Wiles's proof of Fermat's Last Theorem is
a proof by British mathematician Andrew Wiles
of a special case of the modularity theorem
for elliptic curves. Together with Ribet's
theorem, it 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, meaning that they were believed
to be impossible to prove using current knowledge.Wiles
first announced his proof on Wednesday 23
June 1993 at a lecture in Cambridge entitled
"Modular Forms, Elliptic Curves and Galois
Representations". 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 that allowed him to correct the
proof to the satisfaction of the mathematical
community. The corrected proof was published
in 1995.Wiles' 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 which were not
available to Fermat.
Together, the two papers which contain the
proof are 129 pages long, and consumed over
seven years of Wiles's research time. John
Coates described the proof as one of the highest
achievements of number theory, and John Conway
called it the proof of the [20th] century.
Wiles' path to proving Fermat's Last Theorem,
by way of proving the modularity theorem for
the special case of semistable elliptic curves,
established powerful modularity lifting techniques
and opened up entire new approaches to numerous
other problems. For solving Fermat's Last
Theorem, he was knighted, and received other
honours such as the 2016 Abel Prize. When
announcing that Wiles had won the Abel Prize,
the Norwegian Academy of Science and Letters
described his achievement as a "stunning proof".
== Precursors to Wiles' proof ==
=== 
Fermat's Last Theorem and progress prior to
1980 ===
Fermat's Last Theorem, formulated in 1637,
states that no three distinct positive integers
a, b, and c can satisfy the equation
a
n
+
b
n
=
c
n
{\displaystyle a^{n}+b^{n}=c^{n}}
if n is an integer greater than two (n > 2).
Over time, this simple assertion became one
of the most famous unproved claims in mathematics.
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. It spurred the
development of entire new areas within number
theory. Proofs were eventually found for all
values of n up to around 4 million, first
by hand, and later by computer. But no general
proof was found that would be valid for all
possible values of n, nor even a hint how
such a proof could be undertaken.
=== The Taniyama–Shimura–Weil conjecture
===
Separately 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 elliptic
curves and modular forms. These were mathematical
objects with no known connection between them.
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. This became known as
the Taniyama–Shimura conjecture. 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.
By around 1980, much evidence had been accumulated
to form conjectures about elliptic curves,
and many papers had been written which examined
the consequences if the conjecture was true,
but the actual conjecture itself was unproven
and generally considered inaccessible - meaning
that mathematicians believed a proof of the
conjecture was probably impossible using current
knowledge.
For decades, the conjecture remained an important
but unsolved problem in mathematics. 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 yet another 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
)
.
{\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 this same curve,
now called a Frey curve. He showed that it
was likely that the curve could link Fermat
and Taniyama, since any counterexample to
Fermat's Last Theorem would probably also
imply that an elliptic curve existed that
was not 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.)If the link
identified by Frey could be proven, then in
turn, it would mean that a proof or disproof
of either of Fermat's Last Theorem or the
Taniyama–Shimura–Weil conjecture would
simultaneously prove or disprove the other.
=== Ribet's Theorem ===
To complete this link, it was necessary to
show that Frey's intuition was correct: that
a Frey curve, if it existed, could not be
modular.
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 (which Serre
had noticed early on) became known as the
epsilon conjecture or ε-conjecture (it is
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 Fermat
and Taniyama.
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 a 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
(which Frey's curve has), then that curve
cannot be modular, in the sense that there
cannot exist a modular form which gives rise
to the same Galois representation.
=== Situation prior to Wiles' proof ===
Following the developments related to the
Frey Curve, and its link to both Fermat and
Taniyama, a proof of Fermat's Last Theorem
would follow from a proof of the Taniyama–Shimura–Weil
conjecture — or at least a proof of the
conjecture for the kinds of elliptic curves
that included Frey's equation (known as semistable
elliptic curves).
From Ribet's Theorem and the Frey Curve, 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 semistable elliptic curves,
then by definition every Frey's curve that
existed must be modular.
The contradiction could have only one answer:
if Ribet's Theorem and the Taniyama–Shimura–Weil
conjecture for semistable curves were both
true, then it would mean there could not be
any solutions to Fermat's equation – because
then there would be no Frey curves at all,
meaning no contradictions would exist. This
would finally prove Fermat's Last Theorem.However,
despite the progress made by Serre and Ribet,
this approach to Fermat was widely considered
unusable as well, since almost all mathematicians
saw the Taniyama–Shimura–Weil conjecture
itself as completely inaccessible to proof
with current knowledge. For example, Wiles's
ex-supervisor John Coates stated that it seemed
"impossible to actually prove", and Ken Ribet
considered himself "one of the vast majority
of people who believed [it] was completely
inaccessible".
== Andrew Wiles ==
Hearing of Ribet's 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
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]."
== Announcement and subsequent developments
==
Wiles initially presented his proof in 1993.
It was finally accepted as correct, and published,
in 1995, following the correction of a subtle
error in one part of his original 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. There was a relatively
large amount of press coverage afterwards.After
the announcement, Nick 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. 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. By the end of 1993,
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.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 he 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. Each was inadequate by
itself, but fixing one approach with tools
from the other would resolve the issue and
produce a class number formula (CNF) valid
for all cases that were not already proven
by his refereed paper:
"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 SinghOn
6 October Wiles asked three colleagues (including
Faltings) to review his new proof, and on
24 October 1994 Wiles submitted two manuscripts,
"Modular elliptic curves and Fermat's Last
Theorem" and "Ring theoretic properties of
certain Hecke algebras", 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. 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.
