Date: Mon, 7 Jun 1999 21:11:19 +0100 (BST)
From: Kevin Buzzard <k.buzzard@ic.ac.uk>
> Definitely.  How about if I write it up and ask you lots of questions?
Alright. I've sent you lots of emails about it, so maybe they contain
some answers. Alternatively, if you simply *get started* writing the
paper (maybe we want something like a brief description of what Hecke does
in there somewhere? and/or what was known about Artin before us?),
and describe the computation or something, then
I'll write the section saying why our computation is good enough
to establish a new case---I thought about that bit a lot and I could
probably write that bit quite quickly. See, I feel that if someone presented
me a 1/2-written paper (and you write so *quickly*!) with a note
saying "you write this section" then I might well just sit down and do it
immediately. But I don't feel at all qualified to write down e.g. a brief
explanation of what Hecke was doing, so I keep putting off writing, and
eventually I just thought I'd ask you. Course, I'm not suggesting you
write a man page for Hecke, but it would be nice to give a quick sketch
of the parts of the computation that you thought were interesting.
>    Well, those are my thoughts. I was looking at a recent pre-print of
>    Fujiwara and I reckon that soon our results might be redundant!
>
> Wow, in what sense?  Can you elaborate.  Does it seem maybe Fujiwara
> will do Artin infinitely many cases of Artin's conjecture?
I don't know what he'll do. He has to prove some level-lowering results
for p=2, and he seems to have proved _something_but he doesn't mention
it in his pre-print. I'm confused as to what he's done for p=2 but I know
he's thinking about it because he sent me an email saying he could do
some cases.
> I had a good guess, but I never spent enough time being sure,
> especially because  I rewrote Hecke so that it avoids ever doing
> that computation.  Now it computes the kernel of T2 (say), then
> uses "fast Tp" to compute all successive kernels.
This might be a bad idea in our case because 2 divides the level
a lot so ker(T_2) might well be huge. Or is this still better than
doing what it used to do? The strategic move would be to either
ask the user, or guess, which T_p-a_p you think will have the smallest kernel!
Were there problems though, even with T_2? I think I recall
computing dim(ker(T_2)) and getting different answers on different runs?
>   June 7-11 writing a paper on component groups.  I have a way to
> compute the numbers c_p appearing in the BSD conjecture for quotients
> A_f of J_0(N).  I'm writing it up nicely with examples, but it's
> getting kind of boring.
My experience is, just write and get it over with. I am very bad at
this, hence I have e.g. a paper that I started writing 2 years ago,
never finished, and now not only do I have to do all the boring things
that I kept putting off, but I have to read the paper and try and
remember what I was doing!

I knew that Helena was going to turn 30 around now (it was always her
birthday during exams, and Cambridge exams are going on right now;
also I am a few months older than her and I am 30 now) but I hadn't
realised it was today. Happy birthday Helena!
(schedule deleted)
You don't _want_to know my schedule. It involves lots of exam
marking, writing some lecture for schoolchildren about solving
equations, seeing students every day about projects, checking
the marking of some other exams, and doing lots of other tedious tasks!
No time at all for maths! Having a job is really bad! But one good
thing about it is that in 3 weeks I'll have finished the marking and
the students will have gone home and so maybe I'll get some work
done.
The thing about p=2 is that I think already several things are known
when it comes to level-lowering. Putting these things together with
Mark Dickenson's thesis would give you a theorem, in some cases.
The thing is, Dickenson still has conditions on his result e.g. he
needs the repn to be semi-stable at all primes and blah blah blah.
I knew that it would take some time to iron out all the troubles
so I was in no hurry to do anything. But now hearing that Fujiwara
is thinking about level-lowering just makes me think that in fact
we should get a move on---after all, it won't be too long a paper.
I'll happily write the bits that you don't want to write, but
*one* of us has to sit down and start, and kind of structure the
paper etc, and to be honest you're _much_better at this kind of
thing than I am, I spend ages thinking about what order to put
sections in and blah blah blah, and never get anywhere.
> I will attempt to produce something about Artin in the next few days.
> Helena is going to be in Denmark working with Ian Kimming so she
> keeps asking me about Artin's conjecture so at least she'll be into it
> to.
Yeah, Kiming will be very interested---he's interested in repns to
GL_2(C) and Helena seems to be getting interested in repns to GL_2(finite)
and the paper we could write gives a nice link between them!
Sorry to ask you to start, but I firmly believe that if we both put
our minds to it it could be over within days (I'll find time somehow,
now I am worried about Fujiwara!)
Kevin