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

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 $f:X⟶X$ 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 ${\mathbb{P}}_{\overline{\mathbb{Q}}}^2$. 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 $f$

arXiv comment30 pages