[I hope someone will soon volunteer to paste in these interesting
reports on "what have been doing recently" into a more permanent
record.]
Georg,
Very interesting.
Did you know that although our descriptions of the modular symbol
method sound as if we are working with the space of modular symbols
itself, in fact almost everything we do is in the dual space? By "we"
here I mean both myself (working in the dual space for 20 years) and
William (ditto for 10 or so).
I'm also interested in the integrality question, and my own code is
pretty good at keeping denominators to a minimum (at most 2), though
admittedly that is just for weight 2 and trivial character.
John
On Sat, Jan 29, 2011 at 4:19 PM, Georg S. Weber
<georgswe...@googlemail.com> wrote:
> Hi Sage-Devel,
>
> this January I worked on two topics.
> Firstly, getting Sage working as an integral part of Gentoo Prefix (a
> "hosted" distribution, like Cygwin, MacPorts, ...), under three host
> OSes: 32-bit OS X 10.4, 64-bit OS X 10.6, and Maemo 5 (a Debian-based
> 32-bit Linux on my N900 mobile) --- not with complete success yet, but
> e.g. on OS X 10.4, Sage-on-Prefix at least starts up and is usable.
>
> Secondly, I worked on getting the "dual" notion of the modular symbols
> into Sage.
> I do not have a good name for them yet, for the time being I call them
> "comodular symbols". Technically, such a comodular symbol is a Python
> dictionary whose keys are "all" the 2x2 matrices with entries from ZZ,
> and invariant (in a suitable sense) under some arithmetic subgroup
> Gamma (so we need only ever a finite number of key-value pairs to be
> actually stored). Since every notion is dual, instead of factoring out
> the 2-relations and 3-relations, one has to take the subspace
> invariant under 2- and 3-relations. There are also coboundary symbols
> (Python dicts on the cusps), and ("dual" to the boundary map) a
> coboundary map from these to the comodular symbols, the image of which
> is exactly the Co(?)-Eisenstein subspace of the comodular symbols. And
> so on, etc.pp.
>
> Why do I do this, since from a theoretical point of view, it's "just
> taking the dual(s)"?
>
> On the one hand, that's what half of the papers (from e.g. Greenberg,
> Stevens, Pollack, ...), where "modular symbols" occur, actually mean
> by "modular symbol". And poor me quite regularly messing things up, I
> finally wanted to be able to "put my hands on" these two notions (dual
> to each other) a bit more concretely.
> On the other hand, there are technical reasons to do this.
> Currently, Sage needs to use values from a field when working with
> modular symbols. That's OK for QQ, CC, or finite fileds. But already
> the integers (ZZ) pose a problem. Although I'm pretty confident that
> the current Sage modular symbols code base could be fully
> "integralized" (this implies e.g. finding "good" sets of
> representatives under the 2- and 3-relations), that does not suffice
> for my purposes. Ultimately, I want to work with power series over the
> p-adics as "domain of values" --- and these could only be approximated
> in computations by certain "domains of values" which are rings with
> non-trivial zero-divisors. (Coming not primarily from inevitably
> truncating the power series, but from the coefficent rings being
> essentially all some ZZ mod p^k, where p^k is a prime power, but with
> the power k varying.) For these, the "comodular" computational setting
> seems to have some advantages over the usual (Merel, Stein, ...) "Sage
> modular" computational setting. (Famous last words ... let's see ...)
>
>
> Cheers,
> Georg
>
> --
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