systems with Portraits