Intersecting subvarieties of abelian schemes with group subschemes I

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 $t^{\mathrm{t}\mathrm{h}}$ 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.