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 EditionAmong the challenges that I have been facing with my GELLMU project
are (1) convincing mathematicians that it is possible to use
comfortable like markup in a fully rigorous way to prepare our
articles so that they can have formal inclusion in the markup category
known as href="http:www.w3.orgXML"XML and
(2) then convincing them that high quality typesetting may be
obtained from the ensuing XML document instance
Toward this end I have revisited the markup for the official
notes on my October 1993 Albany seminar presentation and edited what
was source to convert it to like source markup for
the article document type that is part of the GELLMU didactic
markup production system Information about this system and my
reasons for developing it may be found at
http:www.albany.eduhammondgellmuAfter 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 MathematicsThere 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 caseI 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 primeVery 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
representationsWhat 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. RibetThe 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 curvesA 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 infinityAn 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 zeroWhen 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 originFor 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 CWhen 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
$C\mapsto 0$
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 kernelAny 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 ClinearA 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
rra 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 wayThe 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 numbersSince 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 formsThe group SL_{2}(Z) is an infinite group that is generated by the
two elements
rr0 1 1 0 , rr0 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
rr1 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 i$ 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
rra 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 rr1 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 qFor 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 12It 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 2 is the
genus of X The fact that there are no cuspforms of weight
2 for the group (1) matches the previously mentioned fact
that X is P^{1} It is certain of the cuspforms of weight
two for the groups _{0}(N) that, according to the
ShimuraTaniyamaWeil conjecture, parameterize the isogeny classes of
elliptic curves defined over QEuler productsIt will be recalled that the infinite series
_{n 1}^{}1n^{s}
converges for Re(s) 1 and gives rise by analytic continuation
to a meromorphic function (s) in C For Re(s) 1(s) admits the absolutely convergent infinite product
expansion
_{p}11 p^{s},
taken over the set of primes This Euler product may be regarded
as an analtyic formulation of the principle of unique factorization
in the ring Z of integers It is, as well, the product taken
over all the nonarchimedean completions of the rational field Q
(which completions Q_{p} are indexed by the set of primes) of the
Mellin transformThe Mellin transform is, more or less,
Fourier transform on the multiplicative group
Classically, the Mellin transform of f is given formally by
(s) _{0}^{}f(x) x^{s} (dxx)
in Q_{p}_{p}(s) 11 p^{s},
of the canonical Gaussian
density_{p}(x) ll1 if x closure ofZinQ_{p}0 otherwise,
which Gaussian density is equal to its own Fourier transform
For the archimedean completion
$Q$_{}R
of the rational field Q one forms the classical Mellin transform
_{}(s) ^{(s2)}(s2)
of the classical Gaussian density
_{}(x) e^{ x2},
(which also is equal to its own Fourier transform)
Then the function
(s) _{}(s) (s) _{p }_{p}(s)
is meromorphic in C, and satisfies the functional equation
(1s) (s) The connection of Riemanns function with the subject of
modular forms begins with the observation that (2s) is essentially
the Mellin transform of _{I}(x) (ix) 1 where ,
which is a modular form of weight 12 and level 8, is defined in the
upperhalf plane H by the formula
() _{m Z}exp( i m^{2}).
In fact, one of the classical proofs of the functional equation
(iref="eq.zetafnl"eq.zetafnl) is given by applying the Poisson summation
formulaOn the other hand, (iref="eq.zetafnl"eq.zetafnl) may be regarded
directly as a divergent model of the Poisson summation formula.
to the function
$xexp(\pi ix2)$,
while observing that the substitution
$s(12)s$
for (2s) corresponds in the upperhalf plane to
the substitution
$1$
for the theta seriesIf f is a cuspform for a congruence group containing
T rr1 1 0 1 ,
and so, consequently,
$f(1)f()$, then, as previously explained, one has
the Fourier expansion (iref="eq.fourier"eq.fourier)
f() _{m 1}^{}c_{m} e^{2 i m }.
