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

TL;DR

This paper investigates the distribution of splitting primes in families of integer polynomials and applies these results to estimate torsion in class groups, establishing a Central Limit Theorem for prime splitting counts.

Contribution

It proves an average Chebotarev Density Theorem for polynomial families and derives new probabilistic results on prime splitting and class group torsion.

Findings

Establishes a Central Limit Theorem for prime splitting in polynomial families.

Provides estimates for class group torsion and ramified primes.

Develops an average distribution result for splitting primes in number fields.

Abstract

We study the distributions of the splitting primes in certain families of number fields. The first and main example is the family Pn,N of integer polynomials 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 Central Limit Theorem for the number of primes with given splitting type in some ranges. As an application, we deduce some estimates for the torsion in the class groups and for the average of ramified primes.