Wiles' proof of Fermat's Last Theorem

The proof itself is over 150 pages long and consumed seven years of Wiles' research time.[1] John Coates described the proof as one of the highest achievements of number theory, and John Conway called it the proof of the century.[2] For solving Fermat's Last Theorem, he was knighted, and received other honours.

Progress of the previous decades

Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy 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 ideas posed by Yutaka Taniyama. 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 Taniyama–Shimura–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.[3] 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 \( a^n + b^n=c^n \! \)is an 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 parametrise 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 n greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. As such, a proof or disproof of either of Fermat's Last Theorem or the Taniyama–Shimura–Weil conjecture would simultaneously prove or disprove the other.[4]

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 of his proposal; the missing part became known as the epsilon conjecture or ε-conjecture (now known as Ribet's theorem). 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 much 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. Equally it could happen that one or more of these conjectures were actually untrue.

Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to show that Frey's intuition was correct: that the above elliptic curve (now known as a Frey curve), if it exists, is always non-modular. Frey did not quite succeed in proving this rigorously; the missing piece (the so-called "epsilon conjecture", now known as Ribet's theorem) was noticed by Jean-Pierre Serre[5]:1 and proved in 1986 by Ken Ribet. Second, it was necessary to prove the modularity theorem—or at least to prove it for the sub-class of cases (known as semistable elliptic curves) which included Frey's equation.

Ribet's theorem—if proved—would show that any solution to Fermat's equation could be used to generate a semistable elliptic curve that was not modular;
The modularity theorem—if proved for Frey's equation—would show that all such elliptic curves must be modular.
The contradiction implies that no solutions can exist to Fermat's equation, thus proving Fermat's Last Theorem.

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 not proved at the time), together with the now proved epsilon conjecture, implies Fermat's Last Theorem. Thus, if the Taniyama–Shimura conjecture is true for semistable elliptic curves, then Fermat's Last Theorem would be true. However this theoretical approach was widely considered unattainable, since the Taniyama–Shimura conjecture was itself widely seen as completely inaccessible to proof with current knowledge.[6]:203–205, 223, 226 For example, Wiles' ex-supervisor John Coates states that it seemed "impossible to actually prove",[6]:226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".[6]:223

Hearing of the 1986 proof of the epsilon conjecture, Wiles decided to begin working exclusively towards a proof of the Taniyama–Shimura conjecture. Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]." [6]:223
Wiles' proof
Overview

Wiles opted to attempt to 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) He realised 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 being treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebras".

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

Faltings diagram

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 extending approaches which had recently been developed by Kolyvagin and Flach,[7] which he adopted after the Iwasawa method failed.[8] 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).
General approach and strategy

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

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

to

\( \mathrm{GL}_2(\mathbf{Z}/l^n \mathbf{Z}) , \)

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

\( \mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q}) \rightarrow \mathrm{GL}_2(\mathbf{Z}/l^n \mathbf{Z}).: \)

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.
Structure of Wiles' proof

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

Introduction

443

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

Chapter 3

517 Estimates for the Selmer group

Chapter 4

525 1. The ordinary CM case
533 2. Calculation of η

Chapter 5

541 Application to elliptic curves

Appendix

545 Gorenstein rings and local complete intersections

Gerd Faltings subsequently provided some simplifications to the 1995 proof, primarily in switch from geometric constructions to rather simpler algebraic ones.[9][10] The book of the Cornell conference also contained simplifications to the original proof.[5]
Reading and notation guide

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

One might want to first read an email Ken Ribet sent in 1993,[11][12] Hesselink's quick review of top-level issues, which gives just the elementary algebra and avoids abstract algebra,[13] 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.[14] F. Q. Gouvêa provides an award-winning review of some of the required topics.[15][16][17][18] Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist.[19] 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,[5] 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.[20]

The work of almost every mathematician who helped to lay the groundwork for Wiles did so in specialised ways, often creating new specialised concepts and yet more new jargon. Subscripts and superscripts are used extensively in the equations because of the numbers of concepts that Wiles is sometimes dealing within 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.
Qunr is the unramified extension of Q. A related but more specialised 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, Grössencharacter, L-function, abelian variety, Jacobian, Néron model, Gorenstein ring, torsion subgroup (including torsion points on elliptic curves here[21] and here[22]), 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.

