Vol. 295, No. 1, 2018

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

Joseph H. Silverman

Vol. 295 (2018), No. 1, 145–190

Let K be a number field, let S be a finite set of places of K, and let RS be the ring of S-integers of K. A K-morphism f : K1 K1 has simple good reduction outside S if it extends to an RS-morphism [b]RS1 RS1. A finite Galois invariant subset X K1(K̄) has good reduction outside S if its closure in RS1 is étale over RS. We study triples (f,Y,X) with X = Y f(Y ). We prove that for a fixed K, S, and d, there are only finitely many PGL2(RS)-equivalence classes of triples with deg(f) = d and PY ef(P) 2d + 1 and X having good reduction outside S. We consider refined questions in which the weighted directed graph structure on f : Y X is specified, and we give an exhaustive analysis for degree 2 maps on 1 when Y = X.

good reduction, dynamical system, portrait, Shafarevich conjecture
Mathematical Subject Classification 2010
Primary: 37P45
Secondary: 37P15
Received: 4 April 2017
Revised: 15 September 2017
Accepted: 15 September 2017
Published: 13 March 2018
Joseph H. Silverman
Department of Mathematics
Brown University
Providence, RI
United States