Lattices with skew-Hermitian forms over division algebras and unlikely intersections

TL;DR

This paper investigates lattices with skew-Hermitian forms over division algebras, establishing structural results and applying them to prove cases of the Zilber-Pink conjecture related to unlikely intersections in certain Shimura varieties.

Contribution

It provides new structural results on lattices with skew-Hermitian forms over division algebras and applies these to prove cases of the Zilber-Pink conjecture under certain conditions.

Findings

Lattices contain orthogonal bases with bounded index for Albert types I and II.

Proved the Zilber-Pink conjecture for curves intersecting special subvarieties of simple PEL type I and II.

Established the Galois orbits conjecture for specific cases of type II.

Abstract

This paper has two objectives. First, we study lattices with skew-Hermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a lattice contains an "orthogonal" basis for a sublattice of effectively bounded index. Second, we apply this result to obtain new results in the field of unlikely intersections. More specifically, we prove the Zilber-Pink conjecture for the intersection of curves with special subvarieties of simple PEL type I and II under a large Galois orbits conjecture. We also prove this Galois orbits conjecture for certain cases of type II.