Re: Langlands and FLT

I've been a little swamped with work the past two days, as my team is ahead
of schedule and has been asked to help a hopelessly behind schedule team
pull their act together, and I've been having to do a lot of last minute
rearchitecting to ensure that we can get their feature points completed for
them. However, in what spare time I've had I've written the following which
is not everything I wanted to get to, but which has some important points
that may help clear up some misconceptions. I apologise for any unfinished
thoughts, but I thought it a good idea to post today to give time for review
while I finish up some of the threads of thought tonight.

In article <y93sl6kow4x.fsf@xxxxxxxxxxxxxxxxxxxx>,
Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx> wrote:
!! galathaea <galathaea@xxxxxxxxxx> writes:
!!
!! > In article <y93bqdarwqc.fsf@xxxxxxxxxxxxxxxxxxxx>,
!! > Allan Adler <ara@xxxxxxxxxxxxxxxxxxxx> wrote:
!!
!! > > 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.
!!
!! Ordinary representations are, among other things, reducible. Accordingly,
!! if V is the finite flat k-vector scheme associated to rho0, we have an
!! exact sequence, not necessarily unique, 0 -> V' -> V -> V'' -> 0 of finite
!! flat k-vector schemes. Wiles requires that there is such an exact sequence
!! in which V' is associated to an unramified character Gal(Qpbar/Qp)->Gl(1,k)
!! or, what is the same, V' is the spectrum of a direct sum of rings of
!! integers of unramified extensions of Qp.
!!
!! What you suggested about generalizations to finite flat group schemes
!! of reduction conditions on elliptic curves is interesting and I'd like
!! to understand it, but what I'm trying to do at this point is just to rewrite
!! Wiles' definitions completely in terms of their associated finite flat
!! representations. With the definition of ordinary representation, that
!! wasn't hard to do, although there are still some details I haven't mastered.
!!
!! In the case of flat representations, the first requirement is that the
!! representation is not ordinary, but this requirement can be waived.
!! So, it really isn't a requirement, just a convention that is usually
!! observed, and let us explicitly remove it. The only remaining requirement
!! then is that the determinant of rho0, restricted to the inertia group I_p
!! at p, is the character omega, which Wiles calls the Teichmueller character
!! giving the action on p-th roots of unity.
!!
!! An automorphism sigma of Qpbar acts on the group of p-th roots of unity by
!! raising each p-th root of unity x to the same power, say x^a. So, the action
!! on the p-th roots of unity in effect sends sigma to a mod p, i.e. to an
!! element of the multiplicative group of the field F_p with p-elements. So,
!! this gives us a representation of Gal into GL(1,F_p) and therefore into GL(1,k)
!! and maybe that is what is meant by the Teichmueller character. Wiles requires
!! that det rho0 agree with the Teichmuller on I_p. The kernel of this character
!! on Gal should just be the subgroup of Gal fixing the p-th roots of unity and
!! the associated fixed field should be the extension Q_p(zeta_p) where zeta_p
!! denotes a primitive p-th root of unity. If we replace Qp by the maximal
!! unramified extension Qpnr of Qp, then the character on I_p have for its
!! kernel the subgroup of I_p fixing the field Qpnr(zeta_p).

Teichmuller characters over Fp generate the character group of Fp*, so they
are convenient and have good universal properties. Any character over Fp is an
integral power of omega, and it describes a fixed point of the mod p quotient
(for all a e Fp*, w(a) = a mod p).

For each place p, the inertia groups obey the following exact sequence
(written multiplicatively)

1 --> Ip --> Gal(Qbarp, Qp) --> Gal(Fbarp, Fp) --> 1

where Fbarp is Zbarp / lambda for some maximal ideal lambda of Zbarp.

