Weighted Discriminants and Mass Formulas for Number Fields

TL;DR

This paper introduces weighted discriminants and explores when their associated counting functions satisfy mass formulas for number fields, extending existing results and classifying such functions for specific groups.

Contribution

It generalizes the concept of discriminants to weighted versions, extends Kedlaya's results on mass formulas, and classifies all such functions for certain groups.

Findings

Proper counting functions for finite groups have mass formulas outside primes dividing group order.

Finitely many weighted discriminant counting functions have mass formulas for all primes when the group is an -group.

Complete enumeration of such functions for groups D4 and Q8.

Abstract

We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $\Gamma$ has a mass formula for the set of primes not dividing $|\Gamma|$. We also prove that if $\Gamma$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $\Gamma$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting functions for $\Gamma=D_4$ and $\Gamma=Q_8$.