#11464: Maximal orders in QuaternionAlgebra of prime discriminant has a bug
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Reporter: mmasdeu | Owner:
craigcitro
Type: defect | Status: closed
Priority: major | Milestone: sage-5.0
Component: modular forms | Resolution: fixed
Keywords: maximal order, quaternion algebra | Work issues:
Report Upstream: N/A | Reviewers: David
Loeffler
Authors: Marc Masdeu | Merged in:
sage-5.0.beta9
Dependencies: | Stopgaps:
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Old description:
> In Sage 4.7 (and older) the following code enters an infinite loop:
> {{{
> #!python
> A.<i,j,k>=QuaternionAlgebra(17)
> O=A.maximal_order()
> }}}
> The maximal_order() function is essentially a formula, except for when
> the prime (here 17) is 1 mod 8. In that case there is a mistake in how
> the formula ought to be.
>
> It looks like the invariants as returned when starting from a Brandt
> module are in opposite order as when a QuaternionAlgebra is created by
> hand, and the code makes assumptions on that ordering.
New description:
In Sage 4.7 (and older) the following code enters an infinite loop:
{{{
#!python
A.<i,j,k>=QuaternionAlgebra(17)
O=A.maximal_order()
}}}
The maximal_order() function is essentially a formula, except for when the
prime (here 17) is 1 mod 8. In that case there is a mistake in how the
formula ought to be.
It looks like the invariants as returned when starting from a Brandt
module are in opposite order as when a QuaternionAlgebra is created by
hand, and the code makes assumptions on that ordering.
--
Comment (by daniels):
It looks like the new code can still produce incorrect results if p is 1
mod 8, e.g.:
{{{
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
}}}
so the "order" is not actually multiplicatively closed.
The problem is that Pizer's formulas expect the invariants of the algebra
to have a certain form, e.g.,
{{{
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)
}}}
seems to be correct. 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: R = A.maximal_order()
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
}}}
I'm not sure what the best way is to fix this: Just throw a
NotImplementedError if the invariants are not of the required form or try
to compute an isomorphism?
(PS: I've almost finished implementing a more general algorithm of Voight
that would also work for non-prime discriminants, so perhaps not too much
effort should be expended on this)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11464#comment:4>
Sage <http://www.sagemath.org>
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