!! Instead of talking about det rho0, we can look at the second exterior
!! power /\ V of the 2-dimensional k-vector scheme V; presumably this can
!! be defined directly in terms of V without reference to det rho0. This
!! second exterior power will again be the spectrum of a direct sum of rings
!! of integers of fields and the condition about det rho0 should say something
!! to the effect that each of these summands is contained in Qpnr(zeta_p)
!! and that its composite with Qpnr is Qpnr(zeta_p). (I'm writing this off
!! the top of my head and such details need to be checked.)
!!
!! Anyway, now that I look again at Wiles' definition, I see that he says that
!! the notion of flat representation occurs in [Se1] (according to the preprint),
!! which is perhaps a typographical error for [Ser1], which is a paper of Serre,
!! "Sur les representations modulaire de degre 2 de Gal(Qbar/Q)", Duke Math. J.
!! 54 (1987) 179-230. I have a copy of this paper somewhere and I'll try to look
!! at this paper tomorrow to see whether Serre explains it better than Wiles does.
!! According to Wiles, Serre uses the term "finite" instead of "flat".
!!
!! >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?
!!
!! Are you saying that Wiles also uses the terminology "ordinary" and "flat"
!! to describe types of reduction of elliptic curves? Or are you saying that
!! somewhere in the literature, this terminology is used in a single reference?
!! If so, can you tell me exactly where in Wiles' paper, or whatever paper it
!! is?

If an elliptic curve has supersingular reduction at p, then the residual
representation rhobar_{E,p} | Dp is absolutely irreducible. This can be
demonstrated as follows:

Let Tp be the Tate module over E. It consists of vectors (t1, t2, ...) of points
of E over Zp such that p^i t_i = 0 and p t_i = t_(i-1). This is a free module of
dimension 2 over Zp.

Let Vp be Tp (x) Q. It is the module of vectors (v0, v1, ...) of points such that
v0 is arbitrarily chosen of order a power of p and p v_i = v_(i-1). This is a
free module of dimension 2 over Qp.

If rhobar_{E,p} | Dp were not absolutely irreducible, then all elements sigma
of the Galois group would act on a given w elementOf Tp (let w = (w1, w2, ...))
as a one dimensional module over Zp, ie. sigma w would be a p-adic multiple of
w for all sigma e Gal. However, there are only p^n such multiples, and there
is a theorem in supersingular elliptic curve reduction that the ramification
index [K(wn) : K] >= C p^(2n) at places of supersingular reduction, so there
must be some sigma w that will not fit in the p^n points of the p-adic multiples
of w. Thus the assumption is wrong, rhobar_{E,p} | Dp is absolutely irreducible,
and w and sigma w form a basis for the 2d space.

As you mention above, one of the big distinguishing assumptions between the
ordinary case and the flat case is the reducibility assumption. Every place
that Wiles investigates the flat case it seems he is looking at the irreducibility
at places of supersingular reduction. I don't know of any places where the
description is made explicit, but all reviews of Wiles' work seem to make
the implicit association of these terms, and use them loosely interchangeably.

At least, that's been my impression.

!! >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.
!!
!! That assumes the correctness of your hypothesis that the terminology
!! "ordinary" and "flat" for representations is just a renaming of the
!! corresponding concepts for reduction of elliptic curves. That hypothesis
!! hasn't been demonstrated. Nevertheless, as I said above, detailed examination
!! of illuminating examples is always healthy.

Let me give this a second try then. If anything, the repetition will help me.

First, I will write Wiles' definitions, as found in his short (109p) paper
"Modular elliptic curves and Fermat's Last Theorem".

first in intro:
rho0: Gal(Qbar/Q) -> GL(2, Fbarp)

" (I) rho0 is ordinary (at p) by which we mean that there is a one-dimensional
subspace of Fbarp^2 stable under a decomposition group at p and such that
the action on the quotient space is unramified and distinct from the
action on the subspace.

