# proof of FLT

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

1. ### James WongGuest

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

2. ### Arturo MagidinGuest

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,
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
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
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

3. ### James WongGuest

Thanks for your help. And sorry about the multiposting. It was only until

James Wong, Feb 1, 2011
4. ### William ElliotGuest

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
5. ### Arturo MagidinGuest

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
6. ### William ElliotGuest

That I did as I've not the background
for such a theory of intersections.

William Elliot, Feb 3, 2011
7. ### Arturo MagidinGuest

Yet, you thought youself in a position ot make fun of the term.

What an ass.

Arturo Magidin, Feb 3, 2011
8. ### Arturo MagidinGuest

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
9. ### William ElliotGuest

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
10. ### Arturo MagidinGuest

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
11. ### William ElliotGuest

No thanks.

William Elliot, Feb 4, 2011