Counting abelian number fields with restricted ramification type

TL;DR

This paper counts abelian number fields with specific ramification restrictions, providing explicit formulas and interpreting results through the lens of the Batyrev-Manin conjecture, including distribution properties.

Contribution

It introduces a new counting method for abelian number fields with restricted tame inertia and formulates an explicit leading constant, connecting to the Batyrev-Manin conjecture.

Findings

Derived an explicit formula for counting abelian number fields with ramification restrictions.

Established equidistribution of such number fields with respect to local conditions.

Reinterpreted the counting problem as a question about integral points on BG.

Abstract

We count abelian number fields ordered by arbitrary height function whose generator of tame inertia is restricted to lie in a given subset of the Galois group, and find an explicit formula for the leading constant. We interpret our results as a version of the Batyrev-Manin conjecture on $BG$ and rephrase our result on number fields with restricted ramification type in terms of integral points on $BG$. We also prove that such number fields are equidistributed with respect to suitable collections of infinitely many local conditions.