#14489: _S_class_group_and_units is mathematically incorrect
---------------------------------+------------------------------------------
Reporter: robharron | Owner: davidloeffler
Type: defect | Status: needs_review
Priority: critical | Milestone: sage-5.10
Component: number fields | Resolution:
Keywords: S-class group | Work issues:
Report Upstream: N/A | Reviewers: Peter Bruin
Authors: Robert Harron | Merged in:
Dependencies: | Stopgaps:
---------------------------------+------------------------------------------
Old description:
> The output of _S_class_group_and_units is incorrect, and hence the output
> of selmer_group as well, in some cases where S contains non-principal
> ideals. Here's an example:
>
> {{{
> sage: K.<a> = NumberField(x^3 - 381 * x + 127)
> sage: S = tuple(K.primes_above(13))
> sage: K.selmer_group(S, 2)
> [-7/13*a^2 - 140/13*a + 36/13,
> 14/13*a^2 + 267/13*a - 85/13,
> 7/13*a^2 + 127/13*a - 49/13,
> -1,
> 1/13*a^2 + 20/13*a - 7/13,
> 1/13*a^2 - 19/13*a + 6/13,
> 121,
> 10/13*a^2 + 44/13*a - 4555/13]
> }}}
>
> It's fairly easy, using Sage, to see that the S-2-Selmer group of K is an
> 8-dimensional F_2-vector space. The list of length 8 that is returned is
> supposed to be a basis of this (or rather a set of representatives in
> K^×^). But the S-2-Selmer group is a subgroup of K^×^ mod squares, so 121
> is the zero vector and hence the output is not linearly independent. The
> problem lies in the following:
>
> {{{
> sage: K._S_class_group_and_units(S)
> ([-7/13*a^2 - 140/13*a + 36/13,
> 14/13*a^2 + 267/13*a - 85/13,
> 7/13*a^2 + 127/13*a - 49/13,
> -1,
> 1/13*a^2 + 20/13*a - 7/13,
> 1/13*a^2 - 19/13*a + 6/13],
> [(Fractional ideal (11, a - 2), 2, 121),
> (Fractional ideal (19, 1/13*a^2 - 45/13*a - 332/13),
> 2,
> 10/13*a^2 + 44/13*a - 4555/13)])
> }}}
>
> The 121 in there is supposed to be such that (11, a-2)^2^ = (121). But
> (11, a-2)^2^ is *not* principal (in fact, (11, a-2) is a generator of the
> cyclic subgroup of order 6). It is just in the subgroup of the class
> group generated by the primes in S.
>
> I'll shortly upload a patch that fixes this (partially suggested to me by
> Zev Klagsbrun).
New description:
The output of _S_class_group_and_units is incorrect, and hence the output
of selmer_group as well, in some cases where S contains non-principal
ideals. Here's an example:
{{{
sage: K.<a> = NumberField(x^3 - 381 * x + 127)
sage: S = tuple(K.primes_above(13))
sage: K.selmer_group(S, 2)
[-7/13*a^2 - 140/13*a + 36/13,
14/13*a^2 + 267/13*a - 85/13,
7/13*a^2 + 127/13*a - 49/13,
-1,
1/13*a^2 + 20/13*a - 7/13,
1/13*a^2 - 19/13*a + 6/13,
121,
10/13*a^2 + 44/13*a - 4555/13]
}}}
It's fairly easy, using Sage, to see that the S-2-Selmer group of K is an
8-dimensional F_2-vector space. The list of length 8 that is returned is
supposed to be a basis of this (or rather a set of representatives in
K^×^). But the S-2-Selmer group is a subgroup of K^×^ mod squares, so 121
is the zero vector and hence the output is not linearly independent. The
problem lies in the following:
{{{
sage: K._S_class_group_and_units(S)
([-7/13*a^2 - 140/13*a + 36/13,
14/13*a^2 + 267/13*a - 85/13,
7/13*a^2 + 127/13*a - 49/13,
-1,
1/13*a^2 + 20/13*a - 7/13,
1/13*a^2 - 19/13*a + 6/13],
[(Fractional ideal (11, a - 2), 2, 121),
(Fractional ideal (19, 1/13*a^2 - 45/13*a - 332/13),
2,
10/13*a^2 + 44/13*a - 4555/13)])
}}}
The 121 in there is supposed to be such that (11, a-2)^2^ = (121). But
(11, a-2)^2^ is *not* principal (in fact, (11, a-2) is a generator of the
cyclic subgroup of order 6). It is just in the subgroup of the class group
generated by the primes in S.
I'll shortly upload a patch that fixes this (partially suggested to me by
Zev Klagsbrun).
Apply: trac_14489_fix_S_class_group_and_units.patch
--
Comment (by pbruin):
I would tend to give this a positive review. The patch fixes the problem,
I believe the code is correct, all tests pass, and the documentation
builds correctly. I have a few comments, though:
1. The changes in the docstrings are not very clearly written. The
sentences are very long and contain awkward constructions ("a fractional
ideal representative of the S-class group generator whose order (in the
S-class group) is order"; "principal generator").
2. The new docstring of _S_class_group_and_units in number_field.py
suggests that to obtain a principal ideal, gen^order^ can be multiplied by
any fractional ideal J whose class is in the subgroup of the class group
generated by ideals in S. However, the condition is more strict: J must
be in the subgroup of the ''ideal group'' generated by ideals in S.
3. In general (not because of this patch), the code for computing Selmer
groups is a bit convoluted. Conceptually, the computation of the
generators for principal ideals of the form gen^order^ belongs in
selmer_group, not _S_class_group_and_units. In my opinion it would be
more correct to return, as S-class group generators, pairs (gen, order)
instead of triples (gen, order, pr), and leave the computation of the
principal ideal generators to selmer_group.
I made a patch on top of this one to address these issues. An additional
reviewer would probably be needed, but I think it should not be too hard
to review. The two patches together have a net effect of simplifying the
Selmer group code somewhat.
Alternatively, I could file a new ticket for the additional changes, if
the second patch would delay closing this ticket too long.
Any opinions about which option would be better?
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14489#comment:12>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/groups/opt_out.
