Computing ideal classes representatives in quaternion algebras

TL;DR

This paper presents an efficient algorithm for computing ideal class representatives in quaternion algebras over totally real fields, improving existing methods by simplifying equivalence checks and not requiring class number knowledge.

Contribution

It introduces a novel algorithm to find ideal class representatives for Bass orders in quaternion algebras, enhancing efficiency and applicability over previous approaches.

Findings

Algorithm efficiently computes ideal class representatives.

No need for class number knowledge in the process.

Successfully applied to an order over [] with level 30.

Abstract

Let $K$ be a totally real number field and let $B$ be a totally definite quaternion algebra over $K$. In this article, given a set of representatives for ideal classes for a maximal order in $B$, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in $B$. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of level 30 in an algebra over the real quadratic field $\Q[\sqrt{5}]$.