The average number of integral points in orbits

Abstract

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

arXiv commentWe strengthen the main result to non-isotrivial families