Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields

TL;DR

The paper develops a general algorithm to classify $R$-lattices in a fixed $K$-module $V=\bigoplus_i K_i^{s_i}$ up to $R$-linear isomorphism by exploiting the maximal order $\mathcal{O}$ and conductor $\mathfrak{f}$ of $R$, providing an effective, finite version of Jordan–Zassenhaus for commutative $\mathbb{Z}$-orders. It then applies this framework to (i) compute conjugacy classes of semisimple integer matrices with prescribed minimal and characteristic polynomials via a Latimer–MacDuffee-type correspondence, and (ii) determine the $\mathbb{F}_q$-isomorphism classes of abelian varieties within a Frobenius-characteristic isogeny class under the ordinary or no-real-root hypotheses, using Deligne–Centeleghe–Stix theory. The main contributions include a general, implementable construction that unifies matrix-conjugacy problems and abelian-variety classifications, together with a labeling scheme for sorting computed objects and a MAGMA implementation. The results have practical impact for explicit arithmetic geometry computations and database integration (e.g., LMFDB), enabling systematic enumeration of isomorphism classes in broad settings beyond prior, more specialized cases.

Abstract

We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $\mathbb{F}_q$-isomorphism classes of abelian varieties over a finite field $\mathbb{F}_q$ which belong to an isogeny class determined by a characteristic polynomial $h$ of Frobenius when $h$ is ordinary, or $q$ is prime and $h$ has no real roots.

Theorems & Definitions (24)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Theorem 3.6
• proof
• Corollary 3.7