Vojta's conjecture, heights associated with subschemes, and primitive prime divisors in arithmetic dynamics

TL;DR

Under Vojta's conjecture, the paper establishes a condition for the ratio of heights related to orbits of a self-morphism on a variety to tend to zero, linking it to the Dynamical Mordell-Lang conjecture and primitive prime divisors.

Contribution

It proposes a new conjecture on height limits in arithmetic dynamics assuming Vojta's conjecture, connecting it to existing conjectures and primitive prime divisor problems.

Findings

A sufficient condition for the height ratio limit to be zero is given.

The conjecture implies the Dynamical Mordell-Lang conjecture for certain endomorphisms.

Applications to primitive prime divisors in orbits are discussed.

Abstract

Assuming Vojta's conjecture, we give a sufficient condition for the limit \[ \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x))} \] is equal to zero, where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$, $h_{H}$ is an ample height function on $X$, and $h_{Y}$ is a global height function associated with a closed subscheme $Y \subset X$ of codimension at least two. Based on this, we propose a conjecture on a sufficient condition for the limit to be zero. We point out that our conjecture implies Dynamical Mordell-Lang conjecture for endomorphisms on $\mathbb{P}^{2}_{\overline{\mathbb{Q}}}$. We also discuss applications of Vojta's conjecture with truncated counting function to the problem of the existence of primitive prime divisors of coordinates of orbits of $f$