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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.19060 |
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| _version_ | 1866912446097129472 |
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| author | Fishman, Lior Lambert, David Merrill, Keith Simmons, David |
| author_facet | Fishman, Lior Lambert, David Merrill, Keith Simmons, David |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_19060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt Fishman, Lior Lambert, David Merrill, Keith Simmons, David Number Theory 11J20 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. |
| title | Diophantine approximation on abelian varieties; a conjecture of M. Waldschmidt |
| topic | Number Theory 11J20 |
| url | https://arxiv.org/abs/2506.19060 |