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Subject: sci.math FAQ: Wiles attack
Summary: Part 6 of many, New version,
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Archive-Name: sci-math-faq/FLT/Wiles
Last-modified: December 8, 1994
Version: 6.2



Wiles' line of attack



   Here is an outline of the first, incorrect proposed proof. The bits
   about Euler system are

          From Ken Ribet:

          Here is a brief summary of what Wiles said in his three
          lectures.

          The method of Wiles borrows results and techniques from lots
          and lots of people. To mention a few: Mazur, Hida, Flach,
          Kolyvagin, yours truly, Wiles himself (older papers by Wiles),
          Rubin... The way he does it is roughly as follows. Start with a
          mod p representation of the Galois group of Q which is known to
          be modular. You want to prove that all its lifts with a certain
          property are modular. This means that the canonical map from
          Mazur's universal deformation ring to its maximal Hecke algebra
          quotient is an isomorphism. To prove a map like this is an
          isomorphism, you can give some sufficient conditions based on
          commutative algebra. Most notably, you have to bound the order
          of a cohomology group which looks like a Selmer group for Sym^2
          of the representation attached to a modular form. The
          techniques for doing this come from Flach; and then the proof
          went on to use Euler systems a la Kolyvagin, except in some new
          geometric guise. [This part turned out to be wrong and
          unnecessary].

          If you take an elliptic curve over Q , you can look at the
          representation of Gal on the 3-division points of the curve. If
          you're lucky, this will be known to be modular, because of
          results of Jerry Tunnell (on base change). Thus, if you're
          lucky, the problem I described above can be solved (there are
          most definitely some hypotheses to check), and then the curve
          is modular. Basically, being lucky means that the image of the
          representation of Galois on 3-division points is GL(2,Z/3Z) .

          Suppose that you are unlucky, i.e., that your curve E has a
          rational subgroup of order 3. Basically by inspection, you can
          prove that if it has a rational subgroup of order 5 as well,
          then it can't be semistable. (You look at the four non-cuspidal
          rational points of X_0(15) .) So you can assume that E[5] is
          ``nice''. Then the idea is to find an E' with the same
          5-division structure, for which E'[3] is modular. (Then E' is
          modular, so E'[5] = E[5] is modular.) You consider the modular
          curve X which parameterizes elliptic curves whose 5-division
          points look like E[5] . This is a twist of X(5) . It's
          therefore of genus 0, and it has a rational point (namely, E ),
          so it's a projective line. Over that you look at the
          irreducible covering which corresponds to some desired
          3-division structure. You use Hilbert irreducibility and the
          Cebotarev density theorem (in some way that hasn't yet sunk in)
          to produce a non-cuspidal rational point of X over which the
          covering remains irreducible. You take E' to be the curve
          corresponding to this chosen rational point of X .


          From the previous version of the FAQ:

          (b) conjectures arising from the study of elliptic curves and
          modular forms. - The Taniyama-Weil-Shmimura conjecture.

          There is a very important and well known conjecture known as
          the Taniyama-Weil-Shimura conjecture that concerns elliptic
          curves. This conjecture has been shown by the work of Frey,
          Serre, Ribet, et. al. to imply FLT uniformly, not just
          asymptotically as with the ABC conjecture.

          The conjecture basically states that all elliptic curves can be
          parameterized in terms of modular forms.

          There is new work on the arithmetic of elliptic curves. Sha,
          the Tate-Shafarevich group on elliptic curves of rank 0 or 1.
          By the way an interesting aspect of this work is that there is
          a close connection between Sha, and some of the classical work
          on FLT. For example, there is a classical proof that uses
          infinite descent to prove FLT for n = 4 . It can be shown that
          there is an elliptic curve associated with FLT and that for n =
          4 , Sha is trivial. It can also be shown that in the cases
          where Sha is non-trivial, that infinite-descent arguments do
          not work; that in some sense ``Sha blocks the descent''.
          Somewhat more technically, Sha is an obstruction to the
          local-global principle [e.g. the Hasse-Minkowski theorem].


          From Karl Rubin:







          It has been known for some time, by work of Frey and Ribet,
          that Fermat follows from this. If u^q + v^q + w^q = 0 , then
          Frey had the idea of looking at the (semistable) elliptic curve
          y^2 = x(x - a^q)(x + b^q) . If this elliptic curve comes from a
          modular form, then the work of Ribet on Serre's conjecture
          shows that there would have to exist a modular form of weight 2
          on Gamma_0(2) . But there are no such forms.

