Calculating The Local Ideal Class Monoid and Gekeler Ratios

TL;DR

This paper develops algorithms to compute the local ideal class monoid and Gekeler ratios for polynomial rings over finite fields, aiding the enumeration of isogeny classes of Drinfeld modules.

Contribution

It introduces new algorithms for computing the ideal class monoid, overorders, and Gekeler ratios in the context of function fields, advancing the computational tools in this area.

Findings

Algorithms successfully compute the ideal class monoid and overorders.

The product of local Gekeler ratios can be explicitly calculated.

These computations facilitate the enumeration of isogeny classes of Drinfeld modules.

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.