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

TL;DR

This paper proves cases of the Zilber-Pink conjecture for certain PEL types by establishing parameter height bounds, assuming the Large Galois Orbits conjecture, advancing the understanding of unlikely intersections in algebraic geometry.

Contribution

It provides the first parameter height bounds for PEL types III and IV, supporting the Zilber-Pink conjecture under specific assumptions.

Findings

Proved the Zilber-Pink conjecture for PEL types III and IV.

Established parameter height bounds for arithmetic components.

Assumed the Large Galois Orbits conjecture for the results.

Abstract

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.