Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture
Number Theory
2012-09-18 v1 Dynamical Systems
Abstract
A primitive prime divisor of an element a_n of a sequence (a_1,a_2,a_3,...) is a prime P that divides a_n, but does not divide a_m for all m < n. The Zsigmondy set Z of the sequence is the set of n such that a_n has no primitive prime divisors. Let f : X --> X be a self-morphism of a variety, let D be an effective divisor on X, and let P be a point of X, all defined over the algebraic closure of Q. We consider the Zsigmondy set Z(X,f,P,D) of the sequence defined by the arithmetic intersection of the f-orbit of P with D. Under various assumptions on X, f, D, and P, we use Vojta's conjecture with truncated counting function to prove that the set of points f^n(P) with n in Z(X,f,P,D) is not Zariski dense in X.
Keywords
Cite
@article{arxiv.1209.3491,
title = {Primitive Divisors, Dynamical Zsigmondy Sets, and Vojta's Conjecture},
author = {Joseph H. Silverman},
journal= {arXiv preprint arXiv:1209.3491},
year = {2012}
}
Comments
18 pages