Uniform bounds for preperiodic points in families of twists

Abstract

Let ϕ \phi be a morphism of P N \mathbb {P}^N defined over a number field K . K. We prove that there is a bound B B depending only on ϕ \phi such that every twist of ϕ \phi has no more than B B K K -rational preperiodic points. (This result is analogous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q \mathbb {Q} , we find the bound B B explicitly.