Re: Langlands and FLT

Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx> writes:

Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx> writes:
Is the tensor product of the algebra A (of Gal equivariant functions from
k^2 to Zpbar) with Zpbar the algebra of all functions from k^2 to Zpbar?
Here Zpbar is the ring of integers of Qpbar.

I think the answer to this is no.

I still think so.

let r1,...,rn be a basis for R over Zp. K tensor Qpbar over Qp is isomorphic
to the direct sum of n copies of Qpbar via the map that sends x in Qp to
(s1(x),...,sn(x)). If we look at the same map on R tensor Zpbar over Zp
into the direct sum of n copies of Zpbar, it has a matrix M whose i,j-th
entry is si(rj). The square of the determinant of M is the discriminant
of the basis r1,...,rn over Qp and that suggests that ramification of K
over Qp is an obstacle to extending the group scheme over Qp associated to
rho0 to a finite flat group scheme over Zp. In particular, it suggests that
a necessary and sufficient condition for the group scheme to extend over Zp
is that the fixed field of the stabilizer in Gal of each element of k^2 be
unramified over Qp.

Strictly speaking, what I wrote doesn't support these last two suggestions.
What I think is actually proved is that A tensor Zpbar over Zp is the
algebra of functions from k^2 to Zpbar if and only if the fixed field of
the stabilizer in Gal of each element of k^2 is unramified over Qp.

Since subfields and composites of unramified extensions are unramified,
the latter condition is equivalent to the assertion that the fixed field
of ker(rho0) is unramified over Qp. In that case, the image of rho0
is isomorphic to the Galois group over Qp of a finite unramified extension
K of Qp and is therefore cyclic. So, in this case, rho0 is completely
described by specifying the element rho0(Frob) of GL(2,k). I'll consider
that later.

In case ker(rho0) has an unramified fixed field, then I think the algebra
A of Gal-equivariant functions from k^2 to Zpbar does have a spectrum which
is a finite flat group scheme over Zp. As I indicated in an earlier posting,
there is a natural "comultiplication" m from A to B, where B is the algebra of
Gal-equivariant functions from k^2 x k^2 to Zpbar, given by composition
with the addition from k^2 x k^2 to k^2. However, a genuine comultiplication
should map A to A tensor A over Zp. There is a natural map from A tensor A
over Zp to B, which takes f tensor g to the function f(x)g(y) on k^2 x k^2.
If it is an isomorphism, we can use its inverse to regard m as being valued
in A tensor A over Zp, and that is reassuring for the construction of a
group law on the proposed group scheme Spec(A). If we tensor m with Zpbar
over Zp, and if A tensor Zpbar over Zp equals the algebra of all functions
from k^2 to Zpbar, then I think we get an isomorphism, and that m itself
is therefore an isomorphism. So, that's why I think that my construction
does give a finite flat k-vector scheme over Zp if the fixed field of
ker(rho0) is unramified over Zp.

This doesn't address the question of whether there might be some other
way of extending to a finite flat k-vector scheme over Zp when the fixed
field of ker(rho0) is ramified.

This is certainly not what Wiles has in mind when he talks about flat
representations. His flat representations have determinant omega. Since
the kernel of det(rho0) contains the kernel of rho0, the fixed field of
ker(det(rho0)) is a subfield of the fixed field of ker(rho0) and, in
the situation I described above, where the fixed field of ker(rho0) is
unramified, it follows that the fixed field of ker(det(rho0)) is also
unramified. But the character omega doesn't have an unramified fixed field.
So, my construction definitely does not lead to a flat representation
in Wiles' sense when the fixed field of ker(rho0) is unramified over Qp.

If we take rho0 to be the direct sum of two distinct unramified characters,
then we do get an ordinary representation and, moreover, Spec(A) will be
a finite flat k-vector scheme over Zp.

As shown above, the condition that the fixed field (let's call it K(rho0))
of ker(rho0) be unramified over Zp is necessary and sufficient for A
to become the algebra of functions from k^2 to Zpbar when A is tensored with
Zpbar over Zp. The latter condition was desired in order to prove that
A tensor A over Zp is isomorphic to B, which in turn was desired in order
to get a comultiplication from A to A tensor A over Zp from the natural
map m from A to B. But it is at least conceivable that the natural map
from A tensor A over Zp to B (let's call it c for brevity) is an isomorphism
even if A tensor Zpbar over Zp is not the algebra of all functions from k^2
to Zpbar. So, that offers some hope for this construction even in cases
where K(rho0) is ramified over Qp. Furthermore, we don't even really need for
A tensor A over Zp to be isomorphic to B. We only need for the image in B
of A tensor A over Zp to contain the image of m: A -> B. So that offers a
little more hope.

I've been looking at the maps c and m, in the hope of being able to compare
their images in B. That seems to involve examination of double cosets of
stabilizers of elements of k^2 but it seems to be somewhat complicated
at this stage and I don't understand it yet.

This particular construction Spec(A) is, admittedly, arbitrary but it is
readily available and convenient to describe. It might serve as a handy
example or as a lab rat. In some cases, what Wiles is describing in rather
abstract terms might actually be an example of this concrete construction.
I don't know. But even to prove that it is no good for anything makes it
necessary to learn a certain amount of the relevant theory, which I need to
do anyway. It also involves playing with the rat, which is often a lot more
fun than fighting one's way through the relevant literature. So, I'm inclined
to keep looking at this construction, even if it is ultimately a waste of time.

Just to be perfectly clear about what actual progress has been made in
understanding Wiles' paper: I still don't even properly understand the
notions of ordinary and flat representations, which are the first step
in Wiles' paper.

I think that instead of working with the preprint, I should obtain a copy of
Wiles' paper from JSTOR and work with the published version.
--
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.
.