Unlikely intersections with Hecke translates of a special subvariety

TL;DR

This paper advances the understanding of the Zilber-Pink conjecture in Shimura varieties by proving new cases involving Hecke translates of special subvarieties, both conditionally and unconditionally, using sophisticated arithmetic and transcendence techniques.

Contribution

It proves new cases of the Zilber-Pink conjecture for intersections with Hecke translates, extending previous results to broader settings with conditional and unconditional proofs.

Findings

Conditional proof of Zilber-Pink for curves and Hecke translates

Unconditional proof for intersections with Hecke correspondences on moduli spaces

Extension of Habegger and Pila's results to new contexts

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-W\"ustholz isogeny theorem, comparison between Faltings and Weil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.