A Chebotarev Density Theorem over Local Fields

TL;DR

This paper extends the Chebotarev density theorem to local fields, establishing a functional equation for densities of points with specific splitting types and proving a conjecture on p-adic polynomial factorization densities.

Contribution

It introduces a new framework using admissible pairs to compute p-adic splitting densities and proves a conjecture on factorization densities, including wild primes.

Findings

Established a Chebotarev-type density theorem over local fields.

Proved a functional equation reflecting Poincaré duality in étale cohomology.

Confirmed the conjecture on p-adic polynomial factorization densities for tamely ramified primes.

Abstract

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.