On the probabilities of local behaviors in abelian field extensions

TL;DR

This paper investigates the probabilities of local behaviors in abelian field extensions over a number field, showing that these probabilities are well-behaved and largely independent when ordered by conductor, with notable differences when ordered by discriminant.

Contribution

It provides explicit probability formulas for local completions in abelian extensions ordered by conductor and compares these with Chebotarev's theorem, highlighting independence and asymptotics.

Findings

Probabilities of local splitting are well-behaved and mostly independent.

Asymptotics for counting G-extensions with bounded conductor are established.

Independence of probabilities fails when extensions are ordered by discriminant.

Abstract

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime p of K, we determine the probability that p splits into r primes in a random G-extension of K that is unramified at p. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev's density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.