Orbital integrals and ideal class monoids for a Bass order

TL;DR

This paper provides a closed formula for counting fractional ideals of Bass orders in number fields, utilizing orbital integrals and local-global methods to enumerate overorders.

Contribution

It introduces a novel explicit formula for the number of fractional ideals of Bass orders, connecting orbital integrals with ideal class monoids.

Findings

Derived a closed formula for fractional ideals count

Connected orbital integrals with ideal class enumeration

Provided explicit enumeration method for orders containing a Bass order

Abstract

A Bass order is an order of a number field whose fractional ideals are generated by two elements. The majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is fourth-power-free in $\mathbb{Z}$, is a Bass order. In this paper, we will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. Since $R$ is a Bass order, this is the same as the number of overorders of $R$. We will also explain the explicit enumeration of all orders containing $R$. Our method is based on the local-global argument and the exhaustion argument, using orbital integrals for $\mathfrak{gl}_n$ as a mass formula.