Wiles's proof of Fermat's Last Theorem

Sir Andrew John Wiles

Wiles' proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap. The widely accepted version of the proof was released by Andrew Wiles in September 1994, and published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th century techniques not available to Fermat.

The proof itself is over 100 pages long and consumed seven years of Wiles' research time. Among other honors for his accomplishment, he was knighted.

Progress of the previous decades

Fermat's Last Theorem states that no nontrivial integer solutions exist for the equation

$a^{n}+b^{n}=c^{n}\!$

if n is an integer greater than two.

In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on some ideas that Yutaka Taniyama posed. In the West it became well known through a 1967 paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Shimura-Taniyama-Weil conjecture. It states that every rational elliptic curve is modular.

On a separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve. The curve consists of all points in the plane whose coordinates (x, y) satisfy the relation

$y^{2}=x(x-a^{n})(x+b^{n}).\,$

Such an elliptic curve would enjoy very special properties, which are due to the appearance of high powers of integers in its equation and the fact that aⁿ + bⁿ = cⁿ is a nth power as well.

In 1982–1985, Gerhard Frey called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. Again, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrize coordinates x and y of the points on it. Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero a, b, c and p greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction.

In 1985, Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture. Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the Taniyama–Shimura conjecture. Although in the preceding twenty or thirty years a lot of evidence had been accumulated to form conjectures about elliptic curves, the main reason to believe that these various conjectures were true lay not in the numerical confirmations, but in a remarkably coherent and attractive mathematical picture that they presented. Moreover, it could have happened that one or more of these conjectures were actually false.

In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture. (His article was published in 1990.) He demonstrated that, just as Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still unproven at the time), together with the now proven epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama–Shimura conjecture holds for a class of elliptic curves called semistable elliptic curves, then Fermat's Last Theorem would be true.

General approach of proof

Given an elliptic curve E over the field Q of rational numbers $E({\bar {Q}})$, for every prime power $l^{n}$, there exists a homomorphism from the absolute Galois group

$\mathrm {Gal} ({\bar {Q}}/Q)$

to

${\mathit {GL}}_{2}(Z/l^{n}Z)$,

the group of invertible 2 by 2 matrices whose entries are integers ($\mod l^{n}$). This is because $E({\bar {Q}})$, the points of E over ${\bar {Q}}$, form an abelian group, on which $\mathrm {Gal} ({\bar {Q}}/Q)$ acts; the subgroup of elements x such that $l^{n}x=0$ is just $(Z/l^{n}Z)^{2}$, and an automorphism of this group is a matrix of the type described.

Less obvious is that given a modular form of a certain special type, a Hecke eigenform with eigenvalues in Q, one also gets a homomorphism from the absolute Galois group

$\mathrm {Gal} ({\bar {Q}}/Q)\rightarrow {\mathit {GL}}_{2}(Z/l^{n}Z)$.:

This goes back to Eichler and Shimura. The idea is that the Galois group acts first on the modular curve on which the modular form is defined, thence on the Jacobian variety of the curve, and finally on the points of $l^{n}$ power order on that Jacobian. The resulting representation is not usually 2-dimensional, but the Hecke operators cut out a 2-dimensional piece. It is easy to demonstrate that these representations come from some elliptic curve but the converse is the difficult part to prove.

Instead of trying to go directly from the elliptic curve to the modular form, one can first pass to the ($\mod l^{n}$) representation for some l and n, and from that to the modular form. In the case l=3 and n=1, results of the Langlands-Tunnell theorem show that the (mod 3) representation of any elliptic curve over Q comes from a modular form. The basic strategy is to use induction on n to show that this is true for l=3 and any n, that ultimately there is a single modular form that works for all n. To do this, one uses a counting argument, comparing the number of ways in which one can lift a ($\mod l^{n}$) Galois representation to ($\mod l^{n+1}$) and the number of ways in which one can lift a ($\mod l^{n}$) modular form. An essential point is to impose a sufficient set of conditions on the Galois representation; otherwise, there will be too many lifts and most will not be modular. These conditions should be satisfied for the representations coming from modular forms and those coming from elliptic curves. If the original (mod 3) representation has an image which is too small, one runs into trouble with the lifting argument, and in this case, there is a final trick, which has since taken on a life of its own with the subsequent work on the Serre Modularity Conjecture. The idea involves the interplay between the (mod 3) and (mod 5) representations. See Chapter 5 of the Wiles paper for this 3/5 switch.

