Local isomorphism classes of fractional ideals of orders in étale algebras
Stefano Marseglia
TL;DR
This work addresses the problem of classifying local isomorphism (weak equivalence) classes of fractional $R$-ideals in orders of étale algebras, where non-maximal orders admit multiple genera. Building on a decomposition $W(R)=\bigsqcup_S \overline{W}(S)$ and a key reduction to overorders with at most one non-invertible maximal ideal, the author proves a main theorem that expresses $\overline{W}(R)$ in terms of linear-algebra data over residue fields: for each representative $J_i$ of $\overline{W}(T)$ with $T=(\mathfrak p:\mathfrak p)$, the weak classes correspond to orbits of subspaces ${\mathcal V}_i$ under the free action of $U=T_{{\mathfrak p}}^{\times}/R_{{\mathfrak p}}^{\times}$, giving $\overline{W}(R) \cong \bigsqcup_{i=1}^n {\mathcal V}_i/U$. This leads to two recursive algorithms, WRbar and WR, which compute $\overline{W}(R)$ and $W(R)$ efficiently by reducing to finite linear-algebra problems over residue fields and controlled extensions to overorders. The results enable fast computation of ideal-class monoids and, via known correspondences, large-scale enumeration of abelian varieties over finite fields, achieving substantial speedups over previous methods. The methods are implemented in Magma and applied to hundreds of thousands of abelian varieties, with significant computational savings confirmed by concrete timings. The work thereby provides a practical, scalable framework for arithmetic investigations in non-maximal orders and their applications to algebraic geometry over finite fields.
Abstract
We study the local isomorphism classes, also known as genera or weak equivalence classes, of fractional ideals of orders in étale algebras. We provide a classification in terms of linear algebra objects over residue fields. As a by-product, we obtain a recursive algorithm to compute representatives of the classes, which vastly outperforms previously known methods.