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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