Wiles' proof

Shortly after learning of the proof of the epsilon conjecture, it was clear that a proof that all rational semistable elliptic curves are modular would also constitute a proof of Fermat's Last Theorem. Wiles decided to conduct his research exclusively towards finding a proof for the Taniyama-Shimura conjecture. Many mathematicians thought the Taniyama-Shimura conjecture was inaccessible to prove because the modular forms and elliptic curves seem to be unrelated.

Wiles opted to attempt to "count" and match elliptic curves to counted modular forms. He found that this direct approach was not working, so he transformed the problem by instead matching the Galois representations of the elliptic curves to modular forms. Wiles denotes this matching (or mapping) that, more specifically, is a ring homomorphism:

$R_{n}\rightarrow T_{n}.$

R is a deformation ring and T is a Hecke ring.

Wiles had the insight that in many cases this ring homomorphism could be a ring isomorphism. (Conjecture 2.16 in Chapter 2, §3) Wiles had the insight that the map between R and T is an isomorphism if and only if two abelian groups occurring in the theory are finite and have the same cardinality. This is sometimes referred to as the "numerical criterion". Given this result, one can see that Fermat's Last Theorem is reduced to a statement saying that two groups have the same order. Much of the text of the proof leads into topics and theorems related to ring theory and commutation theory. The Goal is to verify that the map R → T is an isomorphism and ultimately that R=T. This is the long and difficult step. In treating deformations, Wiles defines four cases, with the flat deformation case requiring more effort to prove and is treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebra".

Gerd Faltings, in his bulletin, on p. 745. gives this commutative diagram:

  --> T -> T/m
 /          ^
R           |
 \          |
  --> Z3 -> F3

or ultimately that R = T , indicating a complete intersection. Since Wiles cannot show that R=T directly, he does so through Z3, F3 and T/m via lifts.

In order to perform this matching, Wiles had to create a class number formula (CNF). He first attempted to use horizontal Iwasawa theory but that part of his work had an unresolved issue such that he could not create a CNF. At the end of the summer of 1991, he learned about a paper by Matthias Flach, using ideas of Victor Kolyvagin to create a CNF, and so Wiles set his Iwasawa work aside. Wiles extended Flach's work in order to create a CNF. By the spring of 1993, his work covered all but a few families of elliptic curves. In early 1993, Wiles reviewed his argument beforehand with a Princeton colleague, Nick Katz. His proof involved the Kolyvagin-Flach method, which he adopted after the Iwasawa method failed. In May 1993 while reading a paper by Mazur, Wiles had the insight that the 3/5 switch would resolve the final issues and would then cover all elliptic curves (again, see Chapter 5 of the paper for this 3/5 switch). Over the course of three lectures delivered at Isaac Newton Institute for Mathematical Sciences on June 21, 22, and 23 of 1993, Wiles announced his proof of the Taniyama–Shimura conjecture, and hence of Fermat's Last Theorem. There was a relatively large amount of press coverage afterwards.

After announcing his results, Katz was a referee on his manuscript and he asked Wiles a series of questions that led Wiles to recognize that the proof contained a gap. There was an error in a critical portion of the proof which gave a bound for the order of a particular group: the Euler system used to extend Flach's method was incomplete. Wiles and his former student Richard Taylor spent almost a year resolving it. Wiles indicates that on the morning of September 19, 1994 he realized that the specific reason why the Flach approach would not work directly suggested a new approach with the Iwasawa theory which resolved all of the previous issues with the latter and resulted in a CNF that was valid for all of the required cases. On 6 October Wiles sent the new proof to three colleagues including Faltings. The new proof was published and, despite its size, widely accepted as likely correct in its major components.

