  Motivated by the incidence problems between points and flats of a symplectic
polar space, we study a large class of submodules of the space of functions on
the standard module of a finite symplectic group of odd characteristic. Our
structure results on this class of submodules allow us to determine the
$p$-ranks of the incidence matrices between points and flats of the symplectic
polar space. In particular, we give an explicit formula for the $p$-rank of the
generalized quadrangle ${\rm W}(3,q)$, where $q$ is an odd prime power.
Combined with the earlier results of Sastry and Sin on the 2-rank of ${\rm
W}(3,2^t)$, it completes the determination of the $p$-ranks of ${\rm W}(3,q)$.