The Mellin transform (s) of f_{I} leads to the Dirichlet series
(s) _{m 1}^{}c_{m} m^{s},
which may be seen to have a positive abscissa of convergence
One is led to the questions:
For which cuspforms f does the associated Dirichlet series
(s) admit an analytic continuation with functional equationFor which cuspforms f does the associated Dirichlet series
(s) have an Euler product expansionFor the modular group(1) the Dirichlet series associated
to every cuspform of weight w admits an analtyic continution with
functional equation under the substitution
$sws$
Since (1) is generated by the two matrices T and
W rr0 1 1 0
and since the functional equation of a modular form f relative to
T is reflected in the formation of the Fourier series
(iref="eq.fourier"eq.fourier), the condition that an absolutely convergent series
(iref="eq.fourier"eq.fourier) is a modular form for (1) is the functional
equation for a modular form relative solely to W This is
equivalent to the (properly formulated) functional equation for the
associated Dirichlet series together with a growth
condition For the group _{0}(N), with N 1, the question
of a functional equation is more complicated since, although T is
available, there is no reason for a cuspform to satisfy a law of
transformation relative to W But note that for any the
set of cuspforms of given weight for which the associated Dirichlet
series have analytic continuations satisfying a given finite set of
functional equations is a vector space On the other hand, there is
no reason to believe, even for level 1, that the cuspforms admitting
an Euler product expansion form a vector spaceIn a nutshell the cuspforms admitting Euler products are those which
arise as eigenforms for an arithmetically defined commutative algebra
of semisimple operators on the space of cuspforms of a given weight
introduced by E. Hecke The theory of Hecke operators is reasonably
simple for level 1 but somewhat more complicated in general (see, e.g.,
Shimuras book shimiaf)Observing that the formula
ds^{2}dx^{2} dy^{2}y^{2},forxiy H,
gives a (the hyperbolic) SL_{2}(R)invariant metric in H with
associated invariant measure
ddx dyy^{2},
one introduces the Petersson (Hermitian) inner product in the space of
cuspforms of weight w for with the definition:
f,g _{H}f()g()Im()^{w} d()
(Integration over the quotient H makes sense since the
integrand
f()g() y^{w}
is invariant.)For the modular group (1) the n^{th} Hecke operator
$T(n)T$_{w}(n) is the linear endomorphism of the space
of cuspforms of weight w arising from the following considerations
Let S_{n} be the set of 2 2 matrices in Z with determinant
n For
M rra b c d S_{n}
and for a function f in H one defines
(M _{w} f)() det(M)^{w1} (c d)^{w} f() ,
and then, observing that (1) under _{w} acts trivially on the
modular forms of weight w,
one may define the Hecke operator T_{w}(n) by
T_{w}(n)(f) _{M Sn(1)}(M _{w} f)(),
where the quotient S_{n}(1) refers to the action of (1)
by left multiplication on the set S_{n}
One finds for m, n coprime that
T(m n) T(m) T(n) ,
and furthermore one has
T(p^{e1}) T(p^{e})T(p) p^{w1}T(p^{e1}) .
Consequently, the operators T(n) commute with each other, and, therefore,
generate a commutative algebra of endomorphisms of the space of cusp
forms of weight w for (1) It is not difficult to see that
the Hecke operators are selfadjoint for the Petersson inner product
on the space of cuspforms Consequently, the space of cuspforms of
weight w admits a basis of simultaneous eigenforms for the Hecke
algebra A Hecke eigencuspform is said to be
normalized if its Fourier coefficient c_{1} 1 If f is
a normalized Hecke eigencuspform, then
The Fourier coefficient c_{m} of f is the eigenvalue of f
for T(m)The Fourier coefficients c(m) c_{m} of f satisfy
Consequently, the Dirichlet series associated with a simultaneous
Hecke eigencuspform of level 1 and weight w admits an Euler product
(s) _{p}11 c_{p} p^{s} p^{w12s}
For example, when f is the unique normalized cuspform of
level 1 and weight 12 one has
(s) _{p}11 (p) p^{s} p^{112s},
where c_{p}(p) is the function of RamanujanFor the congruence group _{0}(N) a Hecke eigencuspform
of weight w gives rise to a Dirichlet series (s) that
admits an Euler product expansion whose factors at primes p coprime
to N resemble those given by (iref="eq.modeulerp"eq.modeulerp) In order for
(s) to satisfy a functional equation under the substitution
$sws$,one needs to require that the eigencuspform
f admits a functional equation not only with respect to each element
of the group _{0}(N) but also with respect to the substitution
in the upperhalf plane H given by the matrix
W_{N}rr0 1 N 0 .
