Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms
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| Format: | Preprint |
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2026
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| _version_ | 1866915719302610944 |
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| author | Ferrigno, Luca |
| author_facet | Ferrigno, Luca |
| contents | Let $S$ be a smooth irreducible curve over $\overline{\mathbb{Q}}$, and let $\mathcal{A} \to S$ be an abelian scheme with a curve $C \subset \mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In 2020, Barroero and Capuano proved that if $C$ is not contained in a proper subgroup scheme, then the intersection of $C$ with the union of the flat subgroup schemes of $\mathcal{A}$ of codimension at least 2 is finite. In this article, we continue to study this problem by considering the intersections with the algebraic subgroups of the CM fibers, generalizing a previous result of Barroero for fibered powers of elliptic schemes. A key ingredient of the proof is an explicit control of canonical heights under endomorphisms: for an abelian variety $A/\overline{\mathbb{Q}}$, an ample symmetric divisor $D$, and $f \in \mathrm{End}(A)$, we bound explicitly $\widehat{h}_{A, D}(f(P))$ in terms of $\widehat{h}_{A, D}(P)$ by determining the values of $λ\in \mathbb{R}$ for which the divisors $λD - f^* D$ and $f^* D - λD$ are ample. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05919 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms Ferrigno, Luca Number Theory Algebraic Geometry 11G10, 11G15, 11G50, 11U09 Let $S$ be a smooth irreducible curve over $\overline{\mathbb{Q}}$, and let $\mathcal{A} \to S$ be an abelian scheme with a curve $C \subset \mathcal{A}$, both defined over $\overline{\mathbb{Q}}$. In 2020, Barroero and Capuano proved that if $C$ is not contained in a proper subgroup scheme, then the intersection of $C$ with the union of the flat subgroup schemes of $\mathcal{A}$ of codimension at least 2 is finite. In this article, we continue to study this problem by considering the intersections with the algebraic subgroups of the CM fibers, generalizing a previous result of Barroero for fibered powers of elliptic schemes. A key ingredient of the proof is an explicit control of canonical heights under endomorphisms: for an abelian variety $A/\overline{\mathbb{Q}}$, an ample symmetric divisor $D$, and $f \in \mathrm{End}(A)$, we bound explicitly $\widehat{h}_{A, D}(f(P))$ in terms of $\widehat{h}_{A, D}(P)$ by determining the values of $λ\in \mathbb{R}$ for which the divisors $λD - f^* D$ and $f^* D - λD$ are ample. |
| title | Unlikely intersections with CM abelian varieties in a family and explicit bounds for canonical heights under endomorphisms |
| topic | Number Theory Algebraic Geometry 11G10, 11G15, 11G50, 11U09 |
| url | https://arxiv.org/abs/2601.05919 |