On quadratic rational maps with prescribed good reduction

TL;DR

This paper studies quadratic rational maps over number fields with good reduction outside a finite set of places, showing density results in moduli space and establishing non-density results for maps with specific fixed-point structures.

Contribution

It proves Zariski-density of quadratic maps with good reduction outside S in moduli space and establishes non-density results for maps with special unramified fixed points or 2-cycle structures.

Findings

Quadratic maps with good reduction outside S are Zariski-dense in moduli space.

Maps with double unramified fixed-point structure have non-Zariski-density with good reduction outside S.

Similar non-density results hold for maps with unramified 2-cycle structure.

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 $S$ 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.