Calculating The Local Ideal Class Monoid and Gekeler Ratios

Abstract

Let $A = \mathbb{F}_q[T]$, $\mathfrak{p} \subset A$ prime, $f(x) \in A[x]$ irreducible and set $R = A[x]/f(x)$. 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 $\prod_{\mathfrak{p} \subset A} v_\mathfrak{p}(f) = \prod_{\mathfrak{p} \subset A} \lim_{n \rightarrow \infty} \frac{|\{M \in \text{Mat}_r(A/\mathfrak{p}^n)\mid \text{charpoly}(M)=f\}}{|\text{SL}_r(A/\mathfrak{p}^n)|/|\mathfrak{p}|^{n(r-1)}}$. This provides part of an algorithm to compute the weighted size of an isogeny class of Drinfeld modules.