Bound for preperiodic points for maps with good reduction

TL;DR

This paper establishes new bounds on the number of preperiodic points for rational functions over number fields, improving previous results and showing that these bounds depend only on bad reduction places under certain conditions.

Contribution

It provides improved bounds for preperiodic points based on bad reduction places and degree, and demonstrates the sharpness of these bounds with counterexamples.

Findings

New bounds for preperiodic points depending on bad reduction places.

Improved bounds over previous work by Canci and Paladino.

Counterexamples showing bounds are sharp under certain conditions.

Abstract

Let $K$ be a number field and let $\phi$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the places of bad reduction for $\phi$ (including the archimedan places). Let $Per(\phi,K)$, $PrePer(\phi, K)$, and $Tail(\phi,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $\phi$, respectively. The present paper presents two main results. The first result gives a bound for $|PrePer(\phi,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $\phi$. This bound significantly improves a previous bound given by J. Canci and L. Paladino 2014. For the second result, assuming that $|Per(\phi,K)| \geq 4$ (resp. $|Tail(\phi,K)| \geq 3$), we prove bounds for $|Tail(\phi,K)|$ (resp. $|Per(\phi,K)|$) that depend only on the number of places of bad reduction $|S|$ (and not on the degree $d$). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when $|Per(\phi,K)| < 4$ (resp. $|Tail(\phi,K)| < 3$).