An upper bound for the size of the ideal class monoid

The ideal class monoid for an order $R$ in a finite field extension $E\mathrm{/}F$ of a number field, denoted by $\overline{\mathrm{C}\mathrm{l}}(R)$, is a fundamental object to study in number theory which has useful applications in algebraic geometry and topology. In this paper, we describe an upper bound for $\mathrm{\#}\overline{\mathrm{C}\mathrm{l}}(R)$, in terms of the class number of $E$ and (local) orbital integrals for ${\mathfrak{g}\mathfrak{l}}_n$. We also describe an upper bound for the class number of $E$ in terms of the Minkowski bound. When $[E:F]\le 3$ or when $R$ is a Bass order, we refine our upper bound, using a known formula for local orbital integrals in the authors’ previous work. In particular, if $R=\mathbb{Z}[x]\mathrm{/}(x^3-m{x}^2+(m-1)x-1)$ with $m\in \mathbb{Z}$ which arises in a study of Cappell-Shaneson homotopy 4-spheres in topology, then we further refine our upper bound in terms of the discriminants of $R$ and $E$, which is $\frac{2}{3^5}{\mathrm{\Delta }}_R^{\frac{1}{2}}\cdot {\mathrm{\Delta }}_E^{\frac{3}{2}}$, when ${\mathrm{\Delta }}_E>3075$.