CQQbarRZGalGalQPrSpGLSLPSLGFprojlimellZlZlmrplrpmTatedetwpJJtilFermat<apos/>s Last Theorem<brk/>After 356 YearsWilliam F. HammondA Lecture at the Everyone Seminar
University at Albany, October 22, 1993GELLMU Edition with Retrospective Comments
April 21, 2001
Minor revisions: July 15, 2004Comments for the GELLMU Edition
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After the time of the original talk and the subsequent preparation of my
href="http:math.albany.edu:8000mathpershammondoct93.html"
original writeup, there was a time fortunately not long and also not
to have been unexpected in the aftermath of so large a new development
when Andrew Wiless argument underwent some revision in collaboration
with Richard Taylor Questions about its soundness appeared to
have ceased by the fall of 1994, and the work announced by Wiles in
June 1993, as revised, was published in the May 1995 issue of the
Annals of Mathematics
There has also been discussion, at times appearing to approach
controversy, about the name of the conjecture arising from the 1955
meeting in Japan What I termed the ShimuraTaniyamaWeil
conjecture became known as the modular curve conjecture and then,
from the summer of 1999, as the modular curve theorem after the work
of Breuil, Conrad, Diamond, and Taylor in the same vein
as the work of Wiles and Taylor for the semistable case
I list a few references on these matters for the period since my
original talk:
What follows has the same content as the original writeup except
that the title of the Appendix has been changed from Current Status
to Late 1993Early 1994 StatusIntroduction.
The purpose of this expository lecture is to explain the basic ideas
underlying the final resolution of Fermats Last Theorem after 356
years as a consequence of the reported establishment by Andrew Wiles
of a sufficient portion of the ShimuraTaniyamaWeil conjecture
As these notes are being written, the work of Wiles is not available,
and the sources of information available to the author are (1) reports
by electronic mail, (2) the AMS Notices article
ribetnotices of K. Ribet, and (3) a preprint rubinsilvbg by
K. Rubin and A Silverberg based on the June, 1993 lectures of Wiles
at the Newton Institute in Cambridge, England It should be noted
that the fact that Fermats Last Theorem is a consequence of
sufficient knowledge of the theory of elliptic curves has been
fully documented in the publications (ribinv, ribtoul) of K
RibetFermats Last Theorem is the statement, having origin with Pierre
de Fermat in 1637, that there are no positive integers x, y, z such
that
$xnynzn$
for any integer exponent n 2 Obviously, if there are no
positive integer solutions x, y, z for a particular n, then there
are certainly none for exponents that are multiples of n Since
every integer n 2 is divisible either by 4 or by some odd prime
p, it follows that Fermats Last Theorem is true if there are no
solutions in positive integers of the equation
$xnynzn$
when n 4 and when n p for each prime p 2 The cases n
3, 4 are standard fare for textbooks (e.g., see Hardy Wright
hardywright) in elementary number theory Therefore, this
discussion will focus on the case n p where p 3 is prime
Very briefly, the idea is that we now know enough about the
classification of nondegenerate plane cubic curves
$F(x,\; y)0$
in two variables, also known as elliptic curves, with rational
coefficients to know how to enumerate them in a logical way
so that we may conclude that if there were positive integers a, b, c
with
$apbpcp$,
then the curve
y^{2} x (x a^{p}) (x b^{p}),
which is an elliptic curve known as the Frey curve, would fall
inside of the enumeration Because the classification is
enumerative, when one is presented with a particular elliptic curve
with rational coefficients, one knows where to look for the curve in
the classification The curve just written is not to be found within
the classification As a consequence there cannot be positive
integers a, b, c with a^{p} b^{p} c^{p}
The enumerative classification of nondegenerate plane cubic curves
defined by polynomials with rational coefficients has been entirely
conjectural (variously known as the Taniyama Conjecture, the
Weil Conjecture, the TaniyamaShimura Conjecture, )
until June, 1993 This conjecture, even as a conjecture, has served
as an important motivating example for the idea of the Langlands
Program, or perhaps of an extension of that program, that certain
kinds of objects in geometry should give rise to certain group
representations
What seems to be believed todayAs of the time of this
writeup Wiles has stated that a portion of what he announced in June
needs further justification and that he expects to be able to complete
it See the appendix. is that the portion of the enumerative
classification pertaining to semistable elliptic curves has been
proved by Andrew Wiles That the existence of positive integers a,
b, c with
$apbpcp$
would violate the enumerative classification of semistable elliptic
curves was established by 1987 through the work of G. Frey, J.P
Serre, and K. Ribet
The primary purpose of this lecture is to explain the enumerative
classification of elliptic curves and to give a brief indication of
