Computing ideal classes representatives in quaternion algebras

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 $\Q[\sqrt{5}]$.

Corrected version. Section 4 has been rewritten from scratch

References in corpus (3)

• Shimura correspondence for level $p^2$ and the central values of $L$-series II
• Quaternion Orders and Ternary Quadratic Forms
• Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms

Cited by in corpus (3)

• Preimages for the Shimura map on Hilbert modular forms
• On the change of root numbers under twisting and applications
• Ideal classes of orders in quaternion algebras