Gabor Wiese's Course

MSRI 2006 Summer Graduate Workshop in Computational Number Theory

Please download Wiese's Course Textbook, or get this local copy (if you have trouble with the previous one).


It will depend on what I really cover. See below. It will certainly
have to do with my thesis and some of the resulting papers (which I
would not really recommend as first reading):
G. Wiese. On the faithfulness of parabolic cohomology as a Hecke
module over a finite field.  arXiv:math.NT/0511115
G. Wiese. On modular symbols and the cohomology of Hecke triangle
surfaces. arXiv:math.NT/0511113
G. Wiese. Modular Forms of Weight One Over Finite Fields (PhD thesis),
in particular, Chapter IV (computational details). My web page.
Useful are also Sections 4,5,6 (the appendices are a bit out of date) of
S.J. Edixhoven. Comparison of integral structures on spaces of modular
forms of weight two, and computation of spaces of forms mod 2 of
weight one (arXiv:math.NT/0312019), Journal of the Inst. of
Math. Jussieu (2006) 5(1), 1-34.
If I mention Serre's conjecture, I recommend Ribet/Stein and Serre's
original Duke paper, as well as
S.J. Edixhoven. Serre's conjecture. Cornell, Silverman, Stevens:
Modular Forms and Fermat's Last Theorem. Springer.

Possible Topics

It is a bit difficult to have detailed comments, only knowing the
titles of the lectures and not their contents.
I very essentially need the notion of a Hecke algebra: subring in some
endomorphism ring generated by the Hecke operators. So I need that
they commute, that T_p^n can be got from T_p (and dimanond if
applicable) etc. So, I could give a very short lecture on that on the
first day (unless that is part of "classical modular forms I"). That
lecture would also prove the structure theorem of Artin algebras: they
are the direct product of their localisations - that is much nicer
than just talking about eigenforms, in particular, if there is no
basis of eigenforms which is usually the case over F_p (or, of course,
if there are old forms).
I could treat the following:
1) Modular symbols over (rather) arbitrary rings. William, I guess you
want to do that, which is fine, of course. But I need the definition
in particular over a finite field.
2) Manin symbols over (rather) arbitrary rings. I can give a full
proof that they coincide with modular symbols (at least a complete
sketch) if I'm allowed to use some group cohomology. An ad hoc
definition can be given of group cohomology which already leads very
far. One would have to believe some of the standard facts of group
cohomology (but we also have exercises; maybe, I can find a way of
leading the students to an ad hoc proof of the needed facts, in
3) I essentially need the Eichler-Shimura isomorphism, or the Shokurov
pairing (which I think is much more difficult). I could give a proof
of Eichler-Shimura, at least a rather complete sketch. That would give
a full proof of the modular symbols algorithm for computing
(classical) modular forms, if we identify the modular symbols space
with the cohomology group used in Eichler-Shimura (or Shokurov).
William, if you want to stick to the presentation in your book (which
in the version I read some time ago refers to Merel who refers to
Shokurov), then that's fine for me, of course, I just need some result
of that kind.
4) Mod p modular forms. One already gets very far by seeing them as
reductions of classical modular forms, at least, when the weight is at
least 2. Maybe, I present the Katz definition in passing.  Really
working with it would be too difficult (maybe, also for
me...). Deligne-Rapoport etc. would have to be supposed known. I will
nevertheless mention that there are big differences between the two
definitions in weight one. So, I'll present that area in a black box
5) Computing mod p eigenforms. I'll deduce from the modular symbols
algorithm over the rationals that one gets all mod p eigenforms by
using mod p modular symbols. That is actually very easy.
6) Computing mod p Hecke algebras. In the mod p setting it is
desirable not only to know eigenforms, but the whole Hecke algebra, as
it encodes much more information, e.g. on congruences. I'll present p-
adic Hodge theory in a black box way, as it gives a Z_p-analogue of
the Eichler-Shimura isomorphism, so we get a Z_p-analog of the
algorithm for computing Hecke algebras, hence, also an F_p-analog
which is the one that I'll use. There is, however, the restriction to
weights 2 <= k < p. My result that one can also use k=p on the
ordinary locus will be mentioned, but I think it is too difficult to
prove it, since it makes use of Gross paper in Duke on the tameness
criterion (also making use of Deligne-Rapoport).
7) Computing mod p modular forms of weight 1. That part could present
Edixhoven's algorithm realised on mod p modular symbols. It works by
relating the weight one forms with weight p forms by using the
Frobenius and the Hasse invariant. By the results of 6) one can
compute the Hecke algebra in weight p using mod p modular symbols. The
algorithm then tells one how to use that Hecke algebra to get the
weight one forms.
8) Serre's conjecture. The principal motivations (from my point of
view) of looking at mod p modular forms are congruences and more
importantly Serre's conjecture. I could present it in a nutshell. Now
that Serre's conjecture is on the edge of being completely proved, one
should mention how it can be used from a computational point of view:

    _All 2-dimensional representations (mod p, irreducible, odd) of
    Gal(Qbar/Q) can be computed by computing mod p modular forms._

That sentence is still a bit optimistic (as Edixhoven's project on
that is not yet completely finished). But we can alread now rather
efficiently compute the traces of all Frobenius elements of all these
representations just by computing the eigenvalues of Hecke operators
on mod p modular symbols.
Dont' worry: That the numbers run till 8 does not mean that I want 8
lectures!  1) and 5) are very short. 7) could be dropped. In that case
there would be no mention of Katz' definition of modular forms over a
finite field.