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

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.

63 pages, corrections to typos and other minor errors

References in corpus (2)

• Quantitative reduction theory and unlikely intersections
• Effective André-Oort for non-compact curves in Hilbert modular varieties

Cited by in corpus (1)

• Some cases of the Zilber-Pink conjecture for curves in ${\mathcal{A}}_g$