Intersecting subvarieties of abelian schemes with group subschemes I
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915032902664192 |
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| author | Ge, Tangli |
| author_facet | Ge, Tangli |
| contents | In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all flat group subschemes of relative dimension at most $t$, gives a set of bounded total height. Our main tools include the Ax--Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan--Zhang. As two applications, we generalize Silverman's specialization theorem to a higher dimensional base, and establish a bounded height result towards Zhang's ICM Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16108 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersecting subvarieties of abelian schemes with group subschemes I Ge, Tangli Number Theory Algebraic Geometry In this paper, we establish the following family version of Habegger's bounded height theorem on abelian varieties: a locally closed subvariety of an abelian scheme with Gao's $t^{\mathrm{th}}$ degeneracy locus removed, intersected with all flat group subschemes of relative dimension at most $t$, gives a set of bounded total height. Our main tools include the Ax--Schanuel theorem, and intersection theory of adelic line bundles as developed by Yuan--Zhang. As two applications, we generalize Silverman's specialization theorem to a higher dimensional base, and establish a bounded height result towards Zhang's ICM Conjecture. |
| title | Intersecting subvarieties of abelian schemes with group subschemes I |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2411.16108 |