Ample canonical heights for endomorphisms on projective varieties

TL;DR

This paper introduces an 'ample canonical height' for endomorphisms on projective varieties, generalizing previous concepts, and explores its properties, conjectures, and applications in dynamics and number theory.

Contribution

It defines a new ample canonical height for endomorphisms, formulates a related conjecture, and proves it in specific cases, extending the understanding of dynamical systems on varieties.

Findings

Proved the conjecture for varieties with small Picard numbers, abelian varieties, and surfaces.

Showed non-density of preperiodic points over fixed number fields for certain endomorphisms.

Established a dynamical Mordell–Lang type result for intersections of dense orbits.

Abstract

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mordell--Lang type result on the intersection of two Zariski dense orbits of two endomorphisms on a common variety.