Problem with the chosen filtration

[mmasdeu]

The problem is that at various parts of the code the distribution is "normalized", and this is a destructive operation. In principle one should work with "infinite precision" in the moments 0..k-2 (included), since otherwise one can't compute the action of U_p, for example. Thus we "normalize" we are loosing important information!

I propose that normalize never touches the moments 0..k-2. Moreover, the finite approximation modules should also be always at least of dimension k-1, to accomodate the above moments.

The filtration I suggest we use would be as follows: Fil^N defined to be the set of those distributions that evaluate to 0 when applied to a polynomial of degree <= k-2, and after that the usual decreasing requirement. This makes the approximation modules to be isomorphic to a product of some Z_p's for the first part, and then a product of the form Z/p^N x Z/p^{N-1} x ... x Z/p.

I would like to see what Rob Pollack, David Roe and David Loeffler (and possibly others, of course) think about this...

[loefflerd]

I'm not the expert here (Rob: correct me if I'm wrong) but I don't think what you've just said is quite correct. One can always compute the action of Up on anything on which Sigma0(p) acts, and Sigma0(p) certainly acts on the finite approximation modules.

What I think you mean is that one can't compute the action of Up / p^{k-1} on the approximation modules, which is what one needs to do non-ordinary eigenform lifting. As you point out, one can modify the approximation modules by keeping the first few moments to infinite precision; but this is troublesome because these modified approximation modules cannot be stored on a computer, as they are uncountably infinite!

Rather than explicitly modifying our representation of distributions, I think it's better to follow Remark 8.4 of Pollack-Stevens at this point: when computing lambda^{-1} Up on an approximate lifting of an eigensymbol, we fill in the first few moments to the requisite higher accuracy by using the fact that we already know the classical specialization is an eigensymbol. (Here it is essential that we do not immediately normalize the symbol after applying Up, of course; we need to temporarily keep the lifted symbol in a malformed state before filling in the first few moments at higher accuracy to make it well-formed again.)

[mmasdeu]

I still think that what I explained is correct, let me elaborate. I agree with you that U_p acts on a modular symbol with values in a finite-approximation module. However, this distributions are meant to be used more generally (read: quaternionic "modular symbols"). In those cases, we deal with a p-adic automorphic form (as defined in M.Greenberg's thesis, for example). There, on the one hand we have the distribution endowed with an action of our monoid, but on the other we know that the p-adic automorphic form corresponds to a Gamma-invariant harmonic cocycle (I am thinking of the definite case, but the indefinite one raises the same issues). Even when computing the action of U_p one needs to use this last fact. The difference with the modular symbols setting is that Gamma need not be inside the monoid, and thus does not act on the finite approximation modules!

Therefore even in the ordinary case one needs to deal with this issue. I am not happy about how we solved this in our implementation (in btquotients), so I will try to do a cleaner implementation (the current one assumes that the first moments are p-adic approximations of rationals, and we call rational_approximation()).

[rpollack9974]

I like this idea. In the GL_2 case I think it would neatly solve the problem of acting by U_p / lamba.

But I do think in D_k we need the ability to store the first k+1 moments modulo any given accuracy one wants (and not just have them be untouched by actions). For instance, when you p-stabilize a classical modular symbol, you only know the moments up to some arbitrarily high but fixed accuracy.

Here's a possible scheme: we introduce some base ring of the first k+1 moments -- this could be Q if the moments are stored exactly or it could be Qp(p,M) for some large M. Note this M is independent of the current precision_cap in D_k (it just needs to be at least as big as precision_cap).

This idea also seems to help with the issue of representing D_k and Sym^k simultaneously -- and the whole business with is_symk. In this new scheme, if you form the space of distributions with the weight k action and k+1 moments and specify the base ring to be Q, then you literally get Sym^k(Q).

We should think through all of the mathematical issues carefully, but I think this is doable.