Approximation on abelian varieties by its subgroups

TL;DR

This paper provides an algebro-geometric approach to Faltings' diophantine approximation theorem on abelian varieties, extending previous results from rational points over number fields to finitely generated subgroups.

Contribution

It introduces a new formulation and proves a generalized estimate for diophantine approximation on abelian varieties involving finitely generated subgroups.

Findings

Established a lower bound for the distance of points outside a subvariety

Extended Faltings' theorem to finitely generated subgroups

Provided a new algebro-geometric framework for diophantine approximation

Abstract

In this paper, we introduce an algebro-geometric formulation for Faltings' theorem on diophantine approximation on abelian varieties using an improvement of Faltings-Wustholz observation over number fields. In fact, we prove that, for any geometrically irreducible sub-variety E of an abelian variety A and any finitely generated subgroup F of A(C) we have an estimate of the form d_v(E;x) >cH(x)^d for for some constant c where d_v(E;x) denotes the distance of a point x in F outside E and v is a place of K. This was proved before, only for F being the set of rational points of A over a number field.