Incorrect computation of maximal orders in quaternion algebras

[daniels]

The maximal_order function of QuaternionOrder? can produce incorrect results. It is implemented by means of an explicit formula, which expects not just the algebra to be ramified precisely at one finite prime p and infinity, but also expect the invariants a=i² and b=j² to have a special form. Not all the necessary conditions are checked by the code.

E.g. the following is obviously not an order since it contains non-integral elements:

sage: A.<i,j,k> = QuaternionAlgebra(17)
sage: print A.invariants()
(-3, -17)
sage: R = A.maximal_order()
sage: b = R.basis()
sage: print b
(1/2 + 1/2*j, 1/2*i + 1/2*k, -1/3*j - 1/3*k, k)
sage: b[0]*b[1]
9/2*i
sage: (b[0]*b[1]).reduced_norm() 
243/4

But this is ok, because now the invariants are as in Pizer's paper:

sage: A.<i,j,k> = QuaternionAlgebra(-17,-3)
sage: print A.maximal_order().basis()
(1/2 + 1/2*j, 1/2*i + 1/2*k, -1/3*j - 1/3*k, k)

This could be fixed by simply swapping (i,j) in the formula depending on the invariants, but the invariants can deviate from the required form in more than one way, again possibly resulting in failure of the formula:

sage: A.<i,j,k> = QuaternionAlgebra(-17*9,-3)
sage: print A.maximal_order().basis()
(1/2 + 1/2*j, 1/2*i + 1/2*k, -1/3*j - 1/3*k, k)
sage: print (-1/3*j - 1/3*k).reduced_norm()
154/3

(#11464 is related)

[daniels]

The patch does the following:

• Move maximal_order from brandt.py to quaternion_algebra.py.

• Fix maximal_order to raise a NotImplementedError? if the invariants are not of the expected form

• Add checks in ''__init__'' of QuaternionOrder? to verify if the given basis actually is the basis of an order (this would have catched the bug earlier).

• Fix all the doctests that were expecting the mathematically incorrect results.

[daniels]

See also #12861, for a patch extending this one by a more general algorithm.

[daniels]

New patch applies to sage-5.0.beta13 (old one conflicted with #12461)

[aly.deines]

This applies to sage-5.2.rc0 and all tests pass.

This looks good to me!

[jdemeyer]

Please fill in your real name as Author.

[jdemeyer]

Please fill in your real name as Reviewer.