A note on Bremner's conjecture and uniformity
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866913139440746496 |
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| author | Garcia-Fritz, Natalia Pasten, Hector |
| author_facet | Garcia-Fritz, Natalia Pasten, Hector |
| contents | In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang theorem of Gao--Ge--Kühne. Thus, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a much more direct proof of this last statement, using the height-uniform Mordell theorem of Dimitrov--Gao--Habegger. The method is flexible and we give a new application of these ideas to $x$-coordinates in finitely generated multiplicative groups and geometric progressions; connections to a possible semiabelian uniform Mordell--Lang are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04850 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on Bremner's conjecture and uniformity Garcia-Fritz, Natalia Pasten, Hector Number Theory Primary: 11G05, Secondary: 11B25, 14G05 In 1998, Bremner conjectured that elliptic curves over the rationals having long sequences of distinct rational points whose $x$-coordinates are in arithmetic progression, have large rank. This was proved some years ago in a strong form as a consequence of previous work by the authors, by a combination of Nevanlinna theory and the uniform Mordell--Lang theorem of Gao--Ge--Kühne. Thus, if the ranks of elliptic curves over the rationals are uniformly bounded, then so are the lengths of the aforementioned arithmetic progressions. In this note we give a much more direct proof of this last statement, using the height-uniform Mordell theorem of Dimitrov--Gao--Habegger. The method is flexible and we give a new application of these ideas to $x$-coordinates in finitely generated multiplicative groups and geometric progressions; connections to a possible semiabelian uniform Mordell--Lang are also discussed. |
| title | A note on Bremner's conjecture and uniformity |
| topic | Number Theory Primary: 11G05, Secondary: 11B25, 14G05 |
| url | https://arxiv.org/abs/2604.04850 |