Wiles' Proof of Fermat's Last Theorem, and n=2

Discussion in 'Math Research' started by Anthony Natoli, Jun 29, 2004.

  1. Does anyone know of any publications regarding
    how Wiles' Proof (or related modifications or
    extensions) of Fermat's Last Theorem behaves when n=2?

    If I recall correctly, Wiles' Proof specifically begins
    with the condition n>=5. Since n=3 and n=4 are
    separately proven, this condition n>=5 is sufficient
    to prove FLT for all prime n <> 2.

    I am interested in how Wiles' Proof "fails" for n=2,
    to allow the Pythagorean Theorem to hold.
    Does the Taniyama-Shimura (-Weil) Conjecture fail?
    How does the Frey ellipical curve behave for n=2?
    Do the modular forms corresponding to n=2 reduce
    to simple forms?

    Has there been any discussion on this topic?

    Anthony Natoli
    Anthony Natoli, Jun 29, 2004
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  2. I am not an expert, but since no expert has responded,
    here, for what it's worth, is my understanding: To
    get Fermat's Last Theorem, you need to combine Wiles's
    proof that the Frey curve is modular with Ribet's proof
    that the Frey curve is not modular (from which it
    follows that the Frey curve, along with the alleged FLT
    counterexample from which it arises, cannot exist).

    The part that fails for n=2 and n=3 is not the Wiles
    part but the Ribet part. That's because Ribet's
    argument requires that the action of G(Q-bar,Q) on
    the n-torsion in the Frey curve has to be irreducible.
    For this, one invokes a theorem of Mazur that requires
    n > 4. I can explain almost nothing about how Mazur
    uses that assumption.

    I do not know whether the Frey curves for n=2 are
    well understood in general, but they are certainly
    understood in particular cases. For example, the
    Frey curve associated to the equation 3^2 + 4^2 = 5^2
    is the modular curve X_0(15).

    I am sure that there is some clear intuition due to
    Frey about why the Frey curves should be modular for
    n=2 and n=3 but not modular for n equal to a prime
    greater than 3. Here's where we need that expert.

    Steven E. Landsburg
    Steven E. Landsburg, Jun 30, 2004
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  3. When Wiles announced his proof of Fermat's Last Theorem in 1993,
    quite a few of us went around giving talks to mathematical (and
    non-mathematical) audiences about the proof. We certainly
    discussed the natural question of how the proof breaks down for
    exponent 2.

    Recall that the proof (which is a proof by contradiction) starts
    off with the hypothesis that a^p + b^p = c^p, where a, b, and c
    are specific non-zero integers. One forms the elliptic curve E:
    y^2 = x(x-a^p)(x+b^p), considers the action of the Galois group of
    Q on E[p], and proves that this representation arises from a cusp
    form of impossibly low level. In the case when p=2, the group
    E[p] = E[2] consists of points on E with rational coordinates;
    indeed, the 2-division points on E are of the form (r,0), where r
    is a root of the cubic x(x-a^p)(x+b^p). Thus, the action of the
    Galois group of Q on E[p] is the trivial action. My theorem that
    you can lower levels in this case doesn't work: the proof breaks
    down and the result is false.

    -ken ribet
    Kenneth A. Ribet, Jul 2, 2004
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