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

Joseph H. Silverman

Vol. 295 (2018), No. 1, 145–190

DOI: 10.2140/pjm.2018.295.145

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 $\left[b\right]{\mathbb{P}}_{R_S}^1\to {\mathbb{P}}_{R_S}^1$. A finite Galois invariant subset $X\subset {\mathbb{P}}_K^1\left(\bar{K}\right)$ has good reduction outside $S$ if its closure in ${\mathbb{P}}_{R_S}^1$ is étale over $R_S$. We study triples $\left(f,Y,X\right)$ with $X=Y\cup f\left(Y\right)$. We prove that for a fixed $K$, $S$, and $d$, there are only finitely many ${PGL}_2\left(R_S\right)$-equivalence classes of triples with $deg\left(f\right)=d$ and ${\sum }_{P\in Y}{e}_f\left(P\right)\ge 2d+1$ 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 $Y=X$.

Keywords

good reduction, dynamical system, portrait, Shafarevich conjecture

Mathematical Subject Classification 2010

Primary: 37P45

Secondary: 37P15

Milestones

Received: 4 April 2017

Revised: 15 September 2017

Accepted: 15 September 2017

Published: 13 March 2018