In his 1995 108 page article, Wiles divides the subject matter up into the following chapters (preceded here by page numbers):

443 Introduction

Chapter 1

455 1. Deformations of Galois representations

472 2. Some computations of cohomology groups

475 3. Some results on subgroups of GL₂(k)

Chapter 2

479 1. The Gorenstein property

489 2. Congruences between Hecke rings

503 3. The main conjectures

517 Chapter 3 : Estimates for the Selmer group

Chapter 4

525 1. The ordinary CM case

533 2. Calculation of η

541 Chapter 5 : Application to elliptic curves

545 Appendix: Gorenstein rings and local complete intersections

Gerd Faltings provided some simplifications to the 1995 proof, primarily in switch from geometric constructions to rather simpler algebraic ones. The book of the Cornell conference also contained simplifications to the original proof.

Culmination of the work of many

Because Wiles had incorporated the work of so many other specialists, it had been suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what Wiles has done. The number is likely much larger now with the 10-day conference and book organized by Cornell et al., which has done much to make the full range of required topics accessible to graduate students in number theory. The paper provides a long Bibliography and Wiles mentions the names of many mathematicians in the text. The list of some of the many other mathematicians whose work the proof incorporates includes Felix Klein, Robert Fricke, Adolf Hurwitz, Erich Hecke, Barry Mazur, Dirichlet, Richard Dedekind, Robert Langlands, Jerrold B. Tunnell, Jun-Ichi Igusa, Martin Eichler, André Bloch, Tosio Kato, Ernst S. Selmer, John Tate, P. Georges Poitou, Henri Carayol, Emil Artin, Jean-Marc Fontaine, Karl Rubin, Pierre Deligne, Vladimir Drinfel'd and Haruzo Hida.

Subsequent developments

In 1998, the full modularity theorem was proven by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor using many of the methods that Andrew Wiles used in his 1995 published papers.

A computer science challenge given in 2005 is "Formalize and verify by computer a proof of Fermat's Last Theorem, as proved by A. Wiles in 1995."

Reading and notation guide

The Wiles paper is over 100 pages long and often uses the peculiar symbols and notations of group theory, algebraic geometry, commutative algebra, and Galois theory.

One might want to first read the 1993 email of Ken Ribet, Hesselink's quick review of top-level issues gives just the elementary algebra and avoids abstract algebra., or Daney's web page which provides a set of his own notes and lists the current books available on the subject. Weston attempts to provide a handy map of some of the relationships between the subjects. F. Q. Gouvêa provides an award-winning review of some of the required topics. Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist. For those in search of a commercially available book to guide them, he recommended that those familiar with abstract algebra read Hellegouarch, then read the Cornell book, which is claimed to be accessible to "a graduate student in number theory". Note that not even the Cornell book can cover the entirety of the Wiles proof.

The work of almost every mathematician who helped to lay the groundwork for Wiles did so in specialized ways, often creating new specialized concepts and yet more new jargon. In the equations, subscripts and superscripts are used extensively because of the numbers of concepts that Wiles is sometimes dealing with in an equation.

• See the glossaries listed in Lists of mathematics topics#Pure mathematics, such as Glossary of arithmetic and Diophantine geometry . Daney provides a proof-specific glossary.
• See Table of mathematical symbols and Table of logic symbols
• For the deformation theory, Wiles defines restrictions (or cases) on the deformations as Selmer (sel), ordinary(ord), strict(str) or flat(fl) and he uses the abbreviations list here. He usually uses these as a subscript but he occasionally uses them as a superscript. There is also a fifth case: the implied "unrestricted" case but note that the superscript "unr" is not an abbreviation for unrestricted.
• Q^unr is the unramified extension of Q. A related but more specialized topic used is crystalline cohomology. See also Galois cohomology.
• Some relevant named concepts: Hasse-Weil zeta function, Mordell–Weil theorem, Deligne-Serre theorem
• Grab bag of jargon mentioned in paper: cover and lift, finite field, isomorphism, surjective function, decomposition group, j-invariant of elliptic curves, Abelian group, Grossencharacter, L-function, abelian variety, Jacobian, Néron model, Gorenstein ring, Torsion subgroup (including torsion points on elliptic curves here and here ), Congruence subgroup, eigenform, Character (mathematics), Irreducibility (mathematics), Image (mathematics), dihedral, Conductor, Lattice (group), Cyclotomic field, Cyclotomic character, Splitting of prime ideals in Galois extensions (and decomposition group and inertia group), Quotient space, Quotient group

