Calculating The Local Ideal Class Monoid and Gekeler Ratios

Let $A={\mathbb{F}}_q[T]$, $\mathfrak{p}\subset A$ prime, $f(x)\in A[x]$ irreducible and set $R=A[x]\mathrm{/}f(x)$. Denote its completion by $R_{\mathfrak{p}}$. The ideal class monoid $\text{ICM}(R_{\mathfrak{p}})$ is the set of fractional $R_{\mathfrak{p}}$ ideals modulo the principal $R_{\mathfrak{p}}$ ideals. We provide an algorithm to compute $\text{ICM}(R_{\mathfrak{p}})$. In the process we also get algorithms to compute the overorders and weak equivalence classes of $R_{\mathfrak{p}}$. We then use the algorithms to compute the product of local Gekeler ratios ${\prod }_{\mathfrak{p}\subset A}{v}_{\mathfrak{p}}(f)={\prod }_{\mathfrak{p}\subset A}{\lim }_{n\to \mathrm{\infty }}\frac{\mathrm{\mid }\{M\in {\text{Mat}}_r(A\mathrm{/}{\mathfrak{p}}^n)\mid \text{charpoly}(M)=f\}}{\mathrm{\mid }{\text{SL}}_r(A\mathrm{/}{\mathfrak{p}}^n)\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathfrak{p}{\mathrm{\mid }}^{n(r-1)}}$. This provides part of an algorithm to compute the weighted size of an isogeny class of Drinfeld modules.