We introduce quaternion algebras, which are noncommutative analogues of quadratic field extensions and hence are ubiquitous in mathematics. We will give a demonstration of existing algorithms for computing with quaternion algebras, as currently implemented in Magma, and then discuss possible future directions in this area from both the theoretical and practical points of view.
To implement these algorithms in SAGE, we need:
- maximal orders for rings of integers
- pseudo-bases (algorithms for dedekind rings)
- places of a number field
- p-adics
- completions
- linear algebra over Z
- hermite normal forms
- fast algorithms for enumerating elements of a lattice of a given norm (most pressing)
- group theory
To run the Magma demo magma-quat-demo.m, you need the latest version of Magma, v.2.13-4 or -5. You can also run them from the Magma notebook, http://magma.maths.usyd.edu.au/calc/.
Some quick references for algorithms for quaternion algebras:
- Montserrat Alsina, Pilar Bayer, "Quaternion orders, quadratic forms, and Shimura curves"
David Kohel, "Endomorphism rings of elliptic curves over finite fields", available at http://www.maths.usyd.edu.au/u/kohel/res/index.html
- Marie-France Vigneras, "Arithmetique des algebres de quaternions"
John Voight, "Quadratic forms and quaternion algebras: Algorithms and arithmetic, available at http://www.ima.umn.edu/~voight/
- John Voight, "Algorithms for quaternion algebras", in preparation!
