# Wiles's proof of Fermat's Last Theorem

Andrew Wiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the Modularity Theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, seen as virtually impossible to prove using current knowledge. Wiles first announced his proof on Wednesday 23 June 1993 at a lecture in Cambridge entitled "Elliptic Curves and Galois Representations."[1] However, in September 1993 the proof was found to contain an error. One year later, on Monday 19 September 1994, in what he would call "the most important moment of [his] working life," Wiles stumbled upon a revelation, "so indescribably beautiful... so simple and so elegant," that allowed him to correct the proof to the satisfaction of the mathematical community. The correct proof was published in May 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 150 pages long and consumed seven years of Wiles's 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 such as the 2016 Abel Prize.

## Progress of the previous decades

Fermat's Last Theorem

Fermat's Last Theorem, formulated in 1637, states that no three distinct positive integers a, b, and c can satisfy the equation

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

if n is an integer greater than two (n ≥ 3).

Previous partial solutions for specific integer exponents

Between its publication and Andrew Wiles' eventual solution over 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values of n > 2, or for specific cases. Proofs were found for values of n up to around 4 million, first by hand, and later by computer. But no general proof was found, nor even a hint how such a proof could be undertaken.

The Taniyama–Shimura–Weil conjecture

Completely separate from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between two completely different kinds of advanced mathematical objects then being studied, known as elliptic curves and modular forms.

Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways. They conjectured that every rational elliptic curve is also modular. In the West this conjecture became well known through a 1967 paper by André Weil, who gave conceptual evidence for it; thus, it is sometimes called the Taniyama–Shimura–Weil conjecture or the Taniyama–Shimura conjecture.

By around 1980, much evidence had been accumulated to form conjectures about elliptic curves, the main reason to believe that these various conjectures about elliptic curves 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.

However no proof existed of this conjecture, and no approach to a proof seemed to exist that looked promising. In this way, the conjecture remained for decades, an important and unsolved problem in mathematics, for which no proof was seen as likely or accessible.

(Note - Around 50 years after first being proposed, the conjecture was finally proven and renamed the Modularity Theorem, largely as a result of Andrew Wiles' work described below)

Frey's curve

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 (xy) satisfy the relation

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

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 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.

In plain English, Frey had shown that there were good reasons to believe that any set of numbers (a, b, c, n) capable of disproving Fermat's Last Theorem, could also (probably) be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore if the Taniyama–Shimura–Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, so Fermat's Last Theorem would have to be true as well.

(Mathematically, 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 provided a partial proof that a Frey curve could not be modular. 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–Weil conjecture. However his partial proof came close to confirming the link between the semistable case and Fermat's Last Theorem.

Attacking Fermat's Last Theorem by using Frey's curve

Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to extend Serre's partial proof and show that Frey's intuition was correct: that the above elliptic curves (now known as a Frey curve), if they exist, cannot be modular. The missing piece (the so-called "epsilon conjecture") had been noticed by Jean-Pierre Serre.[5]:1. Second, it was necessary to prove the Taniyama–Shimura–Weil conjecture — or at least to prove it for the kinds of elliptic curves that included Frey's equation (known as semistable elliptic curves) .

• If the epsilon conjecture were true, then any 4 numbers able to be used to disprove Fermat's Last Theorem could also be used to make a semistable elliptic curve ("Frey's curve") that could never be modular;
• But if the Taniyama–Shimura–Weil conjecture were also true for Frey's curves, then by definition every Frey's curve that existed must be modular.
• The contradiction could have only one answer - if the epsilon conjecture and the Taniyama–Shimura–Weil conjecture were both proved to be true, then it would mean there couldn't be any solutions to Fermat's equation - because then there would be no Frey curves at all, meaning no contradictions would exist. This would mean Fermat's claim written over 350 years ago was correct, and finally prove Fermat's Last Theorem.
Ribet's Theorem

