2022/12/01 by G Asvin, Yifan Wei, G, Asvin +3
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2212.00294
openalex publication_date 2022/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Poincaré duality in étale cohomology. As a consequence, we prove a conjecture of Bhargava, Cremona, Fisher, and Gajović 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 Möbius 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.