A. Weil (weilmathann) showed that the cuspforms of weight 2
for the group _{0}(N) satisfying the appropriate functional
equation under the mapping of H given by W_{N} correspond precisely
to Dirichlet series with certain growth conditions that admit analytic
continuations as meromorphic functions in C satisfying a finite
number of twisted functional equationsThe reader will have noticed that it is not extremely easy to
characterize the cuspforms of weight 2 that conjecturally
(ShimuraTaniyamaWeil) parameterize the isogeny classes of elliptic
curves defined over the rational field Q The Euler product is
an extremely important part of the characterization since the
Dirichlet series given by such an elliptic curve, as will be made
explicit in the next section, is, by its very nature, an Euler
product Weil conjectures explicitly that the Dirichlet series with
Euler product given by each elliptic curve defined over Q satisfies
these conditions, i.e., is the Dirichlet series associated to some
W_{N}compatible Hecke eigencuspform for the group _{0}(N),
where N is the conductor of E This has led to efforts,
related to the Langlands program to understand the
W_{N}compatible Hecke eigencuspforms in a more intrinsic way as
objects of representation theory over Q (see, e.g., the survey of
Gelbart gelbartadv)Elliptic curves over the rational field QLet E be an elliptic curve defined over Q One may clear
denominators from its cubic equation, if necessary, in order to arrive
at an equation with integer coefficients having no common factor
While the Weierstrass normal form (iref="eq.swnormform"eq.swnormform) is available to
represent the isomorphism class of any elliptic curve over a field of
characteristic different from 2 and 3, one needs the generalized
Weierstrass form
y^{2} a_{1} x y a_{3} y x^{3} a_{2} x^{2} a_{4} x a_{6}
over an arbitrary field, and, moreover, for each elliptic curve E
defined over Q there is a best possible equation (e.g., see
Silverman silverman) of the form (iref="eq.gwnormform"eq.gwnormform) with
integer coefficients called the Neron model of E With an
abuse of notation E will denote the Neron model, which may be regarded
as a curve over Z (One would want to call it an elliptic curve
over Z if it were smooth over Z, i.e., if it had good
reduction at each prime p; the fact that every Neron model has bad
reduction at least once corresponds under the dictionary to the
fact that there are no cuspforms of weight two and level 1.)
It then may be observed that for each prime p the Neron model gives
rise to a cubic equation over the finite field F_{p} For all
but a finite number of p the equation over F_{p} is nonsingular
over F_{p}, i.e., determines an elliptic curve E_{p} defined
over F_{p} One says in this case that E has good
reduction mod p Following Tate (tatesurvey) one
introduces
b_{2} a_{1}^{2} 4 a_{2}, b_{4} a_{1} a_{3} 2 a_{4}, b_{6} a_{3}^{2} 4 a_{6}, and b_{8} b_{2} a_{6} a_{1} a_{3} a_{4} a_{2} a_{3}^{2} a_{4}^{2}.
Then one has
b_{2}^{2} b_{8} 8 b_{4}^{3} 27 b_{6}^{2} 9 b_{2} b_{4} b_{6}.
The nonvanishing of mod p is necessary and sufficient for
E to have good reduction mod p It follows that a prime p
divides if and only if E does not have good reduction mod
p If p is a prime for which E has bad reduction, then
there is a single singular point of the reduced curve E_{p} and
either (a) E_{p} has distinct tangent lines at the singular point
(semistable reduction) or (b) E_{p} has a single tangent line
occurring with multiplicity 2E is called semistable if
it has either good or semistable reduction at each prime The
conductor of E is the integer N defined by
N _{p}p^{p},
where
_{p}ll0 ifEhas good reduction at p .1 ifEhas semistable reduction atp.2 _{p} 2 otherwise.
