The canonical arithmetic height of subvarieties of an abelian variety over a finitely generated field

TL;DR

This paper defines the canonical height for subvarieties of an abelian variety over finitely generated fields and characterizes when this height is zero, linking it to translations of abelian subvarieties by torsion points.

Contribution

It introduces a new definition of the canonical height for subvarieties over finitely generated fields and establishes a characterization of zero height subvarieties.

Findings

Canonical height is zero if and only if the subvariety is a translation of an abelian subvariety by a torsion point.

Provides a foundational tool for studying subvarieties in arithmetic geometry over finitely generated fields.

Abstract

This paper is the sequel of our paper "Arithmetic height functions over finitely generated fields" (cf. math.NT/9809016). In this paper, we define the canonical height of subvarieties of an abelian variety over a finitely generated field over Q. We also prove that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point.