Orbital integrals and ideal class monoids for a Bass order

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.