  We compute the $p$-adic densities of points with a given splitting type along
a (generically) finite map, analogous to the classical Chebotarev theorem over
number fields and function fields. Under some mild hypotheses, we prove that
these densities satisfy a functional equation in the size of the residue field.
This functional equation is a direct reflection of Poincar\'e duality in
\'etale cohomology. As a consequence, we prove a conjecture of Bhargava,
Cremona, Fisher, and Gajovi\'c on factorization densities of p-adic
polynomials.
  The key tool is the notion of admissible pairs associated to a group, which
we use as an invariant of the inertia and decomposition action of a local field
on the fibers of the finite map. We compute the splitting densities by M\"obius
inverting certain p-adic integrals along the poset of admissible pairs. The
conjecture on factorization densities follows immediately for tamely ramified
primes from our general results. We reduce the complete conjecture (including
the wild primes) to the existence of an explicit "Tate-type" resolution of the
"resultant locus" over the integers and complete the proof of the conjecture by
constructing this resolution.