The nonnegative integer _{p} cannot be positive unless p is
2 or 3 Tautologically, E is semistable if and only if its
conductor N is squarefreeOne defines the Lseries of E by
L(E,s) _{p N}11 c_{p} p^{s}_{p N}11 c_{p} p^{s} p^{1 2s},
where c_{p} is defined when E has good reduction mod p by the
formula
c_{p} p 1 E(F_{p}),
and c_{p} is defined when E has bad reduction mod p by
c_{p}ll1 if_{p} 1and the tangents are defined overF_{p}1 if_{p} 1 with irrational tangents0 if_{p} 1
One observes readily that the Lfunction of E codifies information
about the number of points of E in the finite field F_{p} Quite
generally for an algebraic variety defined over Q the analogous
codification of information obtained by counting points in the various
reductions mod p of the variety yields the HasseWeil zeta
function, which reflects cohomological information about E
The Lseries of E is the essential part, corresponding to cohomology
in dimension 1, of the HasseWeil zeta function of E The
HasseWeil zeta function is a special case of the general notion
(Serre serrepurdue) of zeta function for a scheme of
finite type over ZOne observes that L(E,s) resembles, at least insofar as one
considers its Euler factors for primes p corresponding to good
reductions of E, the Dirichlet series associated to a cuspform of
weight 2 that admits an Euler product expansion The observation of
this resemblance is the beginning of an appreciation of the
ShimuraTaniyamaWeil conjecture One is led to ask to what extent
the two classes of Dirichlet series with Euler products coincide The
conjecture states that the Lfunction of an elliptic curve defined
over Q with conductor N arises from a cuspform for the group
_{0}(N) that is compatible with the substitution in the
upperhalf plane H given by W_{N} Isogenous elliptic curves have
the same Lfunction, and, conversely (cf. Tate tateendom and
Faltings storrsconf) two elliptic curves with the same
Lfunction must be isogenous Thus, the idea of the conjecture is
that the isogeny classes of elliptic curves defined over Q with
conductor N are in bijective correspondence with the set of Hecke
eigencuspforms for the group _{0}(N) of level N compatible
with the extension of that group by the substitution arising from
W_{N} having rational Fourier coefficients and not arising from
levels dividing NThe Shimura mapShimura (shimnagoya, shimiaf, shimjmsj) showed
for a given W_{N}compatible Hecke eigencuspform f of weight 2 for
the group _{0}(N) with rational Fourier coefficients how to
construct how to construct an elliptic curve E_{f} defined over Q
such that the Dirichlet series (s) associated with f is the
same as the Lfunction L(E_{f},s) Thus, the ShimuraTaniyamaWeil
conjecture becomes the statement that Shimuras map from the set of
such cuspforms to the set of elliptic curves defined over Q is
surjective up to isogeny A rough description of the Shimura map
followsLet be a congruence subgroup of SL_{2}(Z) and let
X() denote the compact Riemann surface H^{} The
inclusion of in (1) induces a branched covering X() X(1) P^{1}.
One may use the elementary RiemannHurwitz formula from combinatorial
topology to determine the Euler number, and consequently the genus,
of X() The genus is the dimension of the space of cuspforms
of weight 2 Even when the genus is zero one obtains embeddings
of X() in projective spaces P^{r} through holomorphic maps
(f_{0}(), f_{1}(), , f_{r}()) ,
where
$f$_{0}, f_{1}, , f_{r}
is a basis of the space of modular forms of weight w with w
sufficiently large For example, any w 12 will suffice for
(1) For _{0}(N) (but not for arbitrary ) one
may find a basis of the space of modular forms of weight w having
rational Fourier coefficients Using the corresponding projective
embedding one finds a model for $X$_{0}(N)X(_{0}(N))
over Q i.e., an algebraic curve defined over Q in projective
space that is isomorphic as a compact Riemann surface to X_{0}(N)Associated with any complete nonsingular algebraic curve (i.e.,
after Weyl weyl, any compact Riemann surface) X of genus g
is a complex torus, the JacobianJ(X) of X, that is the
quotient of gdimensional complex vector space C^{g} by the lattice
generated by the period matrix, which is the
g 2g matrix in C obtained by integrating each of the
g members _{i} of a basis of the space of holomorphic
differentials over each of the 2g loops in X representing the
members of a homology basis in dimension 1 Furthermore, if one
picks a base point z_{0} in X, then for any z in X, the path
integral from z_{0} to z of each of the g holomorphic
differentials is welldefined modulo the periods of the differential
One obtains a holomorphic map
$XJ(X)$ from the formula
z (_{z0}^{z}_{1}, ,
_{z0}^{z}_{g}) .
This map is, in fact, universal for pointed holomorphic maps from X
to complex tori Furthermore, the Jacobian J(X) is an algebraic
variety that admits definition over any field of definition for X
and z_{0}, and the universal map also admits definition over any such
field The complex tori that admit embeddings in projective space are
the abelian group objects in the category of projective varieties
They are called abelian varieties Every abelian variety is
isogenous to the product of simple abelian varieties: abelian
varieties having no abelian subvarieties Shimura showed that one of
the simple isogeny factors of J(X_{0}(N)) is an elliptic curve E_{f}
defined over Q characterized by the fact that its
onedimensional space of holomorphic differentials induces on X_{0}(N)
via the composition of the universal map with projection on E_{f}
the onedimensional
space of differentials on X_{0}(N) determined by the cuspform f
He showed further that L(E_{f}, s) is the Dirichlet series
(s) with Euler product given by f An elliptic curve E
defined over Q is said to be modular if it is isogenous to
E_{f} for some W_{N}compatible Hecke eigencuspform of weight 2 for
_{0}(N) Equivalently E is modular if and only if L(E,s)
is the Dirichlet series given by such a cuspform The
ShimuraTaniyamaWeil Conjecture states that every elliptic curve
defined over Q is modular Shimura shimnagoya showed that
this conjecture is true in the special case where the Zmodule rank
of the ring of endomorphisms of E is greater than one In this case
the point (notation of section iref="sec.ellipC"sec.ellipC) of the
upperhalf plane corresponding to E(C) is a quadratic imaginary
number, and L(E, s) is a numbertheoretic Lfunction associated
with the corresponding imaginary quadratic number fieldThe hypothetical Frey curveLet p 5 be a prime Based on the assumption, which presumably
is false, that there are nonzero integers a, b, c such that
$apbpcp0$
G. Frey observed that the elliptic curve given by the equation
y^{2} x(x a^{p})(x b^{p}) ,
which is certainly defined over Q, would not be likely to be
modular Thus, if the ShimuraTaniyamaWeil Conjecture were true,
then Fermats Last Theorem would also be true By 1987 it had
been shown through the efforts of Frey, Ribet and Serre that the Frey
curve (iref="eq.frey"eq.frey) is not modular The proof involves the
systematic study of what is known as the adic representation
of an elliptic curve defined over Q which is described in the
next section This same technique is what has been reported to be the
basis of the proof of Wiles that every semistable elliptic curve
defined over Q is modular The Frey curve (iref="eq.frey"eq.frey) has
discriminant (abc)^{p} It is only slightly difficult to
see that it is semistable, and, therefore, that its conductor N is
the squarefree integer abc If the Frey curve is modular, one
is led to a cuspform of weight 2 for _{0}(abc) The theory
of adic representations leads one along a path of reductions
of the level N from the initial level abc
that enables one to conclude that there is a cuspform
of weight 2 for _{0}(2); but the genus of X_{0}(2) is 0,
and, consequently, there is no such cuspformadic representations of Gal(QQ)Let E be an elliptic curve defined over Q Inasmuch as the group
law
$EEE$
is defined over Q it follows that for each integer m the group
(scheme) E[m] of mtorsion points, i.e., for any field K
containing Q the group E[m](K) consisting of all x in E(K)
such that mx 0, is defined by equations with rational
coefficients Consequently, any automorphism of K must carry the
group E[m](K) into itself Since E(C) is the quotient of C by
a lattice, it is clear that E[m](C) is isomorphic to
$ZmZZmZ$; in fact, this latter group is
isomorphic to E[m](K) for each algebraically closed field of
characteristic 0 There is a unique ring homomorphism
$ZmnZZmZ$ for each integer n 1,
and the family of these ring homomorphisms gives rise to an inverse
system in the category of commutative rings If one specializes to
the case m ^{r} where is prime, the projective limit
is the ring Z_{} of adic integers The groups E[m]
form a direct system with respect to the inclusions
$E[m]E[mn]$, but, corresponding to the inverse
system of the groups ZmZ form an inverse system (the Tate
system) with respect to the family of homorphisms
$E[mn]E[m]$
defined by
$xnx$ If one specializes to the case
m ^{r} where is prime, one obtains the projective limit
T_{}(E) proj lim_{r } E[^{r}](Q) Z_{}Z_{},
which is isomorphic to the cohomology module
H^{1}(E, Z_{}) .
