PRAISE FOR AMIR D. ACZEL'S FERMAT'S LAST THEOREM "This is a captivating volume. . . . The brilliant backdoor method used by Mr. Wiles as he reached his solution, along with the debt he owed to many other contemporary mathematicians, is graspable in Mr. Aczel's lucid prose. Equally important is the sense of awe that Mr. Aczel imparts for the hidden, mystical harmonies of numbers, and for that sense of awe alone, his slender volume is well worth the effort." —The New York Times "For more than three centuries, Fermat's Last Theorem was the most famous unsolved problem in mathematics, here's the story of how it was solved. . .. An excellent short history of mathematics, viewed through the lens of one of its great problems— and achievements." —Kirkus Reviews "This exciting recreation of a landmark discovery reveals the great extent to which modern mathematics is a collaborative enterprise. . . . While avoiding technical details, Aczel maps the strange, beautiful byways of modern mathematical thought in ways the layperson can grasp." —Publishers Weekly "Briefly chronicles the history of the famous problem of the title, which was recently solved by a mathematician named Andrew Wiles after he had devoted seven years to the task. . . . Aczel does a superb job of creating in the nonmathematical reader the illusion of comprehension." —The New Yorker
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PRAISE FOR AMIR D. ACZEL'S
FERMAT'S LAST THEOREM
"This is a captivating volume. . . . The brilliant backdoor method
used by Mr. Wiles as he reached his solution, along with the debt he
owed to many other contemporary mathematicians, is graspable in
Mr. Aczel's lucid prose. Equally important is the sense of awe that
Mr. Aczel imparts for the hidden, mystical harmonies of numbers,
and for that sense of awe alone, his slender volume is well worth the
effort."
—The New York Times
"For more than three centuries, Fermat's Last Theorem was the
most famous unsolved problem in mathematics, here's the story of
how it was solved. . . . An excellent short history of mathematics,
viewed through the lens of one of its great problems— and
achievements."
—Kirkus Reviews
"This exciting recreation of a landmark discovery reveals the great
extent to which modern mathematics is a collaborative
enterprise. . . . While avoiding technical details, Aczel maps the
strange, beautiful byways of modern mathematical thought in ways
the layperson can grasp."
—Publishers Weekly
"Briefly chronicles the history of the famous problem of the title,
which was recently solved by a mathematician named Andrew Wiles
after he had devoted seven years to the task. . . . Aczel does a
superb job of creating in the nonmathematical reader the illusion of
comprehension."
—The New Yorker
Pierre de Fermat (1601-1665).
FERMAT'S LAST THEOREMUnlocking the Secret ojan Ancient
signing them with the mayor's name. Months of such
persecution caused Galois' father to lose his
confidence and he became convinced that the world
was out to get him. Slowly losing touch with reality,
he went to Paris. There, in an apartment not far from
where his son was studying, the father killed himself.
The young Galois never recovered from this tragedy.
Obsessed with the lost cause of the 1830 revolution,
and frustrated with the school director, whom he
considered an apologist for the royalists and clerics,
Galois wrote a scathing letter criticizing the director.
He was inspired to do so after three days of rioting in
the streets, when students all over Paris were
revolting against the regime. Galois and his
classmates were kept locked in their school, unable
to scale the tall fence. The angry Galois sent his
blistering letter criticizing the school director to the
Gazette des ficoles. As a result, he was expelled from the
school. But Galois was undaunted—he wrote a second
letter to the Gazette, and addressed the students at
the school to speak up for honor and conscience. He
got no response.
Out of school, Galois started offering private
lessons in mathematics. He wanted to teach his own
mathematical theories, outside the French schools,
when he was all of nineteen years old. But Galois
could not find students to teach—his theories were
too advanced; he was far ahead of his time.
Facing an uncertain future and doomed not to be
able to pursue a decent education, in desperation
Galois joined the artillery branch of the French
National Guard. Among the National Guard, headed
in its past by Lafayette, there were many liberal
elements close to young Galois' political philosophy.
While in the Guard, Galois made one last attempt to
publish his mathematical work. He wrote a paper on
the general solution of equations—today recognized
as the beautiful Galois Theory—and sent it to Simeon-
Denis Poisson (1781-1840) at the French Academy of
Sciences. Poisson read the paper but determined
that it was "incomprehensible." Once again, the
nineteen-year-old was so far ahead of any of the
older French mathematicians of his day that his
elegant new theories went way over their heads. At
that moment, Galois decided to abandon
mathematics and to become a full-time revolutionary.
He said that if a body was needed to get the people
involved with the revolution, he would donate his.
On May 9, 1831, two hundred young republicans
held a banquet in which they protested against the
royal order disbanding the artillery of the National
Guard. Toasts were offered to the French Revolution
and its heroes, as well as to the new revolution of
1830. Galois stood up and proposed a toast. He said
"To Louis Philippe," the Duke of Orleans, who was
now King of the French. While saying this, holding up
his
glass, Galois was also holding up an open pocket-
knife in his other hand. This was interpreted as a
threat on the life of the King, and caused a riot. The
next day Galois was arrested.
In his trial for threatening the life of the King,
Galois' attorney claimed that Galois had actually
said, while holding his knife, "To Louis Philippe, if he
becomes a traitor." Some of Galois' artillery friends
who were present testified to this, and the jury found
him not guilty. Galois retrieved his knife from the
evidence table, closed it and put it in his pocket, and
left the courtroom a free man. But he was not free
for very long. A month later he was arrested as a
dangerous republican, and kept in jail without a
charge while the authorities looked for a charge
against him that would stick. They finally found one—
wearing the uniform of the disbanded artillery. Galois
was tried for this/:harge and sentenced to six months
in jail. The royalists wercpleased to finally put away a
twenty-year-old they considered to be a dangerous
enemy of the regime. Galois was paroled after some
time and was moved to a halfway facility. What
happened next is open to question. While on parole,
Galois met a young woman and fell in love. Some
believe he was set up by his royalist enemies who
wanted to put an end to his revolutionary activities
once and for all. At any rate, the woman with whom
he got involved was of questionable virtue ("une
cocjuette de has e'tage"). As soon as the two became
lovers, a royalist came to "save her honor" and
challenged Galois to a duel. The young
mathematician was left no way out of the mess. He
tried everything he could to talk the man out of the
duel, but to no avail.
The night before the duel, Galois wrote several
letters.
These letters to his friends lend support to the
theory that he was framed by the royalists. He wrote
that he was challenged by two of the royalists and
that they had put him on his honor not to tell his
republican friends about the duel. "I die the victim of
an infamous coquette. It is in a miserable brawl that
my life is extinguished. Oh, why die for so trivial a
thing, why die for something so despicable!" But
most of that last night before the duel, Galois
carefully put down on paper his entire mathematical
theory, and sent it to his friend Auguste Chevalier. At
dawn on May 30, 1832, Galois faced his challenger on
a deserted field. He was shot in the stomach and left
lying in agony alone in the field. No one bothered to
call a doctor. Finally a peasant found him and
brought him to the hospital, where he died the next
morning. He was twenty years old. In 1846, the
mathematician Joseph Liouville edited and published
Galois' elegant mathematical theory in a journal.
Galois' theory would supply the crucial step in the
method used a century and a half later in attacking
Fermat's Last Theorem.
Another Victim
Cauchy's carelessness and arrogance ruined the life
of at least one other brilliant mathematician. Niels
Henrik Abel (1802-1829) was the son of the pastor of
the village of Findo in Norway. When he was sixteen,
a teacher encouraged Abel to read Gauss' famous
book, the Discjuisitiones. Abel was even successful in
filling in some gaps in the proofs of some of the
theorems. But two years later, his father died and
young Abel had to postpone his study of mathematics
and concentrate his efforts on supporting the family.
In spite of the great difficulties he
faced, Abel managed to continue some study of
mathematics and made a remarkable mathematical
discovery at the age of nineteen. He published a
paper in 1824 in which he proved that no solution
was possible for equations of the fifth degree. Abel
had thus solved one of the most celebrated problems
of his day. Yet the gifted young mathematician had
still not been offered an academic position, which he
badly needed to support his family and so he sent his
work to Cauchy for evaluation and possible
publication and recognition. The .paper Abel sent
Cauchy was of extraordinary power and generality.
But Cauchy lost it. When it appeared in print, years
later, it was much too late to help Abel. In 1829 Abel
died from tuberculosis, brought on by poverty and
the strain of supporting his family in dire
circumstances. Two days after his death, a letter
arrived addressed to Abel informing him he had been
appointed professor at the University of Berlin.
The concept of an Abelian Group (now considered a
word and written with a small "a," abelian) is very
important in modern algebra and is a crucial element
in the modern treatment of the Fermat problem. An
abelian group is one where the order of mathematical
operations can be reversed without affecting the
outcome. An abelian variety is an even more abstract
algebraic entity, and its use was also important in
modern approaches to the solution of Fermat's Last
Theorem.
D ed ek in d 's Idea ls
The legacy of Carl Friedrich Gauss continued through
the centuries. One of Gauss' most notable
mathematical successors was Richard Dedekind
(1831-1916), born in the same
town as the great master, in Brunswick, Germany.
Unlike Gauss, however, as a child Dedekind showed
no great interest in or capacity for mathematics. He
was more interested in physics and chemistry, and
saw mathematics as a servant to the sciences. But at
the age of seventeen, Dedekind entered the same
school where the great Gauss got his mathematical
training—Caroline College—and there his future
changed. Dedekind became interested in
mathematics and he pursued that interest in
Gottingen, where Gauss was teaching. In 1852, at the
age of 21, Richard Dedekind received his doctorate
from Gauss. The master found his pupil's dissertation
on the calculus "completely satisfying." This was not
such a great compliment, and in fact Dedekind's
genius had not yet begun to manifest itself.
In 1854, Dedekind was appointed a lecturer in
Gottingen. When Gauss died in 1855 and Dirichlet
moved from Berlin to take his position, Dedekind
attended all of Dirichlet's lectures at Gottingen and
edited the latter's pioneering treatise on number
theory, adding a supplement based on his own work.
This supplement contained an outline of the theory
Dedekind developed for algebraic numbers, which
are defined as solutions of algebraic equations. They
contain multiples of square roots of numbers along
with rational numbers. Algebraic number fields are
very important in the study of Fermat's equation, as
they arise from the solution of various kinds of
equations. Dedekind thus developed a significant
area within number theory.
Dedekind's greatest contribution to the modern
approach to Fermat's last Theorem was his
development of the theory of ideals, abstractions of
Kummer's ideal numbers. A century
after their development by Dedekind, ideals would
inspire Barry Mazur, and Mazur's own work would be
exploited by Andrew Wiles.
In the 1857-8 academic year, Richard Dedekind
gave the first mathematics course on Galois theory.
Dedekind's understanding of mathematics was very
abstract, and he elevated the theory of groups to the
modern level at which it is understood and taught
today. Abstraction made the twentieth-century
approach to Fermat's problem possible. Dedekind's
groundbreaking course on the theories developed by
Galois was a great step in this direction. The course
was attended by two students.
Then Dedekind's career took a strange turn. He left
Gottingen for a position in Zurich, and after five
years, in 1862, he returned to Brunswick, where he
taught at a high school for fifty years. No dne has
been able to explain why a brilliant mathematician
who brought algebra to an incredibly high level of
abstraction and generality suddenly left one of the
most prestigious positions at any European
university to teach at an unknown high school.
Dedekind never married and lived for many years
with his sister. He died in 1916, and maintained a
sharp, active mind to his last day.
Fin de Siecle
At the turn of the nineteenth century, there lived in
France a mathematician of great ability in a
surprisingly wide variety of areas. The breadth of
knowledge of Henri Poincare (1854-1912) extended
beyond mathematics. In 1902 and later, when he was
already a renowned mathematician, Poincare wrote
popular
books on mathematics. These paperbacks, read by
people of all ages, were a common sight in the cafes
and the parks of Paris.
Poincare was born to a family of great achievers.
His cousin, Raymond Poincare, rose to be the
president of France during the First World War.
Other family members held government and public
service positions in France as well.
From a young age, Henri displayed a powerful
memory. He could recite from any page of a book he
read. His absentmind-edness, however, was
legendary. A Finnish mathematician once came all
the way to Paris to meet with Poincare and discuss
some mathematical problems. The visitor was kept
waiting for three hours outside Poincare's study
while the absentminded mathematician paced back
and forth—as was his habit throughout his working
life. Finally, Poincare popped his head into the
waiting room and exclaimed: "Sir, you are disturbing
me!" upon which the visitor summarily left, never to
be seen in Paris again.
Poincare's brilliance was recognized when he was
in elementary school. But since he was such a
universalist—a renaissance man in the making—his
special aptitude for mathematics did not yet
manifest itself. He distinguished himself with his
excellence in writing at an early age. A teacher who
recognized and encouraged his ability treasured his
school papers. At some point, however, the
concerned teacher had to caution the young genius:
"Don't do so well, please...try to be more ordinary."
The teacher had good reason to make this sugges-
tion. Apparently, French educators had learned
something from the misfortunes of Galois half a
century earlier—teachers found that gifted students
often failed at the hands of uninspired examiners.
His teacher was genuinely worried that
Poincare was so brilliant that he might fail those
exams. As a child, Poincare was already
absentminded. He often skipped meals because he
forgot whether or not he had eaten.
Young Poincare was interested in tjie classics and
he learned to write well. As a teenager he became
interested in mathematics and immediately excelled
in it. He would work out problems entirely in his head
while pacing in a room—then sit down and write
everything very hastily. In this he resembled Galois
and Euler. When Poincare finally took his exams, he
almost failed in math, as his elementary school
teacher had feared years earlier. But he did pass,
only because—at seventeen—his renown as a
mathematician was so great that the examiners
didn't dare fail him. "Any student other than Poincare
would have been given a failing grade," the chief
examiner declared as he passed the student who
went on to the Ecole Polytechnique and became the
greatest French mathematician of his time.
Poincare wrote scores of books on mathematics,
mathematical physics, astronomy, and popular
science. He wrote research papers of over five
hundred pages on new mathematical topics he
developed. He made major contributions to topology,
the area started by Euler. However, Poincare's
results were so important that this branch of
mathematics is considered to have truly been
launched only in 1895, with the publication of
Poincare's Analysis Situs. Topology—the study of shapes
and surfaces and continuous functions—was
important in understanding Fermat's problem in the
late twentieth century. But even more essential to
the modern approach to Fermat's Last Theorem was
another area pioneered by Henri Poincare.
FERMAT'S LAST THEOREM
Modular Forms
Poincare studied periodic functions, such as the sines
and cosines of Fourier—not on the number line as
Fourier had done, but in the complex plane. The sine
function, sin x, is the vertical height on a circle with
radius 1 when the angle is x. This function is periodic:
it repeats itself over and over every time the angle
completes a multiple of its period, 360 degrees. This
periodicity is a symmetry. Poincare examined the
complex plane, which contains real numbers on the
horizontal axis and imaginary ones on the vertical one
as shown below.
Here, a periodic function could be conceived as having a periodicity
both along the real axis and along the imaginary axis. Poincare went
even further and posited the existence of functions with a wider
array of symmetries. These were functions that remained
unchanged when the complex variable z was changed
according to/(z) >j(az+b/cz+d). Here the elements a, b, c,
d, arranged as a matrix, formed an algebraic group. This means
t h a t t h e r e a r e i n f i n i t e l y m a n y p o s s i b l e v a r i a t i o n s . T h e y a l l c o m m u t e w i t h e a c h o t h e r a n d t h e f u n c t i o n / i s invariant u n d e r t h i s g r o u p o f t r a n s f o r m a t i o n s . P o i n c a r e c a l l e d s u c h w e i r d f u n c t i o n s a u t o m o r p h i c f o r m s .
T h e a u t o m o r p h i c f o r m s w e r e v e r y , v e r y s t r a n g e c r e a t u r e s s i n c e t h e y s a t i s f i e d m a n y i n t e r n a l s y m m e t r i e s . P o i n c a r e w a s n ' t q u i t e s u r e t h e y e x i s t e d . A c t u a l l y , P o i n c a r e d e s c r i b e d h i s r e s e a r c h a s f o l l o w s . H e s a i d t h a t f o r f i f t e e n d a y s h e w o u l d w a k e u p i n t h e m o r n i n g a n d s i t a t h i s d e s k f o r a c o u p l e o f h o u r s t r y i n g t o c o n v i n c e h i m s e l f t h a t t h e a u t o m o r p h i c f o r m s h e i n v e n t e d c o u l d n ' t p o s s i b l y e x i s t . O n t h e f i f t e e n t h d a y , h e r e a l i z e d t h a t h e w a s w r o n g . T h e s e s t r a n g e f u n c t i o n s , h a r d t o i m a g i n e v i s u a l l y , d i d e x i s t . P o i n c a r e e x t e n d e d t h e m t o e v e n m o r e c o m p l i c a t e d f u n c t i o n s , c a l l e d m o d u l a r f o r m s . T h e m o d u l a r f o r m s l i v e o n t h e u p p e r h a l f o f t h e c o m p l e x p l a n e , a n d t h e y h a v e a h y p e r b o l i c g e o m e t r y . T h a t i s , t h e y l i v e i n a s t r a n g e s p a c e w h e r e t h e n o n -E u c l i d e a n g e o m e t r y o f B o l y a i a n d L o b a c h e v s k y r u l e s . T h r o u g h a n y p o i n t i n t h i s h a l f - p l a n e , v m a n y " l i n e s " p a r a l l e l t o a n y g i v e n l i n e e x i s t .
P o i n c a r e l e f t b e h i n d t h e s y m m e t r i c a u t o m o r p h i c f o r m s a n d t h e i r e v e n m o r e i n t r i c a t e m o d u l a r f o r m s a n d w e n t o n t o d o o t h e r m a t h e m a t i c s . H e w a s b u s y i n s o m a n y f i e l d s , o f t e n i n a f e w o f t h e m a t o n c e , t h a t h e h a d n o t i m e t o s i t a n d p o n d e r t h e b e a u t y o f h a r d l y i m a g i n a b l e , i n f i n i t e l y s y m m e t r i c e n t i t i e s . B u t u n b e k n o w n s t t o h i m , h e h a d j u s t s o w e d a n o t h e r s e e d i n t h e g a r d e n t h a t w o u l d e v e n t u a l l y b r i n g a b o u t t h e F e r m a t s o l u t i o n .
An Unexpected Connection with a Doughnut
I n 1922, t h e A m e r i c a n m a t h e m a t i c i a n L o u i s J.
M o r d e l l d i s c o v e r e d w h a t h e t h o u g h t w a s a v e r y s t r a n g e c o n n e c t i o n b e t w e e n
t h e s o l u t i o n s o f a l g e b r a i c e q u a t i o n s a n d t o p o l o g y . T h e e l e m e n t s o f t o p o l o g y a r e s u r f a c e s a n d s p a c e s . T h e s e s u r f a c e s c o u l d b e i n a n y d i m e n s i o n : t w o d i m e n s i o n s , s u c h a s t h e f i g u r e s i n a n c i e n t G r e e k g e o m e t r y , o r t h e y c o u l d b e i n t h r e e -d i m e n s i o n a l s p a c e , o r m o r e . T o p o l o g y i s a s t u d y o f c o n t i n u o u s f u n c t i o n s t h a t a c t o n t h e s e s p a c e s , a n d t h e p r o p e r t i e s o f t h e s p a c e s t h e m s e l v e s . T h e p a r t o f t o p o l o g y w h i c h c o n c e r n e d M o r d e l l w a s t h e o n e o f s u r f a c e s i n t h r e e -d i m e n s i o n a l s p a c e . A s i m p l e e x a m p l e o f s u c h a s u r f a c e i s a s p h e r e : i t i s t h e s u r f a c e o f a b a l l , s u c h a s a b a s k e t b a l l . T h e b a l l i s t h r e e -d i m e n s i o n a l , b u t i t s s u r f a c e ( a s s u m i n g n o d e p t h ) i s a t w o - d i m e n s i o n a l o b j e c t . T h e s u r f a c e o f t h e e a r t h i s a n o t h e r e x a m p l e . T h e e a r t h i t s e l f i s t h r e e d i m e n s i o n a l : a n y p l a c e o n t h e e a r t h o r i n s i d e i t c a n b e g i v e n b y i t s l o n g i t u d e ( o n e d i m e n s i o n ) , i t s l a t i t u d e ( a s e c o n d d i m e n s i o n ) , a n d i t s d e p t h ( t h e t h i r d d i m e n s i o n ) . B u t t h e s u r f a c e o f t h e e a r t h ( n o d e p t h ) i s t w o - d i m e n s i o n a l , s i n c e a n y p o i n t o n t h e s u r f a c e o f t h e e a r t h c a n b e s p e c i f i e d b y t w o n u m b e r s : i t s l o n g i t u d e a n d i t s l a t i t u d e .
T w o - d i m e n s i o n a l s u r f a c e s i n t h r e e - d i m e n s i o n a l s p a c e c a n b e c l a s s i f i e d a c c o r d i n g t o t h e i r g e n u s . T h e g e n u s i s
t h e n u m b e r o f h o l e s i n t h e s u r f a c e . T h e g e n u s o f a s p h e r e i s z e r o s i n c e t h e r e a r e n o h o l e s i n i t . A d o u g h n u t h a s o n e h o l e i n i t . T h e r e f o r e t h e g e n u s o f a d o u g h n u t ( m a t h e m a t i c a l l y c a l l e d a t o r u s ) i s o n e . A h o l e m e a n s a h o l e t h a t r u n s c o m p l e t e l y t h r o u g h t h e s u r f a c e . A c u p w i t h t w o h a n d l e s h a s t w o h o l e s t h r o u g h i t . T h e r e f o r e i t i s a s u r f a c e o f g e n u s t w o .
A s u r f a c e o f o n e g e n u s c a n b e t r a n s f o r m e d b y a c o n t i n u o u s f u n c t i o n i n t o a n o t h e r s u r f a c e o f t h e s a m e g e n u s . T h e o n l y w a y t o t r a n s f o r m a s u r f a c e o f o n e g e n u s t o o n e o f a d i f f e r e n t g e n u s
is by closing or opening some holes. This cannot be
done by a continuous function, since it will require
some ripping or fusing together, each of which is a
mathematical discontinuity.
Mordell discovered a strange and totally unexpected
connection between the number of holes in the
surface (the genus) of the space of solutions of an
equation and whether or not the
e q u a t i o n h a d a f i n i t e n u m b e r o r a n i n f i n i t e n u m b e r o f s o l u t i o n s . I f t h e s u r f a c e o f s o l u t i o n s , u s i n g c o m p l e x n u m b e r s f o r g r e a t e s t g e n e r a l i t y , h a d t w o o r m o r e h o l e s ( t h a t i s , h a d g e n u s t w o o r h i g h e r ) , t h e n t h e e q u a t i o n h a d o n l y f i n i t e l y m a n y w h o l e - n u m b e r s o l u t i o n s . M o r d e l l w a s u n a b l e t o p r o v e h i s d i s -c o v e r y , a n d i t b e c a m e k n o w n a s M o r d e l l ' s c o n j e c t u r e .
Faltings' Proof
I n 1983, a t w e n t y - s e v e n y e a r o l d G e r m a n m a t h e m a t i c i a n , G e r d F a l t i n g s , a t t h a t t i m e a t t h e U n i v e r s i t y o f W u p p e r t a l , w a s a b l e t o p r o v e t h e M o r d e l l c o n j e c t u r e . F a l t i n g s w a s n o t i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m , c o n s i d e r i n g i t a n i s o l a t e d p r o b l e m i n n u m b e r t h e o r y . Y e t h i s p r o o f , w h i c h u s e d g r e a t i n g e n u i t y a l o n g w i t h t h e p o w e r f u l m a c h i n e r y o f a l g e b r a i c g e o m e t r y d e v e l o p e d i n t h i s c e n t u r y , h a d p r o f o u n d i m p l i c a t i o n s f o r t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m . B e c a u s e t h e g e n u s o f t h e F e r m a t e q u a t i o n f o r n g r e a t e r t h a n 3
w a s 2 o r m o r e , i t b e c a m e c l e a r t h a t i f i n t e g e r s o l u t i o n s t o t h e F e r m a t e q u a t i o n e x i s t e d a t a l l , t h e n t h e s e w e r e f i n i t e ( w h i c h w a s c o m f o r t i n g , s i n c e t h e i r n u m b e r w a s n o w l i m i t e d ) . S o o n a f t e r w a r d s , t w o m a t h e m a t i c i a n s , G r a n v i l l e a n d H e a t h - B r o w n , u s e d F a l t -
i n g s ' r e s u l t t o s h o w t h a t t h e n u m b e r o f s o l u t i o n s o f F e r m a t ' s e q u a t i o n , i f t h e y e x i s t e d , d e c r e a s e d a s t h e e x p o n e n t n i n c r e a s e d . I t w a s s h o w n t h a t t h e p r o p o r t i o n o f e x p o n e n t s f o r w h i c h F e r m a t ' s L a s t T h e o r e m w a s t r u e a p p r o a c h e d o n e h u n d r e d p e r c e n t a s n i n c r e a s e d .
I n o t h e r w o r d s , F e r m a t ' s L a s t T h e o r e m w a s " a l m o s t a l w a y s " t r u e . I f s o l u t i o n s t o F e r m a t ' s e q u a t i o n e x i s t e d ( i n w h i c h c a s e t h e T h e o r e m w a s n o t t r u e ) , t h e n s u c h s o l u t i o n s w e r e f e w a n d
v e r y f a r b e t w e e n . S o t h e s t a t u s o f F e r m a t ' s L a s t T h e o r e m i n 1983 w a s t h e f o l l o w i n g . T h e t h e o r e m w a s p r o v e n f o r n u p t o a m i l l i o n ( a n d i n 1992 t h e l i m i t w a s r a i s e d t o 4 m i l l i o n ) . I n a d d i -t i o n , f o r l a r g e r n, i f s o l u t i o n s e x i s t e d a t a l l t h e n t h e y w e r e v e r y f e w a n d d e c r e a s i n g w i t h n.
The Mysterious Greek General with the Funny Name T h e r e a r e d o z e n s o f e x c e l l e n t b o o k s o n m a t h e m a t i c s p u b -l i s h e d i n F r a n c e , w r i t t e n i n F r e n c h , w i t h t h e a u t h o r l i s t e d a s N i c o l a s B o u r b a k i . T h e r e w a s a G r e e k g e n e r a l n a m e d B o u r b a k i (1816-1897). I n 1862
B o u r b a k i w a s o f f e r e d t h e t h r o n e o f G r e e c e , w h i c h h e d e c l i n e d . T h e g e n e r a l h a d a n i m p o r t a n t r o l e i n t h e F r a n c o - P r u s s i a n w a r , a n d t h e r e i s a s t a t u e o f h i m i n t h e F r e n c h c i t y o f N a n c y . B u t G e n e r a l B o u r b a k i k n e w n o t h i n g a b o u t m a t h e m a t i c s . A n d h e n e v e r w r o t e a b o o k , a b o u t m a t h e m a t i c s o r a n y t h i n g e l s e . W h o w r o t e t h e m a n y v o l u m e s o f m a t h e m a t i c s b e a r i n g h i s n a m e ?
T h e a n s w e r l i e s i n t h e
h a p p y d a y s i n P a r i s o f b e t w e e n t h e t w o W o r l d W a r s . H e m i n g w a y , P i c a s s o a n d M a t i s s e w e r e n o t t h e o n l y p e o p l e w h o l i k e d t o s i t i n c a f e s a n d m e e t t h e i r f r i e n d s a n d s e e p e o p l e a n d b e s e e n . A t t h a t t i m e , a r o u n d t h e s a m e c a f e s o n t h e L e f t B a n k b y t h e U n i v e r s i t y o f P a r i s , t h e r e f l o u r i s h e d a v i b r a n t m a t h e m a t i c a l c o m m u n i t y . P r o f e s s o r s o f m a t h e m a t i c s f r o m t h e u n i v e r s i t y a l s o l i k e d t o m e e t t h e i r f r i e n d s , d r i n k a c a f e a u l a i t o r a p a s t i s i n a g o o d b r a s s e r i e o n t h e B o u l e v a r d S t . M i c h e l b y t h e b e a u t i f u l L u x e m b o u r g G a r d e n s , a n d d i s c u s s . . . m a t h e m a t i c s . S p r i n g t i m e i n P a r i s i n s p i r e d w r i t e r s , a r t i s t s , a n d m a t h e m a t i c i a n s . O n e i m a g i n e s t h a t o n a s u n n y d a y a t a p l e a s a n t c a f e a r o w d y g r o u p o f m a t h e m a t i c i a n s c o n g r e -
ss
g a t e d . F e e l i n g s o f f r a t e r n i t y o v e r c a m e t h e m a s t h e y a r g u e d a n i m a t e d l y o v e r s o m e f i n e p o i n t s o f a t h e o r y . T h e i r r e v e l r y p r o b a b l y d i s t u r b e d H e m i n g w a y w h o w r i t e s h e a l w a y s l i k e d t o w o r k b y h i m s e l f a t a c a f e , a n d h e p r o b a b l y l e f t t o g o t o o n e o f h i s a l t e r n a t e , l e s s f a v o r i t e h a u n t s . B u t t h e m a t h e m a t i c i a n s d i d n ' t c a r e . T h e y v a l u e d e a c h o t h e r ' s c o m p a n y , a n d a c a f e f u l l o f m a t h e m a t i c i a n s — a l l s p e a k i n g t h e s a m e l a n g u a g e o f n u m b e r s a n d s y m b o l s a n d s p a c e s a n d f u n c t i o n s — w a s i n t o x i c a t i n g . " T h i s i s w h a t t h e P y t h a g o r e a n s m u s t h a v e f e l t w h e n t h e y t a l k e d m a t h e m a t i c s , " p e r h a p s o n e o f t h e s e n i o r p e o p l e i n t h e g r o u p s a i d , a s h e l i f t e d h i s g l a s s i n a t o a s t . " Y e s , b u t t h e y d i d n ' t d r i n k P e r n o d , " s a i d a n o t h e r a n d e v e r y o n e l a u g h e d . " B u t w e c o u l d b e l i k e t h e m , " a n s w e r e d t h e f i r s t . " W h y d o n ' t w e f o r m o u r own s o c i e t y ? A s e c r e t o n e , n a t u r a l l y . " T h e r e w e r e v o i c e s o f a p p r o v a l a l l a r o u n d . S o m e o n e s u g g e s t e d t h e y b o r r o w t h e n a m e o f o l d G e n e r a l B o u r b a k i . T h e r e w a s a r e a s o n f o r t h i s s u g g e s t i o n . I n t h o s e d a y s , t h e m a t h e m a t i c s d e p a r t m e n t a t t h e U n i v e r s i t y d f P a r i s h a d a n a n n u a l t r a d i t i o n o f i n v i t i n g a p r o f e s s i o n a l a c t o r w h o p r e s e n t e d h i m s e l f t o t h e a s s e m b l e d f a c u l t y a n d g r a d u a t e s t u d e n t s a s N i c o l a s
B o u r b a k i . H e w o u l d t h e n d o a o n e - m a n s h o w c o n s i s t i n g o f a l o n g m o n o l o g u e o f m a t h e m a t i c a l d o u b l e - t a l k . S u c h p r e s e n t a t i o n s w e r e v e r y e n t e r t a i n i n g s i n c e t h e r i c h n e s s o f m o d e r n m a t h e m a t i c a l t h e o r y m a k e s f o r a v a s t v o c a b u l a r y t h a t i s b o t h d e s c r i p t i v e o f m a t h e m a t i c s a n d h a s d i f f e r e n t m e a n i n g s i n e v e r y d a y l i f e . O n e s u c h w o r d i s " d e n s e . " T h e r a t i o n a l n u m b e r s a r e s a i d t o b e d e n s e w i t h i n t h e r e a l n u m b e r s . T h i s m e a n s t h a t w i t h i n a n y n e i g h b o r h o o d o f r a t i o n a l n u m b e r s a r e i r r a t i o n a l n u m b e r s . B u t " d e n s e " m e a n s m a n y o t h e r t h i n g s i n e v e r y d a y l i f e .
