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arXiv · 2411.16108

Intersecting subvarieties of abelian schemes with group subschemes I

Tangli Ge
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Abstract

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.

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Tangli Ge. 2024-11-25. Intersecting subvarieties of abelian schemes with group subschemes I. https://arxiv.org/abs/2411.16108

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