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

A trace formula for the distribution of rational $G$-orbits in ramified covers, adapted to representation stability