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 $\phi :\mathbb {P}^1\to \mathbb {P}^1$ defined over $K$ which have good reduction at all places outside $S$ form 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 Zariski nondensity 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.