On quadratic rational maps with prescribed good reduction

Abstract

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\varphi :{\mathbb{P}}^1\to {\mathbb{P}}^1$ defined over $K$ which have good reduction at all places outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal{M}}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside $S$. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure.

This version fixes a few LaTeX errors