#### Vol. 295, No. 1, 2018

 Recent Issues Vol. 295: 1  2 Vol. 294: 1  2 Vol. 293: 1  2 Vol. 292: 1  2 Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
Good reduction and Shafarevich-type theorems for dynamical systems with portrait level structures

### Joseph H. Silverman

Vol. 295 (2018), No. 1, 145–190
##### 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 $f:{ℙ}_{K}^{1}\to {ℙ}_{K}^{1}$ has simple good reduction outside $S$ if it extends to an ${R}_{S}$-morphism $\left[b\right]{ℙ}_{{R}_{S}}^{1}\to {ℙ}_{{R}_{S}}^{1}$. A finite Galois invariant subset $X\subset {ℙ}_{K}^{1}\left(\stackrel{̄}{K}\right)$ has good reduction outside $S$ if its closure in ${ℙ}_{{R}_{S}}^{1}$ 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 ${ℙ}^{1}$ when $Y=X$.

##### Keywords
good reduction, dynamical system, portrait, Shafarevich conjecture
Primary: 37P45
Secondary: 37P15