Atkin-Lehner on modular forms.

[Maarten Derickx]

Dear All,

I need to know how Atkin-Lehner act on the following object:

S = CuspForms(GammaH(N,list_of_integers))

it already has a methord .atkin_lehner_operator but it fails even in the most standard case:

sage: S = CuspForms(Gamma0(17))

sage: S.atkin_lehner_operator()

---------------------------------------------------------------------------

NotImplementedError                       Traceback (most recent call last)

...

NotImplementedError: 

It works for modular symbols though. 

Is there already some literature on how to implement this? Or do people have an idea on how to do this?

Thanks,

Maarten

[John Cremona]

It's not immiediately obvious to me what the definition should be in
the case of these generalised level N subgroups. reference?

John

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[Maarten Derickx]

The first time I read about this for Gamma1(N) was in Johan Bosman his thesis, and he refers to: Twists of Newforms and Pseudo-Eigenvalues of W-Operators by Atkin and Lee which can be found here: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0048&DMDID=DMDLOG_0023&LOGID=LOG_0023&PHYSID=PHYS_0229 . The defenition is on the end of the second and the 3th page of this 20 page article. But it should give something compatible for all modular curves between X_0(N) and X_1(N). It's sad that they don't give a moduli interpretation there. But what the moduli interpretation should be is that if q is an integer such that N and N/q are coprime, and one views X1(N) as parameterizing elliptic curves with pairs of points P_1,P_2 of order q and N/q then W_q should send (E,P_1,P_2) to (E/<P_1>,P_1',P_2 + <P_1>), where P_1 in E/<P_1> is the class of all points in E that pair to \zeta_q with P_1. Note that this no longer needs to be an involution, and is also no longer defined over Q, but they still gives an automorphism. For GammaH as in my original question you just look at the automorphism this induces on X_1(N)/(<d_1>,...,<d_n>).  The fact that these give you autormorphism is basically why I am interested in it, because I want to compute these automorphisms on the canonical model of the modular curve. 

But all these generalizations aside I'm also happy with a reference to something which explains how to compute it for Gamma0, because that is already better then what I got now.

[Peter Bruin]

Hoi Maarten,

> The first time I read about this for Gamma1(N) was in Johan Bosman his
> thesis, and he refers to: Twists of Newforms and Pseudo-Eigenvalues of
> W-Operators by Atkin and Lee which can be found
> here: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN356556735_0048&DMDID=DMDLOG_0023&LOGID=LOG_0023&PHYSID=PHYS_0229 .
> The defenition is on the end of the second and the 3th page of this 20
> page article. But it should give something compatible for all modular
> curves between X_0(N) and X_1(N).

Yes, I think that is correct. I have the impression that the matrix
defining w_Q normalises Gamma_H for any subgroup H of (Z/NZ)^*. This
would imply that the Atkin-Lehner operator w_Q acting on the space of
modular forms of given weight for Gamma_1(N) preserves the subspace of
modular forms for Gamma_H.

I didn't verify the above impression, and it may be mistaken, but in any
case there should be no problems with Gamma_0(N) (H = (Z/NZ)^*) and
Gamma_1(N) (H = {1}).

On the other hand, w_Q does not in general preserve the space of modular
forms of given character, because it does not commute with the diamond
operators. In other words (I think), although w_Q normalises Gamma_H,
it does not centralise Gamma_H/Gamma_1(N).

> It's sad that they don't give a moduli interpretation there. But what
> the moduli interpretation should be is that if q is an integer such
> that N and N/q are coprime, and one views X1(N) as parameterizing
> elliptic curves with pairs of points P_1,P_2 of order q and N/q then
> W_q should send (E,P_1,P_2) to (E/<P_1>,P_1',P_2 + <P_1>), where P_1
> in E/<P_1> is the class of all points in E that pair to \zeta_q with
> P_1. Note that this no longer needs to be an involution, and is also
> no longer defined over Q, but they still gives an automorphism. For
> GammaH as in my original question you just look at the automorphism
> this induces on X_1(N)/(<d_1>,...,<d_n>).