the mathematics involved in showing that the Frey curve violates that
classificationElliptic curves
A polynomial f(X,Y) of degree d in two variables with coefficients
in a field k gives rise to what is called an affine plane
curve of degree d: for each field K containing k (more
generally, for each commutative ring that is a kalgebra) one has
the set
C_{0}(K) (x,y) K^{2} f(x,y) 0 ,
and for each klinear homomorphism
$KK$
one has the induced map
$C$_{0}(K) C_{0}(K)
From the polynomial f one obtains a homogeneous polynomial of degree
d in three variables with coefficients in k:
F(X,Y,Z) Z^{d} f(XZ, YZ),
and the projective plane curve of degree d:
C(K) ((x,y,z)) P^{2}(K) F(x,y,z) 0 ,
where P^{N}(K) denotes Ndimensional projective space, which is
the quotient set of $KN10$
obtained by identifying points lying on the same line through the
origin of K^{N1} Since the projective plane P^{2}(K) is the
disjoint union of the affine plane
$((x,y,1))(x,y)K2$
with the (projective) line at infinity((x,y,0)) ((x,y)) P^{1}(K),
it follows that C(K) is the disjoint union of C_{0}(K) with the
finite set of its points lying on the projective line at infinity
An elliptic curve defined overk is the (projective) plane
curve E given by a homogeneous polynomial F of degree 3 in three
variables with coefficients in k such that (i) F is irreducible
over the algebraic closure k of k, (ii) the gradient vector
F is a nonvanishing vector at points of
$k30$
where F vanishes, and (iii) the set E(k) is nonempty
If k is any field, then after an isomorphism (see Silverman
silverman) one may obtain a given elliptic curve E with an
affine equation of the form
y^{2} a_{1} x y a_{3} y x^{3} a_{2} x^{2} a_{4} x a_{6}
Then the homogeneous equation for the
intersection of E(K) with the line at infinity is
x^{3} 0
Thus, in this case, E has a unique point on the line at infinity
If the characteristic of k is different from 2 and
3Thus, one sees that the primes 2 and 3 play a special
role in the theory of elliptic curves. then one may obtain an
equation in Weierstrass normal form:
y^{2} 4 x^{3} g_{2} x g_{3},
which is nonsingular if and only if the cubic polynomial in the
variable x has distinct roots in k
Elliptic curves are the group objects in the category of algebraic
curves that reside in projective space: for each extension field K
of k the set E(K) of Kvalued points of E is an abelian
group The group law on E(K) is characterized by two conditions:
The origin is a given point of E(k) The points obtained by intersecting E(K) with any line in
P^{2}(K), counted with multiplicities, add up to zero
When E is given by an equation in the form (iref="eq.wnormform"eq.wnormform), the
origin is usually taken to be the unique point on the line at
infinity If two distinct points of E(K) are given, they determine
a line in P^{2}(K); the intersection of that line with E(K) is
given by a cubic polynomial in a parameter for the line which has two
roots in K corresponding to the two given points; hence, there is a
third root of that cubic polynomial in K; this root gives rise to a
point of E(K), which is the negative of the sum of the two given
points The negative of a given point of E(K) is obtained as the
third point in the intersection with E(K) of the line through the
given point and the origin
For a given field k the set of homogeneous cubic polynomials in
three variables is a vector space over k having the set of
monomials of degree three in three variables as basis Thus, the
dimension of the space of homogeneous cubics is 10 The linear group
GL_{3}(k) acts on the space of cubics, and two cubic curves in
P^{2} that are related by this action are isomorphic Since
GL_{3}(k) is 9dimensional, one is led to think of the family
of isomorphism classes of elliptic curves as 1dimensional since
nonsingularity is an open conditionElliptic curves over C
When k is the field C of complex numbers, one knows (see, e.g.,
Ahlfors ahlfors) that for each lattice in C the set
of periodic meromorphic functions on the complex line C
is the field C(, ), which is a quadratic extension of the
rational function field C(), where is the function
of Weierstrass Moreover, satisfies the famous Weierstrass
differential equation
(z)^{2} 4(z)^{3} g_{2}()(z) g_{3}() ;
thus, the formula
$z((z),(z))$
defines a holomorphic map from the punctured complex torus
$C\mapsto $
to the affine cubic curve
y^{2} 4x^{3} g_{2}()x g_{3}() ;
it should hardly be necessary to point out that this map extends to a
holomorphic map from the torus C to the corresponding
(projective) elliptic curve by sending the origin of the torus to the
unique point of the elliptic curve on the line at infinity The
classical theory of theta functions (see, e.g., Igusa igusa or
Siegel siegel) leads to a direct demonstration that this map is a
homomorphism from the group law on the complex torus to the group law
previously described for an elliptic curve It is not difficult to
see that the analytic manifold given by any elliptic curve defined
over C arises from some complex torus Indeed each nonsingular
cubic curve E in P^{2}(C) determines a compact connected
complexanalytic group Its universal cover is given by a holomorphic
homomorphism
$CE$ which has some lattice as kernel
