  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.
