On Sun, May 31, 2020 at 6:58 AM Preston Wake <
presto...@gmail.com> wrote:
>
> Thanks for your answer, William! I think I understand now what ModularSymbols is computing. Let me say it in a different way, to see how it compares to the group denoted \bM_k(G;R) in the Definition 1.23 of your book (
https://wstein.org/books/modform/modform/modular_symbols.html). The command ModularSymbols(G,k,base_ring=R) is first computing a free abelian group M given by the Manin symbols, and then it computes a subgroup M' of M such that:
> (i) M' maps to zero in \bM_k(G), and
> (ii) M/M' \otimes \Q is equal to \bM_k(G;\Q)/
> Then it returns the R-module M/M' \otimes R. Currently this is only implemented when R is a field.
>
> How does this output compare to \bM_k(G;R)? Well, we know that the Manin symbols M generate \bM_k(G) integerally (this is Proposition 1.24), so by (i) there is a surjection M/M' -->> \bM_k(G). Since \bM_k(G) is a free ZZ-module of finite rank, (ii) implies that the kernel of this surjection is exactly the torsion subgroup of M/M'. Hence, when R is a characteristic-zero field, then the output is actually equal to \bM_k(G;R), but when R has characteristic p, the output is equal to \bM_k(G;R) if and only if M/M' is p-torsion-free. If M/M' has p-torsion, then probably the output is useless (?).
>
That sounds right.
> Some questions:
> - Why isn't ModularSymbols(G,k,base_ring=R) implemented for rings (like R=ZZ)? It seems like, given what is already there, it wouldn't be too hard to ask it to output (M/M')/(torsion) (although this may be hard for it to actually compute...)
Because it can be hard to compute and I've never needed it for my
research. Other than that, it could be done.
There is some very closely related functionality that is implemented:
sage: M = ModularSymbols(7,8,1).cuspidal_submodule()
sage: M.integral_hecke_matrix(5)
[-320460 285880 67032]
[-212550 189650 44448]
[-626400 558720 131056]
Try M.integral[tab] to find several other related commands. The
above is a matrix for T_5 on Modular symbols over ZZ (mod any
torsion, just to be clear). You could reduce it modulo 5 and that may
be what you're interested in:
sage: T5 = M.integral_hecke_matrix(5).change_ring(GF(5))
sage: T5.fcp()
(x + 4) * x^2
sage: T5.minimal_polynomial().factor()
x * (x + 4)
https://share.cocalc.com/share/df81e09e5b8f16f28b3a2e818dcdd4560e7818ae/support/2020-05-28-modular-over-R.ipynb?viewer=share
> - For characteristic-p fields, why does Sage output M/M' \otimes R instead of (M/M')/(torsion) \otimes R? The obvious reason is the one William says: outputing M/M' \otimes R may be much faster. But isn't this anathema to what Sage normally does? By default, Sage normally makes you do the more mathematically-meaningful thing, rather than the faster thing.
As mathematicians like to say "for historical reasons". Consider that
I implemented all this about 15 years ago before anybody else used
Sage...
>> But for
>> some
>> small values of p, e.g., p=2 or 3, you can end up with an object that
>> is basically useless (e.g., "Hecke operators" on it don't commute).
>
> I don't think this torsion phenomenon is just about small primes. Are you saying that when p is large, the resulting space is not basically useless?
Sorry, I should have been clear that I was only talking about weight 2
and trivial character in that remark.
> For example:
> ModularSymbols(5,16,1); # M has dimension 9
> Mp=ModularSymbols(5,16,1,GF(13)); #Mp has dimension 10;
> Is this because 13 is small, in this case? Or are you saying that this Mp is not "basically useless"? If so, what is its basic use?
Certainly with work you can hope to find a useful vector subspace in
that 10 dimensional space...
> Best,
> Preston
>
>
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--
William (
http://wstein.org)