          To prove the Theorem, start with an elliptic curve E , a prime
          p and let rho_p : Gal(\bar(Q)/Q) --> GL_2(Z/pZ) be the
          representation giving the action of Galois on the p -torsion
          E[p] . We wish to show that a certain lift of this
          representation to GL_2(Z_p) (namely, the p -adic representation
          on the Tate module T_p(E) ) is attached to a modular form. We
          will do this by using Mazur's theory of deformations, to show
          that every lifting which ``looks modular'' in a certain precise
          sense is attached to a modular form.

          Fix certain ``lifting data'', such as the allowed ramification,
          specified local behavior at p , etc. for the lift. This defines
          a lifting problem, and Mazur proves that there is a universal
          lift, i.e. a local ring R and a representation into GL_2(R)
          such that every lift of the appropriate type factors through
          this one.

          Now suppose that rho_p is modular, i.e. there is some lift of
          rho_p which is attached to a modular form. Then there is also a
          hecke ring T , which is the maximal quotient of R with the
          property that all modular lifts factor through T . It is a
          conjecture of Mazur that R = T , and it would follow from this
          that every lift of rho_p which ``looks modular'' (in particular
          the one we are interested in) is attached to a modular form.

          Thus we need to know 2 things:
        (a)
            rho_p is modular
        (b)
            R = T .


          It was proved by Tunnell that rho_3 is modular for every
          elliptic curve. This is because PGL_2(Z/3Z) = S_4 . So (a) will
          be satisfied if we take p = 3 . This is crucial.

          Wiles uses (a) to prove (b) under some restrictions on rho_p .
          Using (a) and some commutative algebra (using the fact that T
          is Gorenstein, basically due to Mazur) Wiles reduces the
          statement T = R to checking an inequality between the sizes of
          2 groups. One of these is related to the Selmer group of the
          symmetric square of the given modular lifting of rho_p , and
          the other is related (by work of Hida) to an L -value. The
          required inequality, which everyone presumes is an instance of
          the Bloch-Kato conjecture, is what Wiles needs to verify.

          [This is the part that turned out to be wrong in the first
          version]. He does this using a Kolyvagin-type Euler system
          argument. This is the most technically difficult part of the
          proof, and is responsible for most of the length of the
          manuscript. He uses modular units to construct what he calls a
          geometric Euler system of cohomology classes. The inspiration
          for his construction comes from work of Flach, who came up with
          what is essentially the bottom level of this Euler system. But
          Wiles needed to go much farther than Flach did. In the end,
          under certain hypotheses on rho_p he gets a workable Euler
          system and proves the desired inequality. Among other things,
          it is necessary that rho_p is irreducible.

          [The new proof replaces the argument above with one using
          commutative algebra and and some clever observations by De
          Shalit to fill in the gap.]

          Suppose now that E is semistable.

          Case 1. rho_3 is irreducible.
          Take p = 3. By Tunnell's theorem (a) above is true. Under these
          hypotheses the argument above works for rho_3 , so we conclude
          that E is modular.

          Case 2. rho_3 is reducible. Take p = 5 . In this case rho_5
          must be irreducible, or else E would correspond to a rational
          point on X_0(15) . But X_0(15) has only 4 noncuspidal rational
          points, and these correspond to non-semistable curves. If we
          knew that rho_5 were modular, then the computation above would
          apply and E would be modular.

          We will find a new semistable elliptic curve E' such that
          rho_(E,5) = rho_(E',5) and rho_(E',3) is irreducible. Then by
          Case I, E' is modular. Therefore rho_(E,5) = rho_(E',5) does
          have a modular lifting and we will be done.

          We need to construct such an E' . Let X denote the modular
          curve whose points correspond to pairs (A, C) where A is an
          elliptic curve and C is a subgroup of A isomorphic to the group
          scheme E[5] . (All such curves will have mod-5 representation
          equal to rho_E .) This X is genus 0, and has one rational point
          corresponding to E , so it has infinitely many. Now Wiles uses
          a Hilbert Irreducibility argument to show that not all rational
          points can be images of rational points on modular curves
          covering X , corresponding to degenerate level 3 structure
          (i.e. im(rho_3) != GL_2(Z/3) ). In other words, an E' of the
          type we need exists. (To make sure E' is semistable, choose it
          5-adically close to E . Then it is semistable at 5, and at
          other primes because rho_(E',5) = rho_(E,5) .)




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