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arXiv · 1208.5794

On quadratic rational maps with prescribed good reduction

Clayton Petsche·Brian Stout
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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.

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Clayton Petsche, Brian Stout. 2012-08-28. On quadratic rational maps with prescribed good reduction. https://arxiv.org/abs/1208.5794

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