Yes, it is an automorphism defined over a subfield of Q(\zeta_q).

> The fact that these give you autormorphism is basically why I am
> interested in it, because I want to compute these automorphisms on the
> canonical model of the modular curve. But all these generalizations
> aside I'm also happy with a reference to something which explains how
> to compute it for Gamma0, because that is already better then what I
> got now.

I thought getting w_Q on cusp forms would be straightforward from the
fact that we have it on modular symbols. Unfortunately, the function
that one might hope would do this, _convert_matrix_from_modsyms in
sage/modular/modsym/cuspidal_submodule.py, has the following
documentation:

Given a space of modular symbols and a matrix T acting on it,
calculate the matrix of the corresponding operator on the
echelon-form basis of the corresponding space of modular forms.

The matrix T *must* commute with the Hecke operators! We use this
when T is either a Hecke operator, or a diamond operator. This will
*not work* for the Atkin-Lehner operators, for instance, when there
are oldforms present.

This might be a technical restriction which exists for speed reasons.

I have my own toy implementation of modular symbols and modular forms in
PARI, which is less optimised for speed. I have not yet implemented
Atkin-Lehner operators, but once I do that, it will probably be easy to
transfer them from modular symbols to cusp forms. I'll have to think
about it. In the meantime maybe someone else will figure out how to fix
this in Sage!

Groeten uit Engeland,

Peter

[John Cremona]

Peter is right, as of course Atkin-Li gives the general theory of A-L
operators but they do not commute with the ordinary Hecke ops (or the
diamond ops) and so do not preserve character spaces or newform
spaces: that is why one has to define "pseudo-eigenvalues" since the
image of a normalised newform under a W is a scalar multiple of (in
genereal) a different newform, with different character, the scalar
being called the A-L pseudo eigenvalue. If there was a general
implementation of these it would have made Barinder's job easier in
the computations he did for our paper http://arxiv.org/abs/1306.6818.
We actually did those computations in two ways, one using these
pseudoeigenvalues as in Atkin & Li, and also using just modular
symbols. But using modular symbols had a different difficulty: we
know how to normalise a newform (coefficient of q in the q-expansion
=1) and this normalization is necessary for the definition of the ps.
eigenvalue to make sense; but in the associated modular symbol space
it is not so clear what the correct "normalised eigenvector" should
be. This does not matter for usual eigenvalues, since obviously one
gets the same eigenvalue however the eigenvector is normalised, but it
does matter for pseudo-eigenvalues. I worked out how to do this for
my side of the computations on that paper, and have some notes on
that, but out situation was not as general as would be needed for a
general implementation of pseudoeigenvalues for an arbitrary Gamma_H.

Perhaps Peter and I (and Maarten) should work this out in full with a
view to haveing a good and general implementation in Sage?

John

[David Loeffler]

Hi folks,

A couple of remarks.

> Note that this [the Atkin--Lehner on X1(N)] no longer needs to be an involution, and is also

> no longer defined over Q, but they still gives an automorphism.

I think w_N, (or more generally w_Q for any factor Q of N such that
(Q, N/Q) = 1), *is* always an involution when considered as a map from
the modular curve to itself. There is a normalization issue when
defining it as a Hecke operator on modular forms spaces of weight k >
2, which means that some authors have w_N^2 = multiplication by
N^{k-2} etc. It is indeed not defined over Q. (Actually it
interchanges two natural rational structures on X1(N) -- there is a
model of X1(N) that classifies elliptic curves with a point of order
N, and a model which classifies modular curves with an embedding of
the group scheme mu_N, and w_N swaps these.)

> I thought getting w_Q on cusp forms would be straightforward from the
> fact that we have it on modular symbols. Unfortunately, the function
> that one might hope would do this, _convert_matrix_from_modsyms in
> sage/modular/modsym/cuspidal_submodule.py, has the following
> documentation:

> [...]

