#11598: Change to is_congruence method of modular subgroups
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Reporter: davidloeffler | Owner: craigcitro
Type: defect | Status: needs_work
Priority: major | Milestone: sage-4.7.2
Component: modular forms | Keywords: modular congruence subgroup
Work_issues: | Upstream: N/A
Reviewer: Vincent Delecroix | Author: David Loeffler
Merged: | Dependencies: #11422
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Comment(by davidloeffler):
Replying to [comment:3 vdelecroix]:
> Replying to [comment:2 davidloeffler]:
>
> I was asking myself what should be the definition of congruence
subgroup. In order to use Wohlfart's theorem it is necessary to use the
projective definition and that's why it was implemented that way. For all
standard congruence groups it makes no difference. Could you explain me a
concrete case where the difference matter ?
Here's one: the group of index 24 with {{{S2 =
(1,3,13,15)(2,4,14,16)(5,7,17,19)(6,10,18,22)(8,12,20,24)(9,11,21,23), S3
= (1,14,15,13,2,3)(4,5,6,16,17,18)(7,8,9,19,20,21)(10,11,12,22,23,24).}}}
This is taken from the paper of Kiming, Schuett and Verrill in J. London
Math Soc (2010) which is all about this problem of finding noncongruence
odd subgroups whose projective image is congruence.
> In any case, you should add an example of an odd arithmetic subgroup
which is not of congruence but which becomes one when we add the element
-id.
>
> Do you know a method, given an even subgroup, to build all odd subgroups
it may come from ?
There is one in the K-S-V paper but it uses Farey symbols heavily. I can
think of an algorithm which produces some examples of such subgroups using
the permutation representation (which is how I found the one above) but
it's not totally obvious to me if it produces all of them.
> > Just realised that, since the new algorithm in the odd case is
*extremely* slow, it is better to not call it from the generic test
routine! Here's a new patch.
>
> For that particular question, it should be possible, following Hsu
method (who produces uniform presentation for PSL(Z,Z/pZ)), to get a
congruence test method for odd subgroups.
I've actually thought of another approach, which is slower and less
elegant than Hsu's approach but much better than the patch I just made
:-). Rather than finding generators for {{{Gamma(N)}}}, which is hideously
slow if N is more than about 8, one can just look at the subgroup of
{{{SL(2, Z / NZ)}}} generated by the reductions mod n of the generators of
self. Then self is congruence of level N iff this subgroup has the same
index in {{{SL(2, Z/NZ)}}} as self does in {{{SL(2,Z)}}}; and we can
always take N to be twice the generalised level.
Better still, if G isn't congruence, then this algorithm calculates the
smallest congruence subgroup containing G (the congruence closure), which
is an interesting thing to calculate in itself.
I will do a new patch tomorrow.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11598#comment:4>
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