Constructing non-maximal orders in quaternion algebras

TL;DR

This paper provides explicit bases for non-maximal orders in definite rational quaternion algebras, facilitating computations in modular forms related to elliptic and quaternionic cases.

Contribution

It introduces a method to construct explicit bases for arbitrary level orders in quaternion algebras, expanding computational tools in the field.

Findings

Explicit bases for orders of arbitrary level N>1

Applications to computing spaces of modular forms

Enhanced computational techniques for quaternion algebras

Abstract

We present an explicit basis for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms.