An integral point $P$ of a Grassmannian $Gr(k,n)$ is a $k$-dimensional subspace such that $P \cap \mathbb{Z}^n$ is a rank $k$ sublattice of $\mathbb{Z}^n$. Its height $H$ is given by the determinant of $P \cap \mathbb{Z}^n$.
There is an interesting result of Schmidt which gives an asymptotic of $P(n,k,H)$, the number of primitive rank $k$ sublattices of determinant $\leq H$.
For this kind of counting problem, would it be possible to study it via modular forms—for instance, by constructing a Siegel modular form whose $n$th Fourier coefficients encode the counting information?
For $Gr(2,4)$, viewed as embedded in $\mathbb{P}^5$, there is only one Plücker relation which makes the problem simple. For $Gr(2,5)$, there are already 5 Plücker relations, which seem to complicate the situation.
Any reference related to a similar counting problem is appreciated.
