a computation that I would like to discuss

[Amod Agashe]

Dear all:

I will be at Sage days only on Tuesday evening and all of Wednesday. I
wanted to discuss the question of computing efficiently congruences
between spaces of cusp forms with anyone who might be interested. I
have posted the problem and a possible solution at
http://www.math.fsu.edu/~agashe/cong.html

My student Randy Heaton is working on this problem, and may discuss it
with some of you before I arrive. If there is enough interest, I can
give an informal talk on Wednesday.

Looking forward to seeing everyone soon,

Amod

[William Stein]

On Mon, Sep 14, 2009 at 11:12 AM, Amod Agashe <amod....@gmail.com> wrote:
>
> Dear all:
>
> I will be at Sage days only on Tuesday evening and all of Wednesday. I
> wanted to discuss the question of computing efficiently congruences
> between spaces of cusp forms with anyone who might be interested. I
> have posted the problem and a possible solution at
> http://www.math.fsu.edu/~agashe/cong.html

Isn't what you're asking for *exactly* the same thing as the
intersection_number method for modular symbols spaces?

sage: H = ModularSymbols(43).cuspidal_subspace()
sage: D = H.decomposition(); D
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational
Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of
dimension 7 for Gamma_0(43) of weight 2 with sign 0 over Rational
Field
]
sage: D[0].intersection_number(D[1])
4

That computes the index of H[I_1] + H[I_2] in its saturation.
You could also easily modifying the command to get the group structure.

sage: H = ModularSymbols(54).cuspidal_subspace()
sage: D = H.decomposition(); D
[
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 19 for Gamma_0(54) of weight 2 with sign 0 over Rational
Field,
Modular Symbols subspace of dimension 2 of Modular Symbols space of
dimension 19 for Gamma_0(54) of weight 2 with sign 0 over Rational
Field,
Modular Symbols subspace of dimension 4 of Modular Symbols space of
dimension 19 for Gamma_0(54) of weight 2 with sign 0 over Rational
Field
]
sage: D[0].intersection_number(D[2])
9
sage: D[0].congruence_number(D[2])
3

William

> My student Randy Heaton is working on this problem, and may discuss it
> with some of you before I arrive. If there is enough interest, I can
> give an informal talk on Wednesday.
>
> Looking forward to seeing everyone soon,
>
> Amod
>
>

--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org

[Amod Agashe]

On Mon, Sep 14, 2009 at 11:29 PM, William Stein <wst...@gmail.com> wrote:

Isn't what you're asking for *exactly* the same thing as the
intersection_number method for modular symbols spaces?

Essentially, yes. I did not know that it was implemented. I suppose the only question then is  how efficient the function is, and whether any of my ideas may speed it up. Another issue is: if one just wants to check if a given prime p divides an interesection number, then is there a faster way of doing it in sage currently than first computing the interesection number and then checking if p divides it (the same could be asked about congruence numbers as well, perhaps).

--Amod

[William Stein]

Check out this function I posted here. It might be the ultimate
solution to your problem (?):

http://trac.sagemath.org/sage_trac/ticket/6931

William