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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/2508.06680 |
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| _version_ | 1866909730715205632 |
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| author | Ulmer, Douglas Voloch, José Felipe |
| author_facet | Ulmer, Douglas Voloch, José Felipe |
| contents | Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urzúa over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$ (an ``unlikely intersection''), and more precisely, an exact formula for the weighted number of tangencies between $P$ and elements of the ``Betti foliation''. This work used analytic techniques that apparently do not generalize to positive characteristic. In this paper, we extend their work to characteristic $p$, and we develop a second approach to tangency properties of algebraic curves on a complex elliptic surface, yielding a new family of unlikely intersections with a strong connection to a famous homomorphism of Manin. We also correct inaccuracies in the literature about this homomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06680 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New unlikely intersections on elliptic surfaces Ulmer, Douglas Voloch, José Felipe Algebraic Geometry Consider a Jacobian elliptic surface $E \to C$ with a section $P$ of infinite order. Previous work of the first author and Urzúa over the complex numbers gives a bound on the number of tangencies between $P$ and a torsion section of $E$ (an ``unlikely intersection''), and more precisely, an exact formula for the weighted number of tangencies between $P$ and elements of the ``Betti foliation''. This work used analytic techniques that apparently do not generalize to positive characteristic. In this paper, we extend their work to characteristic $p$, and we develop a second approach to tangency properties of algebraic curves on a complex elliptic surface, yielding a new family of unlikely intersections with a strong connection to a famous homomorphism of Manin. We also correct inaccuracies in the literature about this homomorphism. |
| title | New unlikely intersections on elliptic surfaces |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2508.06680 |