Degeneracy loci and unlikely intersections in abelian schemes

Abstract

Let $\mathcal{A}\to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $\mathcal{X}\subseteq\mathcal{A}$ be a positive-dimensional subvariety. For every integer $t>0$, we prove that, outside the $t$-degeneracy locus of $\mathcal{X}$, at most finitely many points lie in flat group subschemes of relative dimension less than $t$. In particular, this intersection is not Zariski dense whenever $\mathcal{X}$ is $t$-nondegenerate. The proof combines a bounded height theorem with uniform large Galois orbit estimates, o-minimal point counting, and mixed Ax--Schanuel, following the Pila--Zannier strategy. The arithmetic input is an explicit bound for the complexity of endomorphism relations, obtained from geometry of numbers and height estimates for abelian varieties. As applications, we formulate a relative Mordell--Lang conjecture, prove it for section images in powers of certain abelian schemes, and establish injectivity of specialization of the group of sections outside a proper closed subset under variation and dimension hypotheses.