Re: Langlands and FLT

In article <y93bqdarwqc.fsf@xxxxxxxxxxxxxxxxxxxx>,
Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx> wrote:

!! galathaea <galathaea@xxxxxxxxx> writes:
!! > On Aug 13, 8:51 pm, Allan Adler <a...@xxxxxxxxxxxxxxxxxxxx> wrote:
!!
!! > > Actually, let's look at my construction again. The Galois group Gal acts
!! > > on H and we can write H as a union of orbits. Choose one point from each
!! > > orbit. If h is an orbit representative, let Gal_h denote the subgroup of
!! > > Gal that fixes h. Let K_h denote the fixed field of Gal_h. Then as a
!! > > vector space over Qp, the algebra of Gal-equivariant functions from H
!! > > to Qpbar looks like the direct sum of the fields K_h as h runs over the
!! > > chosen set of orbit representatives.
!! >
!! > Yes.
!!
!! > > If we had looked at the ring of Gal-equivariant functions from H to the
!! > > ring of integers of Qpbar, then instead we would get a ring whose
!! > > underlying Zp module would look like the direct sum of the rings R_h,
!! > > where R_h is the ring of integers of K_h; as a Zp module, it is therefore
!! > > finitely generated and torsion free. Since Zp is a principal ideal domain,
!! > > every finitely generated torsion free module over Zp is free. So my
!! > > construction seems to lead to a Zp algebra whose underlying Zp module
!! > > is free and therefore flat. Therefore, the construction does seem to give
!! > > rise to a finite flat group scheme over Zp.
!! > >
!! > > Do you accept this argument?
!! >
!! > Yes, the direct sum of flat modules is flat, and it is sufficient
!! > (and necessary) for abelian rings that they be torsion-free to be flat.
!! > Not only is Zp a PID, its a full discrete valuation ring, which brings
!! > in all that machinery.
!!
!! OK, I'm glad we agree on that. Now, the reason I have insisted on this picky
!! detail is that I think it helps to clarify Wiles' definition of ordinary and
!! flat representations. Before I criticize Wiles, let me fully acknowledge that
!! he did quite enough by proving Fermat's Last Theorem. He was writing for
!! experts, and I am not an expert. Speaking merely as someone who is not an
!! expert and who is nevertheless trying to read the article (or rather the
!! preprint), I have to say that the explanation of ordinary versus flat
!! representations is quite misleading to the non-expert precisely because
!! it is worded in such a way as to suggest to the uninformed that ordinary
!! representations are not associated to finite flat group schemes and that
!! being associated to a finite flat group scheme is a genuine restriction
!! on a representation.
!!
!! Now that we have clarified that EVERY representation
!! rho0: Gal(Qpbar/Qp) -> GL(2,k) is associated to a finite flat group scheme,
!! I propose that we reread Wiles' definitions of ordinary and flat
!! representations and see what the essential points of the distinction
!! really are.
!!
!! The wording of the definitions suggests that some representations arise
!! just as set theoretic functions from Gal to GL(2,k) while others arise
!! by some magic from finite flat group schemes. Actually, it is the
!! other way around: one STARTS with the representation rho0 and there is
!! always an associated finite flat group scheme, if one wants to look
!! at it. So, now we can try to articulate the distinction between ordinary
!! and flat representations in terms of the associated finite flat group
!! scheme in both cases, for example. I'm not saying that this is how one
!! must do it. I'm just saying that this tiny insight we've obtained should
!! be used to make the distinction between ordinary and flat representations
!! seem more natural and less arbitrary.

Thank you for your very natural explanation. These are very rare and precious.

I have understood the distinction between ordinary and flat as just a renaming
of the ordinary and supersingular distinction at a given reduction place p.

E / Fq (q=p^r) is supersingular if the dual of its q^th power Frobenius is
purely inseparable or equivalently its [p]-torsion map is purely inseparable
and = 0. Otherwise, it is ordinary, which implies that E[p] = Z/pZ (Lang calls
this second case "singular or generic").

The reduction of an elliptic curve can be good, multiplicative, or additive at a
place if the reduced curve is nonsingular, has a node, or has a cusp.

So even though all elliptic curves at good reduction have flat representations,
only the supersingular good ones were called "flat" by Wiles. Elliptic curves
with good ordinary reduction or with multiplicative reduction were called "ordinary".
And Wiles did not conquer the additive case. Does this sound similar to what you
understand?

I have understood this use of "flat" as a reminder of the torsion-free properties
at some place p^r, which makes some sense, but his wording (and the wording of
several others who reviewed his work) does leave the impression that the ordinary
case does not carry flat representations. In order to carry out your approach, I
believe we need to detail where these Galois representations come from and what
spaces they occur over.

The basic idea is that n-division points E[n] from n-torsion are representatives of
the endomorphism that takes curve group points x to nx. The automorphism group of
E[n] is isomorphic to GL(2, Z/nZ) and so the Galois representations on this are of
the form

rho_{E,n}: Gal -> GL(2, Z/nZ)

The projective limit representation over all n is given by

rho_E: Gal -> GL(2, Zhat) (where Zhat is the profinite completion of Z)

and the direct system over all powers of a given place p gives

rho_{E, poo}: Gal -> GL(2, Zp) (where poo is the common "p-infinity" notation).

These are the "natural" representations, so where do the representations
Gal -> GL(2, k) come from? Well, whether Z/nZ, Zhat, or Zp, these rings all
have a residue field k that is finite. So there are natural maps (in fact,
these are projections)
GL(2, Z/nz) -> GL(2, k)
GL(2, Zhat) -> GL(2, k)
GL(2, Zp) -> GL(2, k)
and we can look at the "barred" residual representations over GL(2, k) by composing
the various natural representations with these projections.

I don't want to go too far astray if this is not what you had in mind, so I'll stop
here for now, but I would guess that the next step is to understand what properties
of the ordinary representations make them "easier" to work with and why there needs
to be the separate study of supersingular cases.

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

Relevant Pages

  • Re: Langlands and FLT
    ... the algebra of Gal-equivariant functions from H ... rise to a finite flat group scheme over Zp. ... representations is quite misleading to the non-expert precisely because ... being associated to a finite flat group scheme is a genuine restriction ...
    (sci.math)
  • Re: Langlands and FLT
    ... I propose that we reread Wiles' definitions of ordinary and flat ... Ordinary representations are, among other things, reducible. ... since ordinariness and flatness for representations such as rho0 are ...
    (sci.math)