The action of Gal(QQ) on the torsion groups E[m] induces an action
of Gal(QQ) on the projective limit T_{}(E) This action gives
rise to a representation
_{} : Gal(QQ)GL_{2}(Z_{}) ,
which is called the adic representation of E In considering
_{} one is reminded of the action of the automorphism group of a
manifold M on the cohomology H^{}(M) and, more particularly, the
action of Gal(CR) on the cohomology of M when M is an
algebraic manifold in P^{n}(C) defined by equations with real
coefficients, but one must keep in mind that the transformations
of E(Q) arising from the elements of Gal(QQ) are not even
remotely continuous in the classical topology on E(C) More
generally, there is an algebraic way of defining the cohomology
ring H^{}(M, Z_{}) (see Tate tatepurdue) when M is an
algebraic variety with the property that automorphisms fixing the
field of definition act on H^{}(M, Z_{}) An introduction to the
study of _{} may be found in Serres Montreal NotesserremontrealThe canonical ring homomorphism from the ring Z_{} of adic integers
to the field ZZ induces a group homomorphism
$AA$,
called reduction mod from the group GL_{2}(Z_{}) to
the finite group GL_{2}(ZZ) An adic representation of
Gal(QQ) is called modular if it is isomorphic to the
representation _{} arising from the elliptic curve E_{f} that is
the image under the Shimura map of a modular form f A
representation
Gal(QQ)GL_{2}(ZZ)
is called modular if
it is isomorphic to _{} for some modular adic representation
_{} In the extensive detailed study of representations of
Gal(QQ) particular attention has been paid to the question of when a
representation in GL_{2}(Z_{}) is modular and also to the question of
when a representation of GL_{2}(ZZ) is modular Under certain
conditions (see Serre serreduke and Ribet ribinv,
ribtoul) one can show that _{} is modular if _{} is
modular, i.e., _{} is modular if it is congruent mod
to a modular adic representation Such arguments are central both
to the work of Ribet in showing that the ShimuraTaniyamaWeil
conjecture implies Fermats Last Theorem and to the reported work
of Wiles in proving that semistable elliptic curves are modular In
the work of Ribet the basic idea is that the modularity of the Frey
curve, which has squarefree conductor N abc, implies the
existence of a cusp form of weight 2 and level N By using an
argument at the scene of the mod representations, Ribet shows
that one may split each odd prime divisor out of the level N and
arrive at the conclusion that there is a cusp form of weight 2 and
level 2, which is not possiblepagebreakAppendixLate 1993early 1994 StatusAndrew Wiles posted the following announcement in the UseNet
electronic news group called sci.math:FromwilesrugolaPrincetonEDUAndrewWilesNewsgroupsscimathSubjectFermatstatusMessageID1993Dec401365012700PrincetonEDUDate4Dec93013650GMTSendernewsPrincetonEDUUSENETNewsSystemOrganizationPrincetonUniversityLines21OriginatornewsnimasterNntpPostingHostrugolaprincetoneduInviewofthespeculationonthestatusofmyworkontheTaniyamaShimuraconjectureandFermatsLastTheoremIwillgiveabriefaccountofthesituationDuringthereviewprocessanumberofproblemsemergedmostofwhichhavebeenresolvedbutoneinparticularIhavenotyetsettledThekeyreductionofmostcasesoftheTaniyamaShimuraconjecturetothecalculationoftheSelmergroupiscorrectHoweverthefinalcalculationofapreciseupperboundfortheSelmergroupinthesemistablecaseofthesymmetricsquarerepresentationassociatedtoamodularformisnotyetcompleteasitstandsIbelievethatIwillbeabletofinishthisinthenearfutureusingtheideasexplainedinmyCambridgelecturesThefactthatalotofworkremainstobedoneonthemanuscriptmakesitstillunsuitableforreleaseasapreprintInmycourseinPrincetonbeginninginFebruaryIwillgiveafullaccountofthisworkAndrewWilespagebreakahlforsLars V. Ahlfors,
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