Re: [sage-support] Modular Symbols and irregular cusps
Alex Ghitza
Hi Kilian,
I am forwarding this to the sage-nt mailing list as well since you might
get a larger audience.
Best,
Alex
On Sun, 3 Jan 2010 13:51:19 -0800 (PST), Kilian <kki...@googlemail.com> wrote:
> Hello,
>
> i have a problem with sage and modular symbols for Gamma1(4) and odd
> weight k, where the cusp 1/2 is irregular.
>
> According to Merel, there is (for k>2) an exact sequence:
>
> 0-> S_k -> M_k -> B_k -> 0
>
> Here B_k is the boundary space and S_k is the cuspidal subspace.
>
> Let the weight k be 7.
>
> If I compute the appropriate dimensions with SAGE, I get 4,6 and 3
> which can't be. Furthermore, computing the boundary map, gives a
> matrix which is definitely _not_ surjective.
>
> On the other hand, Merel explicitely states that the dimension of B_k
> is the number of cusps, i.e. 3, so the failure must already be in
> Merel's paper, or am I missing something?
>
> I assume that 4 and 6 are correct, as a comparison with the usual
> dimension tables for modular forms suggest.
>
> What is even more confusing is that Merel states that the isomorphism
> between the boundary space and the space B_k(Gamma) is an
> _isomorphism_, whereas in the SAGE sourcecode and in William Stein's
> book it is only stated that it's injective.
>
> Thanks in advance,
> Kilian.
>
--
Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne
-- Australia -- http://www.ms.unimelb.edu.au/~aghitza/
William Stein
>
> Hi Kilian,
>
> I am forwarding this to the sage-nt mailing list as well since you might
> get a larger audience.
>
>
> Best,
> Alex
>
>
> On Sun, 3 Jan 2010 13:51:19 -0800 (PST), Kilian <kki...@googlemail.com> wrote:
>> Hello,
>>
>> i have a problem with sage and modular symbols for Gamma1(4) and odd
>> weight k, where the cusp 1/2 is irregular.
>>
>> According to Merel, there is (for k>2) an exact sequence:
>>
>> 0-> S_k -> M_k -> B_k -> 0
>>
>> Here B_k is the boundary space and S_k is the cuspidal subspace.
>>
>> Let the weight k be 7.
>>
>> If I compute the appropriate dimensions with SAGE, I get 4,6 and 3
>> which can't be. Furthermore, computing the boundary map, gives a
>> matrix which is definitely _not_ surjective.
>>
>> On the other hand, Merel explicitely states that the dimension of B_k
>> is the number of cusps, i.e. 3, so the failure must already be in
>> Merel's paper, or am I missing something?
The B_k in Merel's paper has dimension 2. Merel does not state that
dim(B_k) is the number of cusps in general. That's only true when the
weight is even.
Sage does have a very small bug, which is that it computes the correct
space B_k but embeds it (trivially) in a bigger space. There is no
need to do this, and I can see how it could be confusing. The
correct relations are used, the correct map is computed, it's just
that there are extra 0's tacked on. For example, in your example we
have the following matrix for the boundary map:
[-1 0 0]
[ 0 -1 0]
[ 0 -1 0]
[ 0 -1 0]
[ 0 0 0]
[ 0 1 0]
notice that the extra dimension -- the 0 in the last column -- isn't involved.
The fix for this bug is to remove all the cusp classes that are
equivalent to 0 because of the relation
[Gamma(lambda u; lambda v)] ~ sign(lambda)^k[Gamma (u;v)]
For example, in your example that would be the class (u;v) = (1;2).
See trac 7837
http://trac.sagemath.org/sage_trac/ticket/7837
>> I assume that 4 and 6 are correct, as a comparison with the usual
>> dimension tables for modular forms suggest.
Yes.
>> What is even more confusing is that Merel states that the isomorphism
>> between the boundary space and the space B_k(Gamma) is an
>> _isomorphism_, whereas in the SAGE sourcecode and in William Stein's
>> book it is only stated that it's injective.
Injectivity is all that is needed for the algorithm.
-- William
>>
>> Thanks in advance,
>> Kilian.
>>
>
>
> --
> Alex Ghitza -- Lecturer in Mathematics -- The University of Melbourne
> -- Australia -- http://www.ms.unimelb.edu.au/~aghitza/
>
> --
>
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>
>
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
Kilian
Merel writes in his paper that "if k is even than the B_k is
canonically isomorphic to C[G\P^1(Q)]. In any case the dimension of
B_k equals |G\P^1(Q)|".
Kilian.
P.S.: As part of my PhD thesis I determined (and proved) explicit
bases for nearly all relevant spaces of modular symbols for levels <=5
and all weights k, in terms of winding symbols if possible. If the
level is 1 and 2 there are (slightly different) results of Fukuhara
and Fukuhara and Yang (in classical language) who also include
explicit formulas for all Hecke operators. I don't know if this is of
any use for SAGE. If it is, I could post the results.
As a consequence one gets explicit bases for the space of cusp forms
in terms of products of _two_ eisenstein series for these spaces. As I
am not a computational expert I don't really know if it is
computationaly better to compute the Eisenstein series or to compute
these spaces via modular symbols.
On 4 Jan., 00:44, William Stein <wst...@gmail.com> wrote:
> On Sun, Jan 3, 2010 at 1:57 PM, Alex Ghitza <aghi...@gmail.com> wrote:
>
> > Hi Kilian,
>
> > I am forwarding this to the sage-nt mailing list as well since you might
> > get a larger audience.
>
> > Best,
> > Alex
>
> > -- Australia --http://www.ms.unimelb.edu.au/~aghitza/
>
> > --
>
> > You received this message because you are subscribed to the Google Groups "sage-nt" group.
> > To post to this group, send an email to sag...@googlegroups.com.
> > To unsubscribe from this group, send email to sage-nt+u...@googlegroups.com.
> > For more options, visit this group athttp://groups.google.com/group/sage-nt?hl=en-GB.
>
> --
> William Stein
> Associate Professor of Mathematics
> University of Washingtonhttp://wstein.org- Zitierten Text ausblenden -
>
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