Any two lattices in C are related by a change of real basis for
C, i.e., by a matrix in GL_{2}(R) Consequently, there is only
one realanalytic isomorphism class for the complex torus C
as varies The tori corresponding to two lattices are
complexanalytically isomorphic if and only if the corresponding
reallinear isomorphism of R^{2} satisfies the CauchyRiemann partial
differential equations, i.e., if and only if the Rlinear
isomorphism is Clinear
A lattice may be represented concretely by an ordered basis
$$_{1}, _{2}
If _{2}_{1}, then is not real, and after
permuting the basis members, if necessary, one may assume that
is in the upperhalf planeThe fact that the halfplane
is a model of nonEuclidean geometry led a popular columnist in
November, 1993 to question the validity of the work being discussed
here.H of C Observing that is the image under the
Clinear map
$z$_{1} z
of the lattice with ordered basis
$1,$,
one may assume that is this latter lattice Let E()
be the complex torus C Allowing for change of basis
subject to these assumptions on the basis, one sees that there is an
isomorphism of complexanalytic groups
$E()E()$
if
a bc d ,
for some matrix
rr a b c d SL_{2}(Z)
Conversely, the monodromy principle may be used to show that every
complexanalytic isomorphism among the complex tori E() arises
in this way
The coefficients g_{2} and g_{3} in the Weierstrass normal form
(iref="eq.cwnormform"eq.cwnormform) have very explicit constructions as infinite
series (see, e.g., Ahlfors ahlfors or Serre serrecourse)
determined by the given lattice; from this it is straightforward to
see that g_{w} is a modular form of weight2w: if and
are related by (iref="eq.linfrac"eq.linfrac), then
g_{w}() ()^{2w}g_{w}(), c d .
Consequently, the map
(x,y) (^{2} x, ^{3} y)
carries the curve given by (iref="eq.cwnormform"eq.cwnormform) for
isomorphically to the curve given by (iref="eq.cwnormform"eq.cwnormform) for
The discriminant of the cubic polynomial in the
Weierstrass normal form (iref="eq.cwnormform"eq.cwnormform) is a modular form of
weight 12, which up to a multiplicative constant, is:
() g_{2}^{3} 27 g_{3}^{2}. is a nonvanishing holomorphic function in H The
modular invariant() (serrecourse,shimiaf) is defined by:
() (12 g_{2})^{3};
it is a holomorphic function in the upperhalf plane H with the
property that
() ()
if and only if and are related by (iref="eq.linfrac"eq.linfrac)
Furthermore, assumes every value in C at some point of H
Consequently, the complexanalytic isomorphism classes of complex tori
or, equivalently, the isomorphism classes of elliptic curves defined
over C, are parameterized via in a onetoone manner by the
complex numbers
Since this is an expository discourse, it is hoped that the reader
will not feel patronized by having noted the fact that the
coincidence of (1) the category of elliptic curves over C and
(2) the category of complex tori is the genus one case of the
coincidence (see Weyl weyl) of (i) the category of complete
nonsingular algebraic curves over C and (ii) the category of
compact Riemann surfaces (onedimensional connected complexanalytic
manifolds)
Although the classification of elliptic curves over C via the
function is a result that is both beautiful and useful, and
although two elliptic curves defined over Q that are isomorphic as
curves defined over Q give rise to elliptic curves defined over
C that have the same invariant, it is not true that any
two elliptic curves defined over Q having the same invariant
are isomorphic over Q Thus, the classification of elliptic
curves over C does not lead directly to the desired enumerative
classification of elliptic curves defined over Q but it does
bring to the fore the notion of modular form, which is central
in the study of elliptic curves defined over Q What can be said
easily is that, according to the ShimuraTaniyamaWeil conjecture, the
isogeny classes of elliptic curves defined over Q are
parameterized by certain modular formsModular forms
The group SL_{2}(Z) is an infinite group that is generated by the
two elements
rr 0 1 1 0 , rr 0 1 1 1 ,
which have orders 4 and 6 respectively The action of SL_{2}(Z) on
the upperhalf plane H by linear fractional transformations has kernel
rr 1 0 0 1 ,
and the quotient of SL_{2}(Z) by this kernel is the group
PSL_{2}(Z) It is not difficult to see that the set
H 12 Re() 12, 1
is a fundamental domain for the action of PSL_{2}(Z) on H
More precisely, this set meets each orbit, and the only redundancies
are the boundary identifications arising from the maps
$1$ and $1$
The quotient HPSL_{2}(Z) is not compact since the fundamental
domain is open at the top Beyond that the modular invariant
j induces a bicontinuous biholomorphic isomorphism of the quotient
HPSL_{2}(Z) with the affine line over C Since
$(1)()$, and since for
$qe2\pi $ one has q 1 for H,
there is a holomorphic function in the punctured unit
disk such that (q) () Likewise may
be regarded as function of q, and one may use the calculus of residues
to show that has a simple zero at q 0; hence,
has a simple pole at q 0, or, equivalently,
has a simple pole at (the missing top of the fundamental
domain) Thus, gives rise to a bicontinuous biholomorphic
isomorphism
HPSL_{2}(Z) P^{1}(C) .
