The average number of integral points in orbits

Over a number field $K$, a celebrated result of Silverman states that if $\varphi (z) \in K(z)$ is a rational function whose second iterate is not a polynomial, the set of $S$-integral points in the orbit ${\mathrm{Orb}_{\varphi} (P) = \lbrace \varphi^n (P) \rbrace}_{n \geqslant 0}$ is finite for all $P \in \mathbb{P}^1 (K)$. In this paper, we show that if we vary $\varphi$ and $P$ in a suitable family, the number of $S$-integral points in $\mathrm{Orb}_{\varphi} (P)$ is absolutely bounded. In particular, if we fix $\varphi$ and vary the basepoint $P \in \mathbb{P}^1 (K)$, then we show that $\# (\mathrm{Orb}_{\varphi} (P) \cap \mathcal{O}_{K,S})$ is zero on average. Finally, we prove a zero-average result in general, assuming a standard height uniformity conjecture in arithmetic geometry.

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Dedicated to Joseph H. Silverman on the occasion of his 60th birthday

Received 5 May 2016

Accepted 16 November 2017

Published 7 June 2019