Unlikely intersections with Hecke translates of a special subvariety

Abstract

We prove some cases of the Zilber–Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber–Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed special subvariety, conditional on arithmetic conjectures. Secondly, we prove the conjecture unconditionally for intersections between a curve and the union of Hecke correspondences on the moduli space of principally polarised abelian varieties, subject to some technical hypotheses. This generalises results of Habegger and Pila on the Zilber–Pink conjecture for products of modular curves.

The conditional proof uses the Pila–Zannier method, relying on a point-counting theorem of Habegger and Pila and a functional transcendence result of Gao. The unconditional results are deduced from this using a variety of arithmetic ingredients: the Masser–Wüstholz isogeny theorem, comparison between Faltings andWeil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.