proof of FLT

Discussion in 'Undergraduate Math' started by James Wong, Feb 1, 2011.

  1. James Wong

    James Wong Guest

    Hey everybody,

    I was a math undergrad, and I've decided to learn graduate level math as a
    hobby by myself. In particular, I'd like to be able to read Wiles' proof of
    Fermat's last theorem. What would be the reading list to do so?

    I think it would start with a book like Lang's or Hungerford's Algebra. What
    is the sequence, with Wiles' paper as the last thing to read?

    Thanks,
    James
     
    James Wong, Feb 1, 2011
    #1
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  2. I should warn you: it is likely to be a somewhat foolish errand to
    attempt...
    It would be very hard to list all the material you need.

    A good "very last book" you would want to read before tackling the
    papers by Wiles and Wiles-Taylor is "Modular Forms and Fermat's Last
    Theorem", edited by Gary Cornell, Joe Silverman, and Glenn Stevens,
    published by Springer. It contains 21 "lectures" whose purpose is
    precisely to introduce the ideas that go into the proof of FLT.

    Leafing through it, you would need to know about Elliptic curves (the
    very basics can be picked up from Joe Silverman and John Tate's
    "Rational Points on Elliptic Curves", but it would be far from
    sufficient; you'd want something like Neal Koblitz's "Modular Forms
    and Elliptic Curves" to complement it, probably followed by
    Silverman's "The Arithmetic of Elliptic Curves"). You would need to
    know about Galois representations (so you need some representation
    theory, more than what you can pick up with Lang or Hungerford). You
    would need to know about Cohomology. You would need to know Algebraic
    Geometry beyond the level of Hartshorne's "Algebraic Geometry" (which
    would be your starting point) in order to deal with finite flat group
    schemes. Some material on modularity and the Langlands Reciprocity
    Conjecture. Some Intersection Theory (Fulton's "Intersection Theory"
    once you're done with the Algebraic Geometry). And some algebraic
    number theory.

    Now, what would that mean? The basics, Lang or Hungerford, are the
    equivalent of 1+year graduate course at a good university. The intro
    to Elliptic curves is one more semester, with another year to go
    through the Koblitz/Silverman's "Arithmetic" book; the material from
    Hartshorne is about another year's worth of graduate courses, with an
    extra semester for group schemes. The algebraic number theory (say, at
    the level of Lang's book, or Marcus's) is another year's worth
    (perhaps a semester and a half), and another semester at least for the
    rest.

    A graduate student making a truly prodigious bee-line to the material,
    working in a university with its support structure, would likely take
    3-4 years if starting from scratch (one year for the basics, one year
    for the advanced material, one year for the really advanced
    preparatory stuff) before tackling the papers. For self-study, as a
    hobby, I would expect at least 3 or 4 times as long, at best.
     
    Arturo Magidin, Feb 1, 2011
    #2
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  3. James Wong

    James Wong Guest

    Thanks for your help. And sorry about the multiposting. It was only until
    that I realized alt.math.undergrad was another place I could ask.
     
    James Wong, Feb 1, 2011
    #3
  4. What's intersection theory? A general framwork for how any intersection
    of topologies for a space is a topology for that space, any intersection
    of subgroups of a group is a subgroup and similar for other algebraic
    structures?
    No thanks, I've alreadly given up on simpler proofs, like every
    monotonically normal space is the contiuous image of a linear order space.
    It's less than 100 pages and not computer assisted.

    ----
     
    William Elliot, Feb 2, 2011
    #4
  5. Intersection Theory is a branch of algebraic geometry that deals with
    intersections of varieties. From the introduction of Fulton's book:

    "If A and B are subvarieties of a non-singular variety X, the
    intersection product A.B should be an equivalence class of algebraic
    cycles closely related to the geometry of how A/\B, A, and B are
    situated in X. Two extreme cases have been most familiar. If the
    intersection is *proper*, i.e., dim(A/\B) = dim(A) + dim(B) - dim X,
    then A.B is a linear combination of the irreducible components of A/
    \B, with coefficients the intersection multiplicities. At the other
    extreme, if A=B is a nonsingular variety, the self-intersection
    formula says that A.B is represented by the top Chern class of the
    normal bundle of A in X. In each case, A.B is represented by the cycle
    on A/\B, well-defined up to rational equivalence on A/\B. One
    consequence of the theory developed here is a construction of, and
    formulas for, the intersection product AB as a rational equivalence
    class of cycles on A/\B, regardless of the dimensions of the
    components of A/\B. We call such classes *refined* intersection
    products. Similarly other intersection formuals such as the Giambelli-
    Thom-Porteous formulas for the degeneracy loci of a vector bundle
    homomorphism, are constructed on and related to the geomtry of these
    loci, including the cases where the loci have excess dimensions.

    "To give an idea of the main thrust of the text, we sketch what we
    call the *basic construction*, from which such refined classes are
    derived. To a closed regular imbedding i:X-->Y of codimension d, and a
    morphism f:V-->Y, with V a k-dimensional variety (or any purely k-
    dimensional scheme), this construction produces a rational equivalence
    class of (k-d)-cycles on W=f^{-1}(X). This "intersection class",
    denoted X._Y V, can be formed as follows. Since i is a regular
    imbedding, the normal cone to X in Y is a vector bundle; let N denote
    the pull-back of this bundle to W. The normal cone C to W in V is a k-
    dimensional closed subscheme of N. Using the lengths of local rings of
    C along its irreducible components as coefficients, C determines an
    algebraic k-cycle, denoted [C], on N. One may construct X._Y V by
    intersecting [C] with the zero section of N. Thus a (k-d)-cycle Sum
    m_i[Z_i] on W represents X._Y V if Sum m_[N_{Z_i}] is rationally
    equivalent to [C] on N, where N_{Z_i} is the restriction of N to Z_i."

    Now, I know you were just being your usual self, i.e., being an ass.
    And you probably got lost right after "subvarieties of a nonsingular
    variety" above (if not sooner). So, perhaps, next time you'll think
    twice before trying to be funny and failing so miserably?
     
    Arturo Magidin, Feb 2, 2011
    #5
  6. That I did as I've not the background
    for such a theory of intersections.
     
    William Elliot, Feb 3, 2011
    #6
  7. Yet, you thought youself in a position ot make fun of the term.

    What an ass.
     
    Arturo Magidin, Feb 3, 2011
    #7
  8. And you falsified the quote by making deletions and not marking them.
    So a lying ass, on top of everything.
     
    Arturo Magidin, Feb 3, 2011
    #8
  9. My guess as to what the term indicated was incorrect
    which could have been quickly stated without loosing your cool.
     
    William Elliot, Feb 4, 2011
    #9
  10. Pull the other one, it's got bells on.

    You weren't "guessing", you were being a smart-ass.
    You've earned the contempt through your hard work over the years,
    William. Enjoy it.

    (No, I don't "loose my cool", nor do I even **LOSE** my cool, when I
    exhibit my contempt for ye; all I do is give you what you deserve.)
     
    Arturo Magidin, Feb 4, 2011
    #10
  11. No thanks.
     
    William Elliot, Feb 4, 2011
    #11
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