(II) rho0 is flat (at p), meaning that as a representation of a decomposition
group at p, rho0 is equivalent to one that arises from a finite flat group
scheme over Zp, and det rho0 restricted to an inertia group at p is the
cyclotomic character. "

then in body:
rho0: Gal(Q_Sigma/Q) -> GL(2, k)

" (i) rho0 is ordinary; viz., the restriction of rho0 to the decomposition
group Dp has (for a suitable choice of basis) the form

~ / chi1 * \
rho0 | Dp ~ \ 0 chi2 /

where chi1 and chi2 are homomorphisms from Dp to k* with chi2 unramified.
Moreover we require that chi1 =/= chi2. We do allow here that rho0 | Dp
be semisimple. (If chi1 and chi2 are both unramified and rho0 | Dp is
semisimple then we fix our choices of chi1 and chi2 once and for all.)

(ii) rho0 is flat at p but not ordinary (cf. [Se1] where the terminology
finite is used); viz., rho0 | Dp is the representation associated to
a finite flat group scheme over Zp but is not ordinary in the sense of
(i). (In general when we refer to the flat case we will mean that rho0
is assumed not to be ordinary unless we specify otherwise.) We will
assume also that det rho0 | Ip = omega where Ip is an inertia group
at p and omega is the Teichmuller character giving the action on pth
roots of unity. "

Now if I were to try to rewrite these, then like you I would focus on reducibility.
The one-dimensional subspace criteria is a classical rewrite of the condition
that rho0 is reducible (as used above to demonstrate irreducibility of the
supersingular reduction case). The Teichmuller character is merely a convenient
tool to study the irreducible case, and I believe that it does not add any more
restriction on the flat case.

Eventually, these cases will get expanded into deformation types, and much later
the reducibility conditions get pulled over to Jacobians over the relevant schemes
to study properties of the actions of Hecke operators. All of this is a strong
indication to me that the truly functorial property being analysed is the
reducibility of the representations, and I think approaching things from this
angle will help us out when trying to understand the relationship any of this has
to the Langlands programme. In fact, the Langlands Reciprocity Conjecture
specifically concerns irreducible actions...

!! If you don't mind, I would like to go back to what you said earlier about
!! Prop.2.3.1 of Raynaud's paper and Cor.5.10 of Grothendieck's Expose VII
!! in SGA 7 pt.1. I'd like to understand exactly why we are not contradicting
!! these results. Let's start with Cor.5.10. It says (in English):
!!
!! Let A_K be an abelian scheme over the function field K of a tract S.
!! Let l be a prime number. When l = p, we assume char K = 0, in order to be
!! able to use the theorem of Tate already cited in 5.8. In order that A_K
!! have good reduction on S, it is necessary and sufficient that the
!! pro-l-group of Barsotti-Tate T_l(A_K) "have good reduction on S",
!! i.e. extend to a Barsotti-Tate group on S.
!!
!! When I read Grothendieck's Expose, I didn't do much more than try to get
!! a general impression of what was going on. I didn't insist on understanding
!! everything and didn't focus on many details. So, I don't actually know the
!! formal definition of a "Barsotti-Tate group", either over K or over S. If you
!! can provide the definition and a precise reference to where it is formally
!! defined, that would be very helpful in unraveling this.

What Grothendieck calls the Barsotti-Tate group is more commonly called a
"p-divisible" group (and this is the term Wiles uses). It is the inductive limit
lim-> E[p^r] (or more generally the inductive limit of the p^r-torsion groups of
any scheme).

So Grothendieck is stating the equivalence of two conditions:
A_K has good reduction at all places of S
The p-divisible S-group scheme over A is in the generic fibre of a
p-divisible scheme over S.

I will translate Raynaud's proposition, so we can put more of the pieces
together on this one.

Let G = (Gn)_(n>=0) be a p-divisible group defined on K such that for all
n >= 0, Gn is a prolongation of a finite flat R-group scheme. Then G is
a prolongation of a p-divisible R-group scheme Gscript, unique up to
isomorphism.

So, basically, inductive limits preserve the property of being a prolongation
over finite flat R-group schemes if all its members have the property.

Putting both results together, we see that if all contributors to a
p-divisible limit are prolongations of a finite flat scheme, then the limit is
also such a prolongation and has good reduction.

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

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