Arithmetic statistics of families of integer Sn-polynomials and application to class group torsion

Abstract

We study the distributions of the splitting primes in certain families of number fields. The first and main example is the family $P_{n,N}$ of polynomials $f\in Z\left[X\right]$ monic of degree n with height less or equal then N, and then let N go to infinity. We prove an average version of the Chebotarev Density Theorem for this family. In particular, this gives a Central Limit Theorem for the number of primes with given splitting type in some ranges. As an application, we deduce some estimates for the ℓ-torsion in the class groups and for the average of ramified primes.