Tutorial: Computing With Modular Forms Using Magma
I wrote the first version of all the modular forms code in Magma.
Steve Donnelly has taken over this code.
The following general direction for the future of modular forms in Magma
is based on an email on Sunday from Steve Donnelly.
> If possible could you give me a very brief overview of where you
> see modular forms headed in Magma in the future?
Sure, in fact I've been meaning to send you an update on stuff we've done.
- Allan and I are gradually making speed-ups
(and maybe saving memory), and we'll continue to do that
based on examples people send us.
- The main new feature in this year's release will be
q-expansions of half-integral weight forms (obtained in a
simple-minded way from certain spaces of integral weight,
either k−1/2 or k+1/2). I'll add more functionality later
(Hecke operators, at least).
- Kevin Buzzard said he would let us have the weight one forms
- At some stage we want to build or install a large
database for various weight/level -- I'll ask your advice
about it later.
- I'm currently working on routines, for higher weight spaces, to
get the q-expansions by taking products of lower weight forms.
There are a few other small features including Atkin-Lehner
involutions for some spaces with characters that didn't have them.
The biggest thing I have in mind for the future would be
Hilbert modular forms, and in fact Lassina Dembele has agreed
to visit us at some point. Right now John Voight is here for
a short visit, he has written a bunch of machinery to do with
Shimura curves (computing fundamental domains and so on) which
will probably be released this year.