Announcement and subsequent developments

Wiles' proof was initially presented in 1993. It was finally accepted as correct, and published, in 1995, because of an error in one piece of his initial paper. His work was extended to a full proof of the modularity theorem over the following 6 years by others, who built on Wiles' work.
Announcement and final proof (1993–1995)

During 21–23 June 1993 Wiles announced and presented his proof of the Taniyama–Shimura conjecture for semi-stable elliptic curves, and hence of Fermat's Last Theorem, over the course of three lectures delivered at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England.[1] There was a relatively large amount of press coverage afterwards.[20]

After the announcement, Katz was appointed as one of the referees to review Wiles' manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap. There was an error in one 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. The error would not have rendered his work worthless – each part of Wiles' work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.[6]:289, 296–297 Without this part proved, however, there was no actual proof of Fermat's Last Theorem.

Wiles and his former student Richard Taylor spent almost a year resolving this issue.[23][24] Wiles indicates that on the morning of 19 September 1994 he realised that the specific reason why the Flach approach would not work directly suggested a new approach based on his previous attempts using Iwasawa theory, which resolved the issue and resulted in a CNF that was valid for all of the required cases. On 6 October Wiles asked three colleagues (including Faltings) to review his new proof,[9] and on 24 October 1994 Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[25] and "Ring theoretic properties of certain Hecke algebras",[26] the second of which Wiles had written with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper.

The two papers were vetted and finally published as the entirety of the May 1995 issue of the Annals of Mathematics. The new proof was widely analysed, and became accepted as likely correct in its major components.[27][28] These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
Popular accessibility

Fermat himself famously claimed to "...have discovered a truly marvelous proof of this, which this margin is too narrow to contain",[29] but Wiles's proof is very complex, and incorporates the work of so many other specialists that it was suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what he had done.[1][30] The number is likely much larger now with the 10-day conference and book organised by Cornell et al.,[5] which has done much to make the full range of required topics accessible to graduate students in number theory.

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

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."[31]
Notes

Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times. Retrieved 21 January 2013.
http://www.pbs.org/wgbh/nova/transcripts/2414proof.html
Hellegouarch, Yves (2001). Invitation to the Mathematics of Fermat–Wiles. Academic Press. ISBN 978-0-12-339251-0.
Singh, pp. 194–198; Aczel, pp. 109–114.
G. Cornell, J. H. Silverman and G. Stevens, Modular forms and Fermat's Last Theorem, ISBN 0-387-94609-8
Fermat's Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0
Singh, Simon. Fermat's Last Theorem, 2002, p. 259.
Singh, Simon. Fermat's Last Theorem, 2002, p. 260.
Fermat's Last Theorem at MacTutor
Fermat's Last Theorem 1996
FAQ: Wiles attack June 1993
Fermat's Last Theorem a Theorem at last August 1993
How does Wiles prove Fermat's Last Theorem? by Wim H. Hesselink
Research Summary Topics
A Marvelous Proof Fernando Gouvêa, The American Mathematical Monthly, vol. 101, 1994, pp. 203–222
The Mathematical Association of America's Lester R. Ford Award
Year of Award: 1995
MAA Writing Awards, 1995
Gerd Faltings, The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, Notices of the AMS, 42/7
AMS book review Modular forms and Fermat's Last Theorem by Cornell et al., 1999
"Torsion points on elliptic curves". Mat.uab.es. 13 May 2003. Retrieved 24 February 2014.
[1]
A Year Later, Snag Persists In Math Proof 28 June 1994
26 June – 2 July; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D. 3 July 1994
Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics (Annals of Mathematics) 141 (3): 443–551. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.
Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics (Annals of Mathematics) 141 (3): 553–572. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255.
NOVA Video, The Proof, 28 October 1997. See also Solving Fermat: Andrew Wiles
The Proof of Fermat's Last Theorem Charles Daney, 1996
"Why Pierre de Fermat is the patron saint of unfinished business", Eoin O'Carroll, 17 August 2011, csmonitor.com
History of Fermat's Last Theorem Andrew Granville, 24 June 1993

