Almost all quadratic twists of an elliptic curve have no integral points
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916085439135744 |
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| author | Browning, Tim Chan, Stephanie |
| author_facet | Browning, Tim Chan, Stephanie |
| contents | For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04375 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Almost all quadratic twists of an elliptic curve have no integral points Browning, Tim Chan, Stephanie Number Theory 11D45 (11D25, 11G05) For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E_D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall-Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown's analysis of Selmer group statistics for the congruent number curve. |
| title | Almost all quadratic twists of an elliptic curve have no integral points |
| topic | Number Theory 11D45 (11D25, 11G05) |
| url | https://arxiv.org/abs/2401.04375 |