Graduate students today like to play the same
games of double meaning, and they like to tell the
story of beautiful Poly Nomial who meets the smooth
operator Curly Pi (polynomial, smooth operator, and
curly pi are all mathematical terms).
And so the books these mathematicians wrote
together bore the name Nicolas Bourbaki. A Bourbaki
Seminar was initiated, where mathematical ideas and
theories were discussed not infrequently.
Membership in the society was supposed to be
anonymous, and mathematical results were to be
credited to the society in the name of Bourbaki,
rather than to individual members.
But the members of Bourbaki were not the
Pythagoreans. While the author of the textbooks was
Nicolas Bourbaki, research results such as theorems
and their proofs—which are far more prestigious than
books—were credited to the individual
mathematicians who achieved the results. One of the
first members of Bourbaki was Andre Weil (1906- ),
who later moved to the United States and the
Institute of Advanced Study at Princeton. His name
would never be too far away from the important
conjecture leading to the solution of the Fermat
problem.
Another of the founders of Bourbaki was the
French mathematician Jean Dieudonne, who like most
of the other "French only" members of the society
moved on to greener pastures at the universities in
the United States. Dieudonne, who was the principal
author of many of the books bearing the collective
name of Nicolas Bourbaki, epitomizes the clash
between the Bourbakites' quest for individual
anonymity and their individual egos. Dieudonne once
published a paper bearing the name
o f B o u r b a k i . T h e p a p e r w a s f o u n d t o c o n t a i n a n e r r o r , a n d s o D i e u d o n n e p u b l i s h e d a n o t e t i t l e d : " O n a n E r r o r o f N . B o u r b a k i , " a n d s i g n e d i t J . D i e u d o n n e .
T h e s c h i z o p h r e n i c n a t u r e o f t h e s o c i e t y — i t s m e m b e r s w e r e a l l F r e n c h , b u t m o s t o f t h e m l i v e d i n t h e U n i t e d S t a t e s — m a n i f e s t s i t s e l f i n M r . B o u r b a k i ' s a f f i l i a t i o n . U s u a l l y , p u b l i c a t i o n s o f N i c o l a s B o u r b a k i l i s t h i s a f f i l i a t i o n a s " t h e U n i v e r s i t y o f N a n c a g o , " a f u s i o n o f t h e n a m e o f t h e F r e n c h c i t y o f N a n c y w i t h t h a t o f C h i c a g o . B u t B o u r b a k i p u b l i s h e s o n l y i n F r e n c h , a n d w h e n i t s m e m b e r s m e e t , u s u a l l y a t a F r e n c h r e s o r t c i t y , t h e c o n v e r s a t i o n s a r e n o t o n l y i n F r e n c h , t h e y a r e i n t h e d i a l e c t o f t h e P a r i s i a n s t u d e n t s . T h e c h a u v i n i s m c a r r i e s i n t o t h e s e p a r a t e l i v e s o f t h e s e F r e n c h m a t h e -m a t i c i a n s l i v i n g i n A m e r i c a . A n d r e W e i l , a f o u n d i n g B o u r -b a k i t e , p u b l i s h e d m a n y i m p o r t a n t p a p e r s i n E n g l i s h . B u t h i s Collected Works, w h i c h h a d s o m e b e a r i n g o n t h e c o n j e c t u r e r e l a t e d t o t h e F e r m a t p r o b l e m , w a s p u b l i s h e d i n F r e n c h a n d t i t l e d Oeuvres'. W e i l ' s u n u s u a l a c t i o n s w o u l d h u r t o n e o f t h e p r i n c i p a l p l a y e r s i n o u r d r a m a , a n d W e i l w o u l d n o t r e c o v e r f r o m t h i s d e b a c l e .
T h e m e m b e r s o f B o u r b a k i m u s t b e g i v e n c r e d i t f o r t h e i r c o l l e c t i v e s e n s e o f h u m o r . S o m e f o r t y y e a r s a g o , t h e A m e r i c a n M a t h e m a t i c a l S o c i e t y
r e c e i v e d a n a p p l i c a t i o n f o r i n d i v i d u a l m e m b e r s h i p f r o m M r . N i c o l a s B o u r b a k i . T h e s e c r e t a r y o f t h e S o c i e t y w a s i m p e r t u r b a b l e . H e r e p l i e d t h a t i f M r . B o u r b a k i w a n t e d t o j o i n t h e S o c i e t y h e w o u l d h a v e t o a p p l y a s a n i n s t i -t u t i o n a l m e m b e r ( w h i c h w a s m u c h m o r e e x p e n s i v e ) . B o u r b a k i d i d n o t w r i t e b a c k .
F E R M A T ' S L A S T T H E O R E M Elliptic
Curves
D i o p h a n t i n e p r o b l e m s — t h a t i s , p r o b l e m s r a i s e d b y e q u a t i o n s o f t h e f o r m g i v e n b y D i o p h a n t u s i n t h e t h i r d c e n t u r y — b e g a n t o b e s t u d i e d m o r e a n d m o r e i n t h e t w e n t i e t h c e n t u r y u s i n g m a t h e m a t i c a l e n t i t i e s c a l l e d e l l i p t i c c u r v e s . B u t e l l i p t i c c u r v e s h a v e n o t h i n g t o d o w i t h e l l i p s e s . T h e y w e r e o r i g i n a l l y u s e d c e n t u r i e s e a r l i e r i n c o n n e c t i o n w i t h e l l i p t i c f u n c t i o n s , w h i c h i n t u r n w e r e d e v i s e d t o h e l p c a l c u l a t e t h e p e r i m e t e r o f a n e l l i p s e . A s w i t h m a n y i n n o v a t i v e i d e a s i n m a t h e m a t i c s , t h e p i o n e e r i n t h i s f i e l d w a s n o n e o t h e r t h a n G a u s s .