Computer verification of Wiles's proof of Fermat's Last Theorem

References

Aczel, Amir (1 January 1997). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. ISBN 978-1-56858-077-7. Zbl 0878.11003.
John Coates (July 1996). "Wiles Receives NAS Award in Mathematics" (PDF). Notices of the AMS 43 (7): 760–763. Zbl 1029.01513.
Cornell, Gary (1 January 1998). Modular Forms and Fermat's Last Theorem. ISBN 0-387-94609-8. Zbl 0878.11004. (Cornell, et al.)
Daney, Charles (2003). "The Mathematics of Fermat's Last Theorem". Retrieved 5 August 2004.
Darmon, H. (9 September 2007). "Wiles’ theorem and the arithmetic of elliptic curves" (PDF).
Faltings, Gerd (July 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.
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 (1 January 2001). 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 (7 December 2000). The Fermat Diary. American Mathematical Society. ISBN 978-0-8218-2670-6. Zbl 0955.11002. (see book review
Mozzochi, Charles (6 July 2006). The Fermat Proof. Trafford Publishing. ISBN 1-4120-2203-7. Zbl 1104.11001.
O'Connor, J. J.; Robertson, E. F. (1996). "Fermat's last theorem". Retrieved 5 August 2004.
van der Poorten, Alfred (1 January 1996). Notes on Fermat's Last Theorem. ISBN 0-471-06261-8. Zbl 0882.11001.
Ribenboim, Paulo (1 January 2000). Fermat's Last Theorem for Amateurs. ISBN 0-387-98508-5. 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 (October 1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8. Zbl 0930.00002.
Simon Singh "The Whole Story". Edited version of ~2,000-word essay published in Prometheus magazine, describing Andrew Wiles's successful journey.
Richard Taylor and Andrew Wiles (May 1995). "Ring-theoretic properties of certain Hecke algebras" (PDF). Annals of Mathematics (Annals of Mathematics) 141 (3): 553–572. doi:10.2307/2118560. ISSN 0003-486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030.
Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics (Annals of Mathematics) 141 (3): 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 larger PDF misquotes the volume number as 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's effort to prove Fermat's Last Theorem that broadcast on BBC Horizon and UTV/Documentary as Fermat's Last Theorem
Wiles, Ribet, Shimura–Taniyama–Weil and Fermat's Last Theorem
Are mathematicians finally satisfied with Andrew Wiles's proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?, Scientific American, 21 October 1999

1 Overview
1.1 Subsequent developments and solution
2 Mathematical history
2.1 Pythagoras and Diophantus
2.1.1 Pythagorean triples
2.1.2 Diophantine equations
2.2 Fermat's conjecture
2.3 Proofs for specific exponents
2.3.1 Sophie Germain
2.3.2 Ernst Kummer and the theory of ideals
2.3.3 Mordell conjecture
2.3.4 Computational studies
2.4 Connection with elliptic curves
2.4.1 Taniyama–Shimura–Weil conjecture
2.4.2 Frey's equation / Ribet's theorem
2.4.3 Wiles's general proof
2.5 Subsequent developments
3 Exponents other than positive integers
3.1 Reciprocal Integers (Inverse Fermat Equation)
3.2 Rational exponents
3.3 Negative exponents
3.3.1 n = –1
3.3.2 n = –2
3.3.3 Integer n < –2
4 Base values other than positive integers
5 Monetary prizes
6 See also
7 Footnotes
8 References
9 Bibliography
10 Further reading

All proofs for specific exponents used Fermat's technique of infinite descent,[citation needed] either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ad hoc and tied to the individual exponent under consideration.[97] Since they became ever more complicated as p increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents.[97] Although some general results on Fermat's Last Theorem were published in the early 19th century by Niels Henrik Abel and Peter Barlow,[98][99] the first significant work on the general theorem was done by Sophie Germain.[100]
Sophie Germain

Mathematics Encyclopedia