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This is a Quantitative Study study.

October 2, 2025Open Access

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

Key Points

A conjecture of Waldschmidt is equivalent to a diophantine condition for a certain matrix related to the variety.
Optimality of the weak-type Dirichlet theorem is established for rank 1 elliptic curves over a number field.
The upper bound of a related conjecture is demonstrated in full generality, showcasing new findings in diophantine approximation.
The exploration of diophantine approximation problems contributes meaningfully to the understanding of abelian varieties.

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 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.

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