An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications

Let $X$ be a nonsingular Hermitian variety of even dimension over a finite field, and let $Λ$ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan--Tiep and Shimada, these classes span the middle cohomology