=== Subsequent developments ===
Fermat claimed to 
"...have discovered a truly marvelous proof
of this, which this margin is too narrow to
contain". 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. The complexity of Wiles's
proof motivated a 10-day conference at Boston
University; the resulting book of conference
proceedings aimed to make the full range of
required topics accessible to graduate students
in number theory.As noted above, Wiles proved
the Taniyama–Shimura–Weil conjecture for
the special case of semistable elliptic curves,
rather than for all elliptic curves. Over
the following years, Christophe Breuil, Brian
Conrad, Fred Diamond, and Richard Taylor (sometimes
abbreviated as "BCDT") carried the work further,
ultimately proving the Taniyama–Shimura–Weil
conjecture for all elliptic curves in a 2001
paper. Now proved, the conjecture became known
as the modularity theorem.
In 2005, Dutch computer scientist Jan Bergstra
posed the problem of formalizing Wiles' proof
in such a way that it could be verified by
computer.
== Summary of Wiles' proof ==
Wiles used proof by contradiction, in which
one assumes the opposite of what is to be
proved, and show if that were true, it would
create a contradiction. The contradiction
shows that the assumption must have been incorrect.
The proof falls roughly in two parts. In the
first part, Wiles proves a general result
about "lifts", known as the "modularity lifting
theorem". This first part allows him to prove
results about elliptic curves by converting
them to problems about Galois representations
of elliptic curves. He then uses this result
to prove that all semi-stable curves are modular,
by proving that the Galois representations
of these curves are modular, instead.
== Mathematical detail of Wiles 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:
R
n
→
T
n
.
{\displaystyle R_{n}\rightarrow \mathbf {T}
_{n}.}
R
{\displaystyle R}
is a deformation ring and
T
{\displaystyle \mathbf {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 of the
1995 paper). He realised that the map between
R
{\displaystyle R}
and
T
{\displaystyle \mathbf {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
R
→
T
{\displaystyle R\rightarrow \mathbf {T} }
is an isomorphism and ultimately that
R
=
T
{\displaystyle R=\mathbf {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, gives the
following commutative diagram (p. 745):
or ultimately that
R
=
T
{\displaystyle R=\mathbf {T} }
, indicating a complete intersection. Since
Wiles could not show that
R
=
T
{\displaystyle R=\mathbf {T} }
directly, he did so through
Z
3
,
F
3
{\displaystyle \mathbf {Z} _{3},\mathbf {F}
_{3}}
and
T
/
m
{\displaystyle \mathbf {T} /{\mathfrak {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 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. 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.Wiles'
use of Kolyvagin–Flach would later be found
to be the point of failure in the original
proof submission, and he eventually had to
revert to Iwasawa theory and a collaboration
with Richard Taylor to fix it. 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. (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
(
Q
¯
)
{\displaystyle E({\bar {\mathbf {Q} }})}
, for every prime power
ℓ
n
{\displaystyle \ell ^{n}}
, there exists a homomorphism from the absolute
Galois group
Gal
⁡
(
Q
¯
/
Q
)
{\displaystyle \operatorname {Gal} ({\bar
{\mathbf {Q} }}/\mathbf {Q} )}
to
GL
2
⁡
(
Z
/
l
n
Z
)
,
{\displaystyle \operatorname {GL} _{2}(\mathbf
{Z} /l^{n}\mathbf {Z} ),}
the group of invertible 2 by 2 matrices whose
entries are integers (
mod
ℓ
n
{\displaystyle \mod \ell ^{n}}
). This is because
E
(
Q
¯
)
{\displaystyle E({\bar {\mathbf {Q} }})}
, the points of E over
Q
¯
{\displaystyle {\bar {\mathbf {Q} }}}
, form an abelian group, on which
Gal
⁡
(
Q
¯
/
Q
)
{\displaystyle \operatorname {Gal} ({\bar
{\mathbf {Q} }}/\mathbf {Q} )}
acts; the subgroup of elements x such that
ℓ
n
x
=
0
{\displaystyle \ell ^{n}x=0}
is just
(
Z
/
ℓ
n
Z
)
2
{\displaystyle (\mathbf {Z} /\ell ^{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
Gal
⁡
(
Q
¯
/
Q
)
→
GL
2
⁡
(
Z
/
l
n
Z
)
.
{\displaystyle \operatorname {Gal} ({\bar
{\mathbf {Q} }}/\mathbf {Q} )\rightarrow \operatorname
{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
{\displaystyle \ell ^{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
{\displaystyle \mod \ell ^{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
{\displaystyle \mod \ell ^{n}}
) Galois representation to (
mod
ℓ
n
+
1
{\displaystyle \mod \ell ^{n+1}}
) and the number of ways in which one can
lift a (
mod
ℓ
n
{\displaystyle \mod \ell ^{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. (Again, 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 intersectionsGerd
Faltings subsequently provided some simplifications
to the 1995 proof, primarily in switching
from geometric constructions to rather simpler
algebraic ones. The book of the Cornell conference
also contained simplifications to the original
proof.
=== Overviews available in the literature
===
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. The mathematicians
who helped to lay the groundwork for Wiles
often created new specialised concepts and
technical jargon.
Among the introductory presentations are an
email which Ribet sent in 1993; Hesselink's
quick review of top-level issues, which 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's 1994
article "A Marvelous Proof", which reviews
some of the required topics, won a Lester
R. Ford award from the Mathematical Association
of America. 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". The Cornell book does not cover the
entirety of the Wiles proof.
== Notes
