Parameter Height bounds for the Zilber Pink conjecture for PEL types III and IV

We prove the Zilber-Pink conjecture to the intersection of an irreducible Hodge generic algebraic subvariety $V\subset {\mathcal{A}}_g$ with special subvarieties of all simple PEL types other than $\mathbb{Z}$, under the assumption of the Large Galois Orbits conjecture. In particular, we establish parameter height bounds for the arithmetic ingredients of the Pila-Zannier strategy in the case of Albert types III and IV. This paper is a sequel to Daw and Orr’s paper “Lattices with skew-Hermitian forms over division algebras and unlikely intersections” 2023.