References

• Aczel, Amir (1997-01-01). MFermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. ISBN 9781568580777. Zbl 0878.11003.
• John Coates (1996). "Wiles Receives NAS Award in Mathematics" (PDF). Notices of the AMS. 43 (7): 760–763. Zbl 1029.01513. {{cite journal}}: Unknown parameter |month= ignored (help)
• Cornell, Gary (1998-01-01). Modular Forms and Fermat's Last Theorem. ISBN 0387946098. Zbl 0878.11004. (Cornell, et al.)
• Daney, Charles (2003). "The Mathematics of Fermat's Last Theorem". Retrieved 2004-08-05.
• Darmon, H. (September 9, 2007). "Wiles' theorem and the arithmetic of elliptic curves" (PDF).
• Faltings, Gerd (1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the AMS. 42 (7): 743–746. ISSN 0002-9920. Zbl 1047.11510. {{cite journal}}: Unknown parameter |month= ignored (help)
• Frey, Gerhard (1986). "Links between stable elliptic curves and certain diophantine equations". Ann. Univ. Sarav. Ser. Math. 1: 1–40. Zbl 0586.10010.
• Hellegouarch, Yves (2001-01-01). Invitation to the Mathematics of Fermat-Wiles. ISBN 0-12-339251-9. Zbl 0887.11003. See review
• "The bluffer's guide to Fermat's Last Theorem". (collected by Lim Lek-Heng)
• Mozzochi, Charles (2000-12-07). The Fermat Diary. American Mathematical Society. ISBN 978-0-821-82670-6. Zbl 0955.11002. (see book review
• Mozzochi, Charles (2006-07-06). The Fermat Proof. Trafford Publishing. ISBN 1412022037. Zbl 1104.11001.
• O'Connor, J. J. (1996). "Fermat's last theorem". Retrieved 2004-08-05. {{cite web}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• van der Poorten, Alfred (1996-01-01). Notes on Fermat's Last Theorem. ISBN 0471062618. Zbl 0882.11001.
• Ribenboim, Paulo (2000-01-01). Fermat's Last Theorem for Amateurs. ISBN 0387985085. Zbl 0920.11016.
• Ribet, Ken (1995). "Galois representations and modular forms" (PDF). Discusses various material which is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama–Shimura.
• Singh, Simon (1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8 zbl = 0930.00002. {{cite book}}: Check |isbn= value: invalid character (help); Missing pipe in: |isbn= (help); Unknown parameter |month= ignored (help) ISBN 0-8027-1331-9
• Simon Singh "The Whole Story". Edited version of ~2,000-word essay published in Prometheus magazine, describing Andrew Wiles' successful journey.
• Richard Taylor and Andrew Wiles (1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Annals of Mathematics. 141 (3). Annals of Mathematics: 553–572. doi:10.2307/2118560. ISSN 0003-486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030. {{cite journal}}: Unknown parameter |month= ignored (help)
• Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3). Annals of Mathematics: 443–551. doi:10.2307/2118559. ISSN 0003-486X. JSTOR 2118559. OCLC 37032255. Zbl 0823.11029. See also this smaller and searchable PDF text version. The smaller PDF gets the volume number correct: it is 141, not 142.

External links

• Weisstein, Eric W. "Fermat's Last Theorem". MathWorld.
• "The Proof". The title of one edition of the PBS television series NOVA, discusses Andrew Wiles' effort to prove Fermat's Last Theorem broadcast on BBC Horizon and UTV/Documentary series as Template:Google video 1996
• Wiles, Ribet, Shimura-Taniyama-Weil and Fermat's Last Theorem
• Are mathematicians finally satisfied with Andrew Wiles' proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?, Scientific American, October 21, 1999