论文标题
chebotarev密度定理在本地领域
A Chebotarev Density Theorem over Local Fields
论文作者
论文摘要
我们计算沿（一般）有限的映射给定的分裂类型的$ p $ - 亚种密度，类似于数字字段和功能字段的经典chebotarev theorem。在一些轻度的假设下，我们证明这些密度满足残基场大小的功能方程。该功能方程是对典型二元组的直接反映。结果，我们证明了Bhargava，Cremona，Fisher和Gajović的猜想，涉及P-ADIC多项式的分解密度。 关键工具是与一组相关的可接受对的概念，我们将其用作有限图纤维上局部磁场的惯性和分解作用的不变。我们通过将某些P-ADIC积分沿沿允许对的Poset倒置来计算分裂密度。从我们的一般结果中，对分解密度的猜想立即遵循了腐烂的刺激。我们将完整的猜想（包括野生素数）减少到整数上“结果基因座”的明确“泰特型”分辨率的存在，并通过构造该分辨率来完成猜想的证明。
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.
