An upper bound for the size of the ideal class monoid

Abstract: The ideal class monoid for an order $R$ in a finite field extension $E/F$ of a number field, denoted by $\overline{\mathrm{Cl}}(R)$, 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 $#\overline{\mathrm{Cl}}(R)$, in terms of the class number of $E$ and (local) orbital integrals for $\mathfrak{gl}_n$. We also describe an upper bound for the class number of $E$ in terms of the Minkowski bound. When $[E:F]\leq 3$ 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 $R=\mathbb{Z}[x]/(x<sup>3-mx<sup>2+(m-1)x-1)$ 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 $$\frac{2}{3^5} \Delta_R^{{\frac{1}{2}}\cdot} \Delta_E<sup>{\frac{3}{2}}$$, when $\Delta_E&gt;3075$.