  For any polynomial $p\left(x\right)$ over $\mathbb{F}_{l}$ we determine the
asymptotic density of hyperelliptic curves over $\mathbb{F}_{q}$ of genus $g$
for which $p\left(x\right)$ divides the characteristic polynomial of Frobenius
acting on the $l$-torsion of the Jacobian, and give an explicit formula for
this density. We prove this result as a consequence of more general density
theorems for quotients of Tate modules of such curves, viewed as modules over
the Frobenius. The proof involves the study of measures on $R$-modules over
arbitrary rings $R$ which are finite $\mathbb{Z}_{l}$-algebras. In particular
we prove a result on the convergence of sequences of such measures, which can
be applied to the moments computed in recent work of Lipnowski-Tsimerman to
obtain the above results. We also extend the random model for finite
$R$-modules proposed in that work to such rings $R$, and prove several of its
properties. Notably the measure obtained is in general not inversely
proportional to the size of the automorphism group.
