Intersecting subvarieties of abelian schemes with group subschemes I

TL;DR

This paper extends Habegger's bounded height theorem to families of abelian varieties, using advanced tools like Ax--Schanuel and adelic intersection theory, with applications to Silverman's specialization theorem and Zhang's ICM Conjecture.

Contribution

It introduces a family version of Habegger's theorem, incorporating degeneracy loci and flat group subschemes, advancing height bounds in abelian schemes.

Findings

Established a family bounded height theorem for abelian schemes.

Generalized Silverman's specialization theorem to higher dimensions.

Proved a bounded height result related to Zhang's ICM Conjecture.

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.