In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture, now known as Ribet's theorem. His article was published in 1990. In doing so, Ribet finally proved the link between the two theorems by confirming as Frey had suggested, that proof of the Taniyama–Shimura–Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem:

In mathematical terms, Ribet's theorem showed that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form which gives rise to the same Galois representation).[6]

However despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since the Taniyama–Shimura–Weil conjecture was itself seen as completely inaccessible to proof with current knowledge by almost all mathematicians.[7]:203–205, 223, 226 For example, Wiles's ex-supervisor John Coates states that it seemed "impossible to actually prove",[7]:226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".[7]:223

Andrew Wiles

Hearing of the 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable[8] as well as because of the enticing goal of proving such a long standing problem.

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]." [7]:223

## Wiles's proof

### Overview

Wiles opted to attempt to match elliptic curves to a countable set of 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:

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

${\displaystyle R}$ is a deformation ring and ${\displaystyle T}$ 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)[who?] He realised that the map between ${\displaystyle R}$ and ${\displaystyle T}$ 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, Fermat's Last Theorem is reduced to the statement 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. Wiles's goal was to verify that the map ${\displaystyle R\rightarrow T}$ is an isomorphism and ultimately that ${\displaystyle R=T}$. In treating deformations, Wiles defined four cases, with the flat deformation case requiring more effort to prove and 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 commutative diagram:

or ultimately that ${\displaystyle R=T}$, indicating a complete intersection. Since Wiles could not show that ${\displaystyle R=T}$ directly, he did so through ${\displaystyle Z_{3},F_{3}}$ and ${\displaystyle T/m}$ 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 discovered an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof, which could be used to create a CNF, and so Wiles set his Iwasawa work aside and began working to extend Kolyvagin and Flach's work instead, in order to create the CNF his proof would require.[9] By the spring of 1993, his work had covered all but a few families of elliptic curves, and in early 1993, Wiles was confident enough of his nearing success to let one trusted colleague into his secret. Since his work relied extensively on using the Kolyvagin-Flach approach, which was new to mathematics and to Wiles, and which he had also extended, in January 1993 he asked his Princeton colleague, Nick Katz, to help him review his work for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.[7]:261–265[10]

Wiles' use of Kolyvagin-Flach would later be found to be the point of failure in the original proof submission, who eventually had to create a proof based on Iwasawa theory in collaboration with Richard Taylor. 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 ${\displaystyle E({\bar {\mathbf {Q} }})}$, for every prime power ${\displaystyle l^{n}}$, there exists a homomorphism from the absolute Galois group

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

to

${\displaystyle \mathrm {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} )}$,

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

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

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 ${\displaystyle l^{n}}$ 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 (${\displaystyle \mod l^{n}}$) 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 (${\displaystyle \mod l^{n}}$) Galois representation to (${\displaystyle \mod l^{n+1}}$) and the number of ways in which one can lift a (${\displaystyle \mod l^{n}}$) 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's 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.[11][12] The book of the Cornell conference also contained simplifications to the original proof.[5]

Wiles's 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,[13][14] Hesselink's quick review of top-level issues, which gives just the elementary algebra and avoids abstract algebra,[15] 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.[16] F. Q. Gouvêa provides an award-winning review of some of the required topics.[17][18][19][20] Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist.[21] 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". The Cornell book does not cover the entirety of the Wiles proof.[22]

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.

## Announcement and subsequent developments

Wiles's proof was initially presented in 1993. It was finally accepted as correct, and published, in 1995, because of an error in one part 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's 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.[22]

After the announcement, Katz was appointed as one of the referees to review Wiles's 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 Kolyvagin and Flach's method was incomplete. The error would not have rendered his work worthless – each part of Wiles's 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.[7]:289, 296–297 Without this part proved, however, there was no actual proof of Fermat's Last Theorem.

Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success.[23][24][25] By the end of 1994, rumours had spread that under scrutiny, Wiles' proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.[26]

Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin–Flach approach would not work directly, also meant that his original attempts using Iwasawa theory could be made to work if he strengthened it using his experience gained from the Kolyvagin–Flach approach since then. Fixing one approach with tools from the other approach, would resolve the issue and produce a CNF valid for all cases that were not already proven by his refereed paper.[23][27] He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.[23]

"I was sitting at my desk examining the Kolyvagin–Flach method. It wasn’t that I believed I could make it work, but I thought that at least I could explain why it didn’t work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn’t working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I’d keep coming back to my desk looking to see if it was still there. It was still there. I couldn’t contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much."
— Andrew Wiles, as quoted by Simon Singh[28]

On 6 October Wiles asked three colleagues (including Faltings) to review his new proof,[11] and on 24 October 1994 Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[29] and "Ring theoretic properties of certain Hecke algebras",[30] 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.[31][32] 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 famously[33] claimed to "...have discovered a truly marvelous proof of this, which this margin is too narrow to contain".[34] 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][35] The number might be 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 by Jan Bergstra in 2005 is to "Formalize and verify by computer a proof of Fermat's Last Theorem, as proved by A. Wiles in 1995."[36]

## Notes

1. ^ a b c d Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times. Retrieved 21 January 2013.
2. ^
3. ^ Hellegouarch, Yves (2001). Invitation to the Mathematics of Fermat–Wiles. Academic Press. ISBN 978-0-12-339251-0.
4. ^ Singh, pp. 194–198; Aczel, pp. 109–114.
5. ^ a b c d G. Cornell, J. H. Silverman and G. Stevens, Modular forms and Fermat's Last Theorem, ISBN 0-387-94609-8
6. ^ https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm
7. Fermat's Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0
8. ^ http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html
9. ^ Singh p.259-262
10. ^ Singh, pp. 239–243; Aczel, pp. 122–125.
11. ^ a b Fermat's Last Theorem at MacTutor
12. ^ Fermat's Last Theorem Archived 28 February 2009 at the Wayback Machine. 1996
13. ^ FAQ: Wiles attack June 1993
14. ^ Fermat's Last Theorem a Theorem at last August 1993
15. ^ How does Wiles prove Fermat's Last Theorem? by Wim H. Hesselink
16. ^
17. ^ A Marvelous Proof Fernando Gouvêa, The American Mathematical Monthly, vol. 101, 1994, pp. 203–222
18. ^ The Mathematical Association of America's Lester R. Ford Award
19. ^ "News - ps.uci.edu" (PDF).
20. ^
21. ^ Gerd Faltings, The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, Notices of the AMS, 42/7
22. ^ a b AMS book review Modular forms and Fermat's Last Theorem by Cornell et al., 1999
23. ^ a b c Singh, pp. 269–277.
24. ^ A Year Later, Snag Persists In Math Proof 28 June 1994
25. ^
26. ^ Singh, pp. 175-185.
27. ^ Aczel, pp. 132–134.
28. ^ Singh p.186-187 (text condensed).
29. ^ Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. Annals of Mathematics. 141 (3): 443–551. JSTOR 2118559. OCLC 37032255. doi:10.2307/2118559.
30. ^ Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. Annals of Mathematics. 141 (3): 553–572. JSTOR 2118560. OCLC 37032255. doi:10.2307/2118560. Archived from the original on 27 November 2001.
31. ^ NOVA Video, The Proof, 28 October 1997. See also Solving Fermat: Andrew Wiles
32. ^ The Proof of Fermat's Last Theorem Archived 10 December 2008 at the Wayback Machine. Charles Daney, 1996
33. ^ Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn (2013). Modular Forms and Fermat’s Last Theorem (illustrated ed.). Springer Science & Business Media. p. 549. ISBN 978-1-4612-1974-3. Extract of page 549
34. ^ "Why Pierre de Fermat is the patron saint of unfinished business", Eoin O'Carroll, 17 August 2011, csmonitor.com
35. ^ History of Fermat's Last Theorem Andrew Granville, 24 June 1993
36. ^