Re: Langlands and FLT


galathaea writes:

yes i would benefit from a review at this time too

OK.

I appreciate your comments but I don't understand yet how they clarify
what Wiles actually wrote. In the intro, he talks about representations
of Gal(Qbar/Q) into GL(2,Fpbar) and describes briefly what flat representations
are. Of course, Gal(Qbar/Q) is compact and GL(2,Fpbar) is discrete and the
representation is continuous, so the image really lies in GL(2,k) where k
is a finite field. In Chap.I, Section 1, he looks at a representation rho0
of Gal(QSigma/Q) into GL(2,k), where QSigma is the maximal extension of
Q unramified outside of the finite set Sigma which is assumed to include
the prime p. The Galois group of Qpbar over Qp can still be regarded as
a subgroup of Gal(QSigma/Q). He then gives a more precise definition
of what it means for rho0 to be flat.

I don't understand what it means for rho0 to be flat. In case we take
Sigma to be the singleton {p}, rho0 is just a representation of Gal(Qpbar/Qp).
If, as you say, we start with an elliptic curve E, then the representation
on the Tate module is isomorphic to a representation into GL(2,Zp). When
we reduce that modulo powers p^r of p, we don't get a representation
into GL(2,k) unless r=1 and in that case the image really lies in GL(2,F_p).

So, under what conditions do we get a representation into GL(2,k) that doesn't
really map into GL(2,F_p)? I'm writing this whole posting off the top of my
head and don't guarantee that anything I'm writing is correct. With that
caution, I'll proceed. I took another look at Raynaud's paper and my notes
on it and I find I can pick up where I left off. So, I'll continue reading
it. Now, Raynaud is actually classifying finite flat k-vector schemes, i.e.
finite flat groups schemes that have a structure of k-vector space object.
Suppose that the order of the underlying finite flat group scheme is
p^{2n}, where k has p^n elements. Then morally the finite flat k-vector
scheme is a 2-dimensional vector space over k. One way to get that situation
is to consider an abelian variety X of dimension n over Qp whose endomorphism
ring contains the ring of integers of a totally real algebraic number field
F of degree n over Q and assume that p remains prime in F. If all these
endomorphisms are defined over Qp, then the 2n dimensional p-adic
representation of Gal(Qpbar/Qp) on the Tate module of X commutes with
F and is therefore really a 2-dimensional representation of Gal(Qpbar/Qp)
over the p-adic completion of F. Reducing that representation modulo p will
give a representation into GL(2,k) for a finite field k with p^n elements.

Anyway, something like that...

So far, this can all be subsumed under the heading, "How to give some
examples of representations rho0 arising from finite flat group schemes".

So, now let's go back to trying to parse Wiles' definition of a flat
representation. I still don't understand under what conditions a
representation of Gal(Qpbar/Qp) has the following properties:
(1) It has something to do with a finite flat group scheme G.
(2) That something has the effect of mapping the Galois group into GL(2,k)
via the representation rho0.

Now, how does that work if, instead of giving an example of G, we say what
properties G has that we are using to get (1) and (2)?

It's better not to try to give me an overview of the whole paper at
this point, since I'm not ready for the big picture yet. I just want
to understand what is meant be a flat representation. The trouble with
Wiles' definition is that not all of the terms it uses are defined,
notably the terminology, "associated to a finite flat group scheme over Zp".
How is that terminology defined?

Well, you did give a definition:

a representation \rho: G_K -> Aut(M) is called flat
if there exists a finite flat group scheme H_{\R}
such that the G_K representation H(Kbar) is isomorphic to \rho
ie. this is really a property of H -> Spec(R)

OK, let's try to understand this. If rho is isomorphic to the rep'n of G_K
on H(Kbar), then since rho0 maps into GL(2,k), we are tacitly assuming that
H(Kbar) is isomorphic to the k-vector space k^2. And if the finite flat
group scheme H is supposed to be such that it sends schemes (such as
Spec(Kbar)) to k-vector spaces, then presumably it is a finite k-vector scheme.

As I mentioned in my last posting, any finite group can be the group of
Kbar points of a finite flat group scheme: let G be a finite group
and take the direct sum of copies of Spec(Kbar) indexed by G and
use the operations in G to make the resulting scheme into a group scheme
with underlying algebra the algebra A of functions from G to Kbar. Its group
of Kbar points is G. Now take G to be k^2. We get k^2 as the set of Kbar
rational points. But the Galois group doesn't act via rho0. We can let it
act via rho0 and look at Spec of the fixed algebra of that action, but
that wouldn't be semilinear. So the thing to do seems to be to let it
act as follows (maybe changing some s's to their inverses): if s is in
Gal(Qpbar/Qp) and f is a function from k^2 to Qpbar, let f'=s(f) be the
function defined by f'(v) = s(f(rho0(s)v)). The fixed algebra consists
of all functions f such that f(rho0(s)v) = s^-1 f(v).

It looks like one can always do that, but maybe I'm missing something.

Anyway, the question is this: how is the requirement that rho0 arise
from a finite flat group scheme a restriction on rho0? Can you give
an example of a rho0 which does NOT arise from a finite flat group scheme?

i will continue to clarify both for my own benefit and others
but i'd also like to make a caution:

i am not a classically trained mathematician and make many errors a
professional would not however, wiles' proof is one of the major proofs
i have studied rigorously and i have followed an extensive bibliographic
trail to attempt to understand often this means i have learned things out
of order and in places still miss some basic foundations
hopefully others can come in where i am unclear (or even wrong)

OK, I promise not to sue you. Anyway, I'll probably make enough mistakes
for both of us.
--
Ignorantly,
Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx>
* Disclaimer: I am a guest and *not* a member of the MIT CSAIL. My actions and
* comments do not reflect in any way on MIT. Also, I am nowhere near Boston.
.

Relevant Pages

  • Re: Langlands and FLT
    ... In Chap.I, Section 1, he looks at a representation rho0 ... Now, Raynaud is actually classifying finite flat k-vector schemes, i.e. ... Suppose that the order of the underlying finite flat group scheme is ... examples of representations rho0 arising from finite flat group schemes". ...
    (sci.math)
  • Re: Langlands and FLT
    ... based on having looked at the preprint of Wiles' paper. ... But his notion of flat representation is one that, ... arises from a finite flat group scheme. ...
    (sci.math)
  • Re: Langlands and FLT
    ... suppose I start with Wiles' ... as a representation of the decomposition group, ... arises from a finite flat group scheme (among other ... but I think that a finite flat group scheme ...
    (sci.math)
  • Re: Langlands and FLT
    ... either the review or Wiles' paper, ... as a representation of the decomposition group, ... arises from a finite flat group scheme (among ...
    (sci.math)
  • Re: Langlands and FLT
    ... suppose I start with Wiles' ... as a representation of the decomposition group, ... arises from a finite flat group scheme (among other ... but I think that a finite flat group scheme ...
    (sci.math)