A quantitative dynamical Zhang fundamental inequality and Bogomolov-type problems

TL;DR

This paper develops a quantitative version of Zhang's fundamental inequality for heights in dynamical systems, leading to effective gap principles and uniformity results in the dynamical Bogomolov conjecture across various settings.

Contribution

It introduces a quantitative inequality for heights, extends gap principles to abelian varieties over function fields, and provides explicit constants for polynomial endomorphisms.

Findings

Established a quantitative Zhang inequality for polarizable endomorphisms.

Derived effective gap principles for abelian varieties and polynomial endomorphisms.

Provided explicit constants for uniformity in the dynamical Bogomolov conjecture.

Abstract

We prove a quantitative version of Zhang's fundamental inequality for heights attached to polarizable endomorphisms. As an application, we obtain a gap principle for the N\'eron-Tate height on abelian varieties over function fields of arbitrary transcendence degree and characteristic zero, extending the result of Gao-Ge-K\"uhne. We also establish instances of effective gap principles for regular polynomial endomorphisms of $\mathbb{P}^2$, in the sense that all constants can are explicit. These yield effective instances of uniformity in the dynamical Bogomolov conjecture in both the arithmetic and geometric settings, including examples in prime characteristic.