#11215: Periods for Modular Forms
-------------------------------------+-------------------------------------
Reporter: tdupu | Owner: craigcitro
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-6.3
Component: modular forms | Resolution:
Keywords: | Merged in:
Authors: Taylor Dupuy, | Reviewers: Aly Deines, Peter
Frédéric Chapoton, Peter Bruin | Bruin
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/pbruin/11215-ModularForm_period | f03cb122f8dafcf52767f75fee7075067367bea4
Dependencies: | Stopgaps:
-------------------------------------+-------------------------------------
Changes (by {'newvalue': u'Taylor Dupuy, Fr\xe9d\xe9ric Chapoton, Peter Bruin',
'oldvalue': u'tdupu, Fr\xe9d\xe9ric Chapoton'}):
* status: needs_work => needs_review
* author: tdupu, Frédéric Chapoton => Taylor Dupuy, Frédéric Chapoton,
Peter Bruin
* commit: 192e83a3b91207460353830e0ea8817cfbfe2cc2 =>
f03cb122f8dafcf52767f75fee7075067367bea4
* branch: u/chapoton/11215 => u/pbruin/11215-ModularForm_period
* milestone: => sage-6.3
* reviewer: Aly Deines => Aly Deines, Peter Bruin
Old description:
> We should have a functions/functions which allow for pairing of the
> following
>
> -cuspidal modular symbols for Gamma0(N) with cusp forms for Gamma0(N)
>
> -cusp forms for Gamma0(N) with cusp forms Gamma0(N) (petersson inner
> product)
>
> -group elements of Gamma0(N) with cusp forms for Gamma0(N)
>
> Once this is done what works for cusp forms should be extended to
> Meromorphic Differentials of the second kind.
> They should be implemented so that things like
> {{{
> sage: f = CuspForm(15,2).basis()[0]
> sage: g = Gamma0(15).random_element()
> sage: f.pair(g)
> --some number--
> }}}
> should work. Observe that Gamma0(15).random_element() is not implemented.
> A reference for this is chapter 2 of Cremona.
>
> I'm going to attach some code which doesn't work very well and can be
> improved by looking at Fricke involutions but at least it's a start.
> Maybe some people already have some code for this???
>
> Apply [attachment:trac_11215_pairing_with_atkin_lehner.patch]
New description:
This ticket implements a `period(g)` method for cusp forms of weight 2 for
`Gamma0(n)`. It currently accepts only elements `g` in `Gamma0(n)` as
input.
{{{
sage: f = Newforms(15, 2)[0]
sage: g = Gamma0(15)(matrix([[-4, -3], [15, 11]]))
sage: f.period(g)
2.17298044293747e-16 - 1.59624222213178*I
}}}
In the future we should have functions which allow for pairing of the
following:
-cuspidal modular symbols for Gamma0(N) with cusp forms for Gamma0(N)
-cusp forms for Gamma0(N) with cusp forms for Gamma0(N) (Petersson inner
product)
--
Comment:
The new commit renames the method to `period`, and adds a `prec` argument
and a line of code to deduce a bound on the number of terms of the series
from that. I don't have a reference for this bound; I just computed a
slightly rough but hopefully correct one and put the basic idea in a
comment.
There are various other changes. All computations are now done in
`RealField(prec)` and the associated `ComplexField`. The method now also
checks its input more carefully.
I guess this does not really qualify as a reviewer patch anymore, but it
is probably still OK if one of the existing authors reviews these changes
instead of a "disinterested third party".
--
Ticket URL: <http://trac.sagemath.org/ticket/11215#comment:16>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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