O d d l y , e l l i p t i c c u r v e s a r e n e i t h e r e l l i p s e s n o r e l l i p t i c f u n c t i o n s — t h e y a r e c u b i c p o l y n o m i a l s i n t w o v a r i a b l e s . T h e y l o o k l i k e : y 2
= a x 3 + b x 2 + c x w h e r e a, b, a n d c a r e i n t e g e r s o r r a t i o n a l n u m b e r s ( w e a r e c o n c e r n e d w i t h e l l i p t i c c u r v e s o v e r t h e r a t i o n a l n u m b e r s ) . E x a m p l e s o f s u c h e l l i p t i c c u r v e s a r e s h o w n b e l o w . 1 3
W h e n o n e l o o k s a t t h e r a t i o n a l p o i n t s o n t h e e l l i p t i c c u r v e s — t h a t i s , o n e l o o k s o n l y a t p o i n t s o n t h e c u r v e t h a t a r e r a t i o s o f t w o i n t e g e r s ( n o i r r a t i o n a l n u m b e r s s u c h a s p i o r t h e s q u a r e r o o t o f t w o , e t c . ) , t h e s e n u m b e r s f o r m a g r o u p . T h a t m e a n s t h a t t h e y h a v e n i c e p r o p e r t i e s . T a k e a n y t w o s o l u t i o n s , - t h e y c a n b e " a d d e d " i n a s e n s e t o p r o d u c e a third s o l u t i o n o n t h e c u r v e . N u m b e r t h e o r i s t s h a v e b e c o m e f a s c i n a t e d w i t h t h e e l l i p t i c c u r v e s s i n c e
t h e y c a n a n s w e r m a n y q u e s t i o n s a b o u t e q u a t i o n s a n d t h e i r s o l u t i o n s . E l l i p t i c c u r v e s t h u s b e c a m e o n e o f t h e f o r e m o s t r e s e a r c h t o o l s i n n u m b e r t h e o r y . 1 4
A Strange Conjecture Is about to be Made
I t w a s k n o w n f o r s o m e t i m e b y n u m b e r t h e o r y e x p e r t s t h a t some o f t h e e l l i p t i c c u r v e s t h e y w e r e s t u d y i n g w e r e m o d u l a r .
T h a t i s , t h e s e f e w e l l i p t i c c u r v e s c o u l d b e v i e w e d a s c o n n e c t e d w i t h m o d u l a r f o r m s . S o m e e l l i p t i c c u r v e s c o u l d b e s o m e h o w c o n n e c t e d w i t h t h e c o m p l e x p l a n e a n d t h e s e f u n c t i o n s i n h y p e r b o l i c s p a c e w i t h t h e i r m a n y s y m m e t r i e s . I t w a s n o t c l e a r w h y a n d h o w t h i s w a s h a p p e n i n g . T h e m a t h e m a t i c s w a s e x t r e m e l y c o m p l i c a t e d , e v e n f o r e x p e r t s , a n d t h e i n t e r n a l s t r u c t u r e — t h e b e a u t i f u l h a r m o n i e s t h a t e x i s t e d — w a s l i t t l e u n d e r s t o o d . T h e e l l i p t i c f u n c t i o n s t h a t w e r e i n d e e d m o d u l a r h a d i n t e r e s t i n g p r o p e r t i e s . S o o n s o m e o n e w o u l d m a k e t h e b o l d c o n j e c t u r e t h a t all e l l i p t i c f u n c t i o n s w e r e m o d u l a r .
T o u n d e r s t a n d t h e i d e a o f m o d u l a r i t y , w h i c h e x i s t s
w i t h i n
t h e n o n - E u c l i d e a n s p a c e o f t h e u p p e r c o m p l e x h a l f -p l a n e , w h e r e s y m m e t r i e s a r e s o c o m p l i c a t e d t h a t t h e y c a n h a r d l y b e v i s u a l i z e d , i t i s u s e f u l t o l o o k a t a v e r y s i m p l e e x a m p l e . T h i s i s a n e x a m p l e w h e r e t h e c u r v e o f i n t e r e s t i s n o t a n e l l i p t i c c u r v e , - i n s t e a d o f a c u b i c e q u a t i o n i n t w o v a r i a b l e s , i t i s o n l y a s q u a r e d e q u a t i o n i n t w o v a r i a b l e s : t h e c u r v e i s a s i m p l e c i r c l e . T h e e q u a t i o n o f a c i r c l e w i t h r a d i u s a2
w h o s e c e n t e r i s z e r o i s g i v e n b y : x 2 + y2 =a2. N o w l o o k a t t h e s i m p l e p e r i o d i c f u n c t i o n s : x =a c o s ( , a n d y = a s i n t . T h e s e t w o f u n c t i o n s c a n s t a n d f o r x a n d y i n t h e e q u a t i o n o f t h e c i r c l e . T h e e q u a t i o n o f t h e c i r c l e i s m o d u l a r i n t h i s s e n s e . T h e r e a s o n i s t h e t r i g o n o m e t r i c i d e n t i t y t h a t s a y s t h a t s i n 2 t + c o s 2 t = 1, a n d s u b s t i t u t i n g t h i s f o r -m u l a i n t o t h e e q u a t i o n o f t h e c i r c l e ( e a c h t e r m m u l t i p l i e d b y a) g i v e s a n i d e n t i t y .
A m o d u l a r e l l i p t i c c u r v e i s j u s t a n e x t e n s i o n o f t h i s i d e a t o t h e m o r e c o m p l i c a t e d c o m p l e x p l a n e , w i t h a s p e c i a l n o n - E u c l i d e a n g e o m e t r y . H e r e t h e p e r i o d i c f u n c t i o n s a r e s y m m e t r i e s n o t o n l y w i t h r e s p e c t t o o n e v a r i a b l e , t , a s w i t h t h e s i n e s a n d c o s i n e s o n t h e l i n e— t h e y a r e t h e a u t o m o r p h i c , o r t h e m o d u l a r f o r m s o n t h e c o m p l e x p l a n e , w h i c h h a v e s y m m e t r i e s w i t h r e s p e c t t o m o r e c o m p l i c a t e d
t r a n s f o r m a t i o n s ( f ( z ) —>j(az+b/cz+d)).