> This might be a technical restriction which exists for speed reasons.

Unfortunately it's not. It's a genuine shortcoming of the algorithm,
and arises from the fact that there isn't a canonical isomorphism
between modular forms and modular symbols unless you extend to CC (the
periods of forms with rational q-expansions are not rational). The
algorithm used in Sage amounts to cheating, by first computing the
Hecke *algebra* using modular symbols, and then using the fact that
spaces of cusp forms are canonically isomorphic to the duals of their
own Hecke algebras via the pairing (f, T) --> a_1(T . f). If you look
at the code of _convert_matrix_from_modsyms, you can see that it is
actually decomposing the given matrix with respect to a basis of the
Hecke algebra, and it will either fail or return random garbage if the
given matrix isn't in the Hecke algebra in the first place.

The only way I can see to calculate the matrix of the w_Q on a space
of modular forms, with respect to the q-expansion basis, would be to
do it separately on the subspace corresponding to newforms of level D,
for each D dividing N, by doing the calculation at level D first and
then using formulae for how w_Q interacts with the degeneracy maps
from level D to level N. Even the newform calculation is a headache
when the character is nontrivial (because w_N does not commute with
the T_n's in this case either, as it sends a newform f to a scalar
multiple of its complex conjugate).

One of John's former students, Barinder Banwait, was thinking about
this problem at some stage, because he wanted to compute the action of
SL2(Z / p) on modular forms of level Gamma(p); computing the action of
[0, -1; 1, 0] is equivalent to computing the Atkin--Lehner operator on
forms on some GammaH group of level p^2. I don't know exactly how far
he got with this. John?

David

(PS: This email crossed over with John's, but I thought I might as
well send it anyway.)

On 6 November 2013 01:15, Peter Bruin <P.B...@warwick.ac.uk> wrote:

[Maarten Derickx]

Le mercredi 6 novembre 2013 10:40:24 UTC+1, David Loeffler a écrit :

Hi folks,

A couple of remarks.

> Note that this [the Atkin--Lehner on X1(N)] no longer needs to be an involution, and is also
> no longer defined over Q, but they still gives an automorphism.

I think w_N, (or more generally w_Q for any factor Q of N such that
(Q, N/Q) = 1), *is* always an involution when considered as a map from
the modular curve to itself. There is a normalization issue when
defining it as a Hecke operator on modular forms spaces of weight k >
2, which means that some authors have w_N^2 = multiplication by
N^{k-2} etc. It is indeed not defined over Q. (Actually it
interchanges two natural rational structures on X1(N) -- there is a
model of X1(N) that classifies elliptic curves with a point of order
N, and a model which classifies modular curves with an embedding of
the group scheme mu_N, and w_N swaps these.)

If this is true then I'm really wondering what the moduli interpretation of w_N should be. Because using the moduli interpretation that I gave you definitely sometimes get things that are not involutions.

If you compute W_q^2 using my interpretation on triples (E,P_1,P_2) where P_1 and P_2 are of order q and N/q respectively then you will notice that W_q(E,P_1,P_2)=(E,P_1,qP_2) or in other words, W_q^2 acts as a diamond operator. This seems to agree with the sage implementation of modular symbols in sage, taking for example N=21 and d=3 one gets that W_q^2 = <10> .  <10> is of order 3 so W_q is of order 6. And indeed:

sage: M=ModularSymbols(Gamma1(21),weight=2)

sage: S=M.cuspidal_subspace()

sage: S.atkin_lehner_operator(3).matrix().minpoly()

x^5 + x^4 + x^3 + x^2 + x + 1

[David Loeffler]

Sorry, you are quite right, I was mistaken: w_q is indeed not an
involution in general.

David

[Maarten Derickx]

Yes this would be a good idea. There is also another thing that I was thinking of working out in order to have a good general implementation in Sage. And that is modular forms and modular symbols for general congruence subgroups. Because I have at least two possible future projects to work on which will depend on this.