Ideal classes of orders in quaternion algebras

TL;DR

This paper introduces an algorithm for computing representatives of all right equivalence classes of ideals in orders of quaternion algebras over global fields, extending to more general algebraic structures.

Contribution

It develops a comprehensive algorithm that includes non-invertible ideals and generalizes the theory to broader classes of algebras.

Findings

Algorithm successfully computes all ideal classes in quaternion orders.

Includes non-invertible ideals in the classification.

Extends theoretical framework to more general algebraic structures.

Abstract

We provide an algorithm that, given any order $O$ in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right $O$-ideals, including the non-invertible ones. The theory is developed for a more general kind of algebras.