Lower bounds for the normalized height and non-dense subsets of varieties in an abelian variety

TL;DR

This paper establishes lower bounds for the normalized height of subvarieties in abelian varieties and shows that points of bounded height are non-dense under certain conditions, extending previous results to more general settings.

Contribution

It generalizes lower bounds for normalized heights and non-density results to arbitrary subvarieties of abelian varieties, beyond elliptic curves.

Findings

Non Zariski-density of points of bounded height in certain subvarieties

Extension of previous results to higher-dimensional abelian varieties

Conditional on a conjectural lower bound for normalized height

Abstract

This work is the third part of a series of papers. In the first two we consider curves and varieties in a power of an elliptic curve. Here we deal with subvarieties of an abelian variety in general. Let V be an irreducible variety of dimension d embedded in an abelian variety A, both defined over the algebraic numbers. We say that V is weak-transverse if V is not contained in any proper algebraic subgroup of A, and transverse if it is not contained in any translate of such a subgroup. Assume a conjectural lower bound for the normalized height of V. For V transverse, we prove that the algebraic points of bounded height of V which lie in the union of all algebraic subgroups of A of codimension at least d+1 translated by the points close to a subgroup G of finite rank are non Zariski-dense in V. If G has rank zero, it is sufficient to assume that V is weak-transverse. The notion of closeness is defined using a height function.