Orbital integrals and ideal class monoids for a Bass order

Abstract

A Bass order is an order in a number field for which every fractional ideal can be generated by two elements. Examples include quadratic orders, orders containing the maximal order of a subfield $F$ with $[E:F]=2$, and orders whose discriminant is fourth-power-free. We prove a closed conductor formula for the number of fractional ideals of a Bass order $R$ modulo multiplication by invertible ideals. For a Bass order, this number is also the number of overorders of $R$, and we give an explicit conductor parametrization of all overorders. The proof combines the classification of local Bass overorders with a local--global argument. Orbital integrals give the corresponding weighted mass formulas, including the split local case. We also prove, by a smoothening procedure, a geometric orbital integral theorem for the relevant integral model; this theorem is presented separately in Section 4.