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Ideal classes of orders in quaternion algebras

Stefano Marseglia, Harry Smit

TL;DR

The paper develops a complete framework for computing all right ideal classes of an order $O$ in a quaternion algebra over a global field, including non-invertible ideals. It introduces weak right equivalence to factor out the invertible part and local right equivalence to capture local principalities, then combines these to recover full right equivalence classes, underpinned by a duality theory and localization techniques. The authors provide an algorithmic pipeline, with pseudocode and Magma implementation, that leverages Overorders and the interplay between weak and invertible classes, and extend Brandt matrices to account for non-invertible classes, enabling applications to modular forms and arithmetic geometry. This work broadens computational reach beyond invertible ideals, with potential impact on the study of abelian varieties with quaternionic endomorphisms and related modular forms via generalized Brandt matrices.

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.

Ideal classes of orders in quaternion algebras

TL;DR

The paper develops a complete framework for computing all right ideal classes of an order in a quaternion algebra over a global field, including non-invertible ideals. It introduces weak right equivalence to factor out the invertible part and local right equivalence to capture local principalities, then combines these to recover full right equivalence classes, underpinned by a duality theory and localization techniques. The authors provide an algorithmic pipeline, with pseudocode and Magma implementation, that leverages Overorders and the interplay between weak and invertible classes, and extend Brandt matrices to account for non-invertible classes, enabling applications to modular forms and arithmetic geometry. This work broadens computational reach beyond invertible ideals, with potential impact on the study of abelian varieties with quaternionic endomorphisms and related modular forms via generalized Brandt matrices.

Abstract

We provide an algorithm that, given any order in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right -ideals, including the non-invertible ones. The theory is developed for a more general kind of algebras.
Paper Structure (10 sections, 29 theorems, 88 equations, 3 algorithms)

This paper contains 10 sections, 29 theorems, 88 equations, 3 algorithms.

Table of Contents

  1. Introduction
  2. Lattices
  3. Weak and local equivalence
  4. Weak equivalence
  5. Local equivalence
  6. Comparison between weak and local equivalence
  7. Duality
  8. Algorithmic implementation
  9. Brandt Matrices
  10. Computing invertible ideals (by John Voight)

Key Result

Lemma 2.1

Let $I$, $J$, and $K$ be lattices and $\alpha,\beta \in B^\times$. Then

Theorems & Definitions (71)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • ...and 61 more