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For any polynomial $p\left(x\right)$ over $\mathbb{F}_{l}$ we determine theasymptotic density of hyperelliptic curves over $\mathbb{F}_{q}$ of genus $g$for which $p\left(x\right)$ divides the characteristic polynomial of Frobeniusacting on the $l$-torsion of the Jacobian, and give an explicit formula forthis density. We prove this result as a consequence of more general densitytheorems for quotients of Tate modules of such curves, viewed as modules overthe Frobenius. The proof involves the study of measures on $R$-modules overarbitrary rings $R$ which are finite $\mathbb{Z}_{l}$-algebras. In particularwe prove a result on the convergence of sequences of such measures, which canbe applied to the moments computed in recent work of Lipnowski-Tsimerman toobtain 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 itsproperties. Notably the measure obtained is in general not inverselyproportional to the size of the automorphism group.

Densities of eigenspaces of Frobenius and distributions of R-modules