A nontrivial element of PSL_{2}(Z) has a fixed point in H if and
only if it has finite order, and ones explicit knowledge of the
fundamental domain makes it possible to see that the only elements of
finite order are of order 2 or 3Thus, one sees that the
primes 2 and 3 play a special role in the study of the group
SL_{2}(Z).
A congruence subgroup of SL_{2}(Z) is a subgroup
that contains one of the principal congruence subgroups; the
principal congruence subgroup(N) of levelN is
the set of all elements of SL_{2}(Z) that are congruent
N to the identity matrix The group _{0}(N)
is the congruence subgroup of SL_{2}(Z) consisting of all elements
rr a b c d
for which
$c0N$
It is obvious that each congruence group has finite index in
SL_{2}(Z), and, consequently the quotient H is a noncompact
Riemann surface
Observe that for each level N the
group _{0}(N) contains the parabolic element
T rr 1 1 0 1 ,
which gives rise to the holomorphic map
$1$
that fixes the point
A modular formDetails concerning the discussion
in this section may be found in Shimuras book shimiaf.
of weightw for is a holomorphic
function f in H that satisfies the functional equation
f() (c d)^{w} f(),
and that is holomorphic at each cusp of The role
of cusps for is to provide a slightly larger set H^{}
than H,
H^{} H cusps,
where acts such that H^{} is a compact Riemann
surface containing H as the open complement of a finite set
of points arising from cusps The cusps of are the points of
the closure of the boundary of H in P^{1}(C)$C$ that are fixed by some nontrivial
parabolic element of When SL_{2}(Z), the
set of cusps is
$Q$
In view of (iref="eq.modform"eq.modform) applied to the case
$T$ one sees that a modular form f of any weight
for the group _{0}(N) satisfies
f( 1) f() ,
and, therefore, f defines a holomorphic function in the variable q
e^{2 i } for 0 q 1 The condition in the
definition of modular form that f should be holomorphic
at means that f as a function of q is holomorphic at q
0 Consequently, f admits an absolutely convergent Fourier
expansion
f() _{m 0}^{}c_{m} e^{2 i m },
which is a Taylor series in q
For any cusp of a congruence group one may define the
notion holomorphic at for a modular form f by an
analogous procedure using an arbitrary parabolic element of
that fixes instead of T For a given
congruence group two cusps and are
equivalent if there is some element in such that
$$ A modular form f is holomorphic
at any cusp that is equivalent to another where it is holomorphic
The modular form f is a cuspform if, in addition to being
holomorphic at each cusp, f vanishes at each cusp
For a given congruence group a modular form vanishes at
any cusp that is equivalent to another where it vanishes
The set of modular forms of given weight w forms a
finitedimensional vector space over C in which the set of
cuspforms is a linear subspace of codimension bounded by the
number of equivalence classes of cusps In fact, using
Eisenstein series one may show that the codimension of the
space of cuspforms in the space of modular forms is often equal to
the number of equivalence classes of cusps For example, with the group
$(1)SL$_{2}(Z) there are no modular forms of odd
weight, there is an Eisenstein series of every even weight greater
than 2 that is not a cuspform, and every cusp is equivalent to
Furthermore, since is the only zero of the cusp
form (of the preceding section) in the quotient
$H(1)$
and since is a simple zero of every cuspform for
(1) is divisible by Thus, in this case, there are
no cuspforms of weight less than 12
It is not difficult to see that the cuspforms of weight 2 for a
congruence group correspond to holomorphic differential
1forms (differentials of the first kind) on the compact Riemann
surface
$XH$
Thus, the dimension of the space of cuspforms of weight