Intersecting subvarieties of abelian schemes with group subschemes I

We establish the following family version of Habegger’s bounded height theorem on abelian varieties [Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Invent. Math. 176 (2009a), 405–447]: a locally closed subvariety of an abelian scheme with Gao’s tth degeneracy locus [Gao, Generic rank of Betti map and unlikely intersections, Compositio Math. 156 (2020a), 2469–2509] 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 and Zhang [Adelic line bundles on quasi-projective varieties, Annals of Mathematics Studies, vol. 221 (Princeton University Press, Princeton, NJ, 2026)]. As two applications, we generalize Silverman’s specialization theorem [Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211] to a higher-dimensional base, and establish a bounded height result towards Zhang’s ICM conjecture [Zhang, Small points and Arakelov theory, in Proceedings of the international congress of mathematicians, Vol. II (Berlin, 1998), Extra Vol. II (1998), 217–225].