Singular intersections in families of abelian varieties
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908529473880064 |
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| author | Ottolini, Nicola |
| author_facet | Ottolini, Nicola |
| contents | Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove that the set of points in which $\mathcal{C}$ intersects proper flat subgroup schemes of $\mathcal{A}$ tangentially is finite. The crucial case of elliptic curves already follows from a result by Corvaja, Demeio, Masser and Zannier: in this case we provide an alternative proof using the Pila-Zannier method. Such a proof may lead to an effective result using an effective point-counting theorem. This fits in the framework of the so-called problems of unlikely intersections, and can be seen as a variation of the relative Pink conjecture for abelian varieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_15344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singular intersections in families of abelian varieties Ottolini, Nicola Number Theory Algebraic Geometry 11G10, 11G05, 11G50, 14C17 Let $S$ be a smooth irreducible curve defined over $\overline{\mathbb{Q}}$, let $\mathcal{A}$ be an abelian scheme over $S$ and $\mathcal{C}$ a curve inside $\mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In this paper we prove that the set of points in which $\mathcal{C}$ intersects proper flat subgroup schemes of $\mathcal{A}$ tangentially is finite. The crucial case of elliptic curves already follows from a result by Corvaja, Demeio, Masser and Zannier: in this case we provide an alternative proof using the Pila-Zannier method. Such a proof may lead to an effective result using an effective point-counting theorem. This fits in the framework of the so-called problems of unlikely intersections, and can be seen as a variation of the relative Pink conjecture for abelian varieties. |
| title | Singular intersections in families of abelian varieties |
| topic | Number Theory Algebraic Geometry 11G10, 11G05, 11G50, 14C17 |
| url | https://arxiv.org/abs/2506.15344 |