Hybrid Conjecture in a Mixed Shimura variety

Abstract

The authors previously formulated the hybrid conjecture, unifying Andr\'e-Pink-Zannier and Andr\'e-Oort conjectures, and proved it in Shimura varieties of abelian type. We study its analogue for mixed Shimura varieties, and consider the prime example, the universal abelian scheme $\mathcal{A}_g\to \mathbb{A}_g$. In a radical departure from the Pila-Zannier strategy, typically applied to such questions, we employ instead a combination of equidistribution and o-minimality Our main result strictly includes the following: the Hybrid Conjecture, in particular the Andr\'e-Pink-Zannier and Andr\'e-Oort conjectures, for $\mathbb{A}_g$; the mixed Andr\'e-Oort conjecture for $\mathcal{A}_g$; and Manin-Mumford conjecture for arbitrary abelian varieties. It also yields an analogue of the ``Manin-Mumford in arithmetic pencil", a result of Baldi-Richard-Ullmo, for abelian schemes over a variety. The mixed hybrid conjecture in $\mathcal{A}_g$ also encompasses the Mordell-Lang conjecture. We actually reduce the mixed hybrid conjecture for $\mathbb{A}_g$ to its "mordellic" part. We also prove, Galois-theoretic results: uniform variants on the Ribet's Kummer theory of Abelian varieties, and Serre's theorem on Lang's conjecture.