  A standard observation in algebraic geometry and number theory is that a
ramified cover of an algebraic variety $\widetilde{X}\rightarrow X$ over a
finite field $F_q$ furnishes the rational points $x\in X(F_q)$ with additional
arithmetic structure: the Frobenius action on the fiber over $x$. For example,
in the case of the Vieta cover of polynomials over $F_q$ this structure
describes a polynomial's irreducible decomposition type.
  Furthermore, the distribution of these Frobenius actions is encoded in the
cohomology of $\widetilde{X}$ via the Grothendieck-Lefschetz trace formula.
This note presents a version of the trace formula that is suited for studying
the distribution in the context of representation stability: for certain
sequences of varieties $(\widetilde{X}_n)$ the cohomology, and therefore the
distribution of the Frobenius actions, stabilizes in a precise sense.
  We conclude by fully working out the example of the Vieta cover of the
variety of polynomials. The calculation includes the distribution of cycle
decompositions on cosets of Young subgroups of the symmetric group, which might
be of independent interest.
