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arXiv · 2506.19060

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

Lior Fishman·David Lambert·Keith Merrill·David Simmons
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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.

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Lior Fishman, David Lambert, Keith Merrill, David Simmons. 2025-06-23. Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt. https://arxiv.org/abs/2506.19060

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