Re: Langlands and FLT

On Aug 13, 8:29 am, Allan Adler <a...@xxxxxxxxxxxxxxxxxxxx> wrote:
No offense, but I think that you're not addressing my questions. I think it
is important to answer this question: let rho0 be a representation of
Gal(Qpbar/Qp) into GL(2,k), where k is a finite field; does rho0 always
arise from a finite flat k-vector scheme G over Qp (resp. over Zp)?

This is a yes or no question.

The candidate for G would be the algebra of all Gal(Qpbar/Qp)-equivariant
functions from k^2 to Qpbar (resp. Zpbar, the ring of integers of Qpbar),
where Gal(Qpbar/Qp) acts on k^2 via rho0 and on Qpbar via the Galois action.

I don't think I can proceed until this question is answered. Please try to
suppress the big picture until that has been done. If you don't know the
answer, just say so.

I thought I had answered this question.

I think the answer is no.

I had written:

I believe all one needs is an elliptic curve at a place
with bad reduction over Q_p. There is a theorem which
states that if E[p^r] is the generic fiber of a flat
group scheme over Z_p for all r > 1, then E_/Q_p has
good reduction.

In other words, I believe that if you take a curve at a
place with bad reduction, you can find an r for which the
representation of E[p^r] is not the generic fiber of a
flat group scheme over Z_p.

I do not know how to explicitly construct an example, but
I think this result at least provides a starting point.

I am earnestly trying to answer your questions, which is one
of the reasons I took my time in response. When I rush to
respond I make many more errors of irrelevance because I get
too focussed on my own pursuits. I am trying to understand
your own questions because I suspect they will force me to
tie together the concrete and abstract elements I have such
a tenuous grasp on. Please forgive the excess exposition that
accompanies.

Does this address your issue or am I still completely missing
the point?
Is there an error in reasoning here?

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

.

Relevant Pages

  • Re: Langlands and FLT
    ... is important to answer this question: let rho0 be a representation of ... Galinto GL, where k is a finite field; does rho0 always ... arise from a finite flat k-vector scheme G over Qp (resp. ...
    (sci.math)
  • Re: Langlands and FLT
    ... In Chap.I, Section 1, he looks at a representation rho0 ... Suppose that the order of the underlying finite flat group scheme is ... examples of representations rho0 arising from finite flat group schemes". ...
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