Re: q-expansions at other cusps
William Stein
Sorry, but I've just never got around to dealing with this question.
You're about the 10th person to ask me though.
I cc'd the sage-nt mailing list in case anybody wants to comment.
William
On Tue, Sep 22, 2009 at 10:11 AM, Xavier Xarles <xar...@mat.uab.cat> wrote:
> Hi William!
>
> I need some result on q-expansions of cusps forms (weight 2, Gamma_0(N) ) at
> other cusps different from the infinity one. But I did not find any place
> where this is explicitly done...
> Concretely: Suppose N=p is a prime, and we have some cusp form given by its
> q-expansion (at infinity). Can we know the q-expansion at the cusp 0? Is
> there a general formulae/result? If not, how can it be done (say, for
> example, with SAGE or similar)?
> If similar things are know for N a power of a prime, even better!
>
> Thank you very much in advance for the answer, and also for your nice talks
> here in Barcelona!
>
> Best,
>
> Xavier
--
William Stein
Associate Professor of Mathematics
University of Washington
http://wstein.org
Craig Citro
>> I need some result on q-expansions of cusps forms (weight 2, Gamma_0(N) ) at
>> other cusps different from the infinity one. But I did not find any place
>> where this is explicitly done...
>> Concretely: Suppose N=p is a prime, and we have some cusp form given by its
>> q-expansion (at infinity). Can we know the q-expansion at the cusp 0? Is
>> there a general formulae/result? If not, how can it be done (say, for
>> example, with SAGE or similar)?
>> If similar things are know for N a power of a prime, even better!
>>
The answer is currently "no, there's nothing implemented in Sage at
the moment." Nathan Ryan and I started working on this last summer,
but then we both got busy with other things and haven't gotten back to
it. For the case of squarefree level, this is done by Asai in this
paper:
Asai, Tetsuya
On the Fourier coefficients of automorphic forms at various cusps and
some applications to Rankin's convolution.
J. Math. Soc. Japan 28 (1976), no. 1, 48--61.
(I can email you a scan if you can't find it in your library.)
Beyond that, I don't think anything is known -- I'd be very interested
to hear if anyone on this list knows otherwise. Nathan and I had some
ideas for dealing with at least certain other levels, but it's been a
while since I've looked at any of this.
Hope that helps -- definitely email back if you have more questions,
or if you find something about computing for the case of
non-squarefree level!
-cc
John Cremona
Fourier expansion of forms of arbitrary even weight, level and
charcater at general cusps.
It is here: http://math.univ-lyon1.fr/~delaunay/
John
2009/9/24 Craig Citro <craig...@gmail.com>:
Xavier Xarles
to Delaunay's thesis were very helpful and solved (almost) completely
my problem!
Xavier
2009/9/24 John Cremona <john.c...@gmail.com>:
William stein
On Sep 24 2009, 1:56 pm, Xavier Xarles <xavier.xar...@gmail.com>
wrote:
> Thanks to all for the answers. The references to the paper by Asai and
> to Delaunay's thesis were very helpful and solved (almost) completely
> my problem!
>
> Xavier
Did you implement any relevant code?
William
>
> 2009/9/24 John Cremona <john.crem...@gmail.com>:
>
>
>
> > In Delaunay's thesis (2002) pages 64-75 there is a section about the
> > Fourier expansion of forms of arbitrary even weight, level and
> > charcater at general cusps.
>
> > It is here: http://math.univ-lyon1.fr/~delaunay/
>
> > John
>
> > 2009/9/24 Craig Citro <craigci...@gmail.com>: