Lattices with skew-Hermitian forms over division algebras and unlikely intersections
Number Theory
2023-07-20 v3 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.2111.13056,
title = {Lattices with skew-Hermitian forms over division algebras and unlikely intersections},
author = {Christopher Daw and Martin Orr},
journal= {arXiv preprint arXiv:2111.13056},
year = {2023}
}
Comments
63 pages, corrections to typos and other minor errors