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arXiv · 2408.16199

Orbital integrals and ideal class monoids for a Bass order

Sungmun Cho·Jungtaek Hong·Yuchan Lee
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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.

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Sungmun Cho, Jungtaek Hong, Yuchan Lee. 2024-08-29. Orbital integrals and ideal class monoids for a Bass order. https://arxiv.org/abs/2408.16199

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