Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt

TL;DR

This paper explores Diophantine approximation on abelian varieties, establishing equivalences, proposing weaker conjectures, and proving results for rank 1 elliptic curves over real number fields.

Contribution

It demonstrates the equivalence of Waldschmidt's conjecture to a Diophantine condition, introduces a weaker conjecture, and proves it for rank 1 elliptic curves over real fields.

Findings

Equivalence of Waldschmidt's conjecture to a Diophantine condition.

Establishment of an upper bound for the weaker conjecture.

Proof of the conjecture for rank 1 elliptic curves over real number fields.

Abstract

Following the work of Waldschmidt, we investigate problems in Diophantine approximation on abelian varieties. First we show that a conjecture of Waldschmidt for a given simple abelian variety is equivalent to a well-known Diophantine condition holding for a certain matrix related to that variety. We then posit a related but weaker conjecture, and establish the upper bound direction of that conjecture in full generality. For rank 1 elliptic curves defined over a number field $K \subset \mathbb{R}$, we then obtain a weak-type Dirichlet theorem in this setting, establish the optimality of this statement, and prove our conjecture in this case.