Tokyo, Japan, Early 1950s
I n t h e e a r l y 1950s, J a p a n w a s a n a t i o n e m e r g i n g f r o m t h e d e v a s t a t i o n o f w a r . P e o p l e w e r e n o l o n g e r h u n g r y , b u t t h e y w e r e s t i l l p o o r a n d e v e r y d a y s u r v i v a l w a s a s t r u g g l e f o r t h e a v e r a g e J a p a n e s e . Y e t f a c t o r i e s w e r e b e i n g r e b u i l t f r o m t h e r u b b l e , b u s i n e s s e s r e e s t a b l i s h e d , a n d t h e g e n e r a l m o o d w a s h o p e f u l . U n i v e r s i t y l i f e i n J a p a n a t t h a t t i m e w a s a l s o d i f f i c u l t . C o m
p e t i t i o n a m o n g s t u d e n t s w a s f i e r c e : g o o d g r a d e s m e a n t g o o d j o b s a f t e r g r a d u a t i o n . T h i s w a s e s p e c i a l l y t r u e f o r d o c t o r a l s t u d e n t s i n p u r e m a t h e m a t i c s , s i n c e p o s i t i o n s a t u n i v e r s i t i e s w e r e s c a r c e e v e n t h o u g h t h e p a y w a s l o w . Y u t a k a T a n i y a m a w a s o n e s u c h d o c t o r a l s t u d e n t i n m a t h e m a t i c s . H e w a s b o r n o n N o v e m b e r 12, 1927, t h e y o u n g e s t o f e i g h t c h i l d r e n i n t h e f a m i l y o f a c o u n t r y d o c t o r i n t h e t o w n o f K i s a i , a b o u t 3 0 m i l e s n o r t h o f T o k y o . A t a y o u n g a g e , T a n i y a m a b e g a n t o s t u d y t h e a r e a o f m a t h e m a t i c s i n v o l v i n g c o m p l e x m u l t i p l i c a t i o n o f a b e l i a n v a r i e t i e s . N o t m u c h w a s k n o w n a b o u t t h i s f i e l d a n d T a n i y a m a h a d a v e r y d i f f i c u l t t i m e . T o m a k e t h i n g s w o r s e , h e f o u n d t h e a d v i c e o f o l d e r p r o f e s s o r s a t t h e U n i v e r s i t y o f T o k y o v i r t u a l l y u s e l e s s . H e h a d t o d e r i v e e v e r y d e t a i l o n h i s o w n a n d h e u s e d t o d e s c r i b e e v e r y t a s k i n h i s m a t h e m a t i c a l r e s e a r c h u s i n g f o u r C h i n e s e c h a r a c t e r s t h a t m e a n t " h a r d f i g h t i n g " a n d " b i t t e r s t r u g g l e . " N o t h i n g w a s e a s y i n y o u n g Y u t a k a T a n i y a m a ' s l i f e .
T a n i y a m a l i v e d i n a o n e -r o o m a p a r t m e n t o f 81 s q u a r e f e e t . T h e r e w a s o n l y o n e t o i l e t o n e v e r y f l o o r o f t h e b u i l d i n g , s h a r e d b y a l l t h e r e s i d e n t s o f t h e f l o o r . T o t a k e a b a t h , T a n i y a m a h a d t o g o t o a p u b l i c b a t h h o u s e
s o m e d i s t a n c e a w a y f r o m h i s b u i l d i n g . T h e s h a b b y a p a r t m e n t b u i l d i n g w a s n a m e d " V i l l a T r a n q u i l M o u n t a i n s , " i r o n i c a l l y s o s i n c e i t s t o o d o n a b u s y s t r e e t a n d b y a r a i l r o a d t r a c k o n w h i c h t r a i n s t h u n d e r e d b y e v e r y f e w m i n u t e s . P o s s i b l y s o h e c o u l d c o n c e n t r a t e b e t t e r o n h i s r e s e a r c h , y o u n g Y u t a k a w o r k e d m o s t l y a t n i g h t , o f t e n g o i n g t o b e d a t 6 AM
w h e n t h e n o i s y d a y b e g a n . E x c e p t i n t h e h e a t o f s u m m e r , a l m o s t e v e r y d a y T a n i y a m a w o r e t h e s a m e b l u e - g r e e n s u i t w i t h a m e t a l l i c s h e e n . A s h e e x p l a i n e d t o h i s g o o d f r i e n d G o r o S h i m u r a , h i s
f a t h e r b o u g h t t h e m a t e r i a l v e r y c h e a p l y f r o m a p e d d l e r . B u t b e c a u s e o f t h e l u s t e r , n o o n e i n t h e f a m i l y d a r e d w e a r i t . Y u t a k a , w h o d i d n ' t c a r e h o w h e l o o k e d , f i n a l l y v o l u n t e e r e d a n d h a d t h e m a t e r i a l t a i l o r e d f o r a s u i t w h i c h b e c a m e h i s d a i l y o u t f i t .
T a n i y a m a g r a d u a t e d f r o m t h e U n i v e r s i t y o f T o k y o i n 1953 a n d t o o k o n t h e p o s i t i o n o f " s p e c i a l r e s e a r c h s t u d e n t " a t t h e d e p a r t m e n t o f m a t h e m a t i c s t h e r e . H i s f r i e n d S h i m u r a g r a d u a t e d a y e a r e a r l i e r a n d h a d a s i m i l a r p o s i t i o n i n m a t h e m a t i c s a t t h e C o l l e g e o f G e n e r a l E d u c a t i o n a c r o s s c a m p u s . T h e i r f r i e n d s h i p b e g a n a f t e r o n e o f t h e m w r o t e t h e o t h e r a l e t t e r a s k i n g h i m t o r e t u r n t o t h e l i b r a r y a n i s s u e o f a m a t h e m a t i c a l j o u r n a l w h i c h i n t e r e s t e d t h e m b o t h . T h e y w o u l d o f t e n e a t t o g e t h e r a t i n e x p e n s i v e r e s t a u r a n t s s u p p o s e d l y s e r v i n g W e s t e r n -s t y l e f o o d , s u c h a s t o n g u e s t e w , w h i c h w a s b e c o m i n g p o p u l a r i n J a p a n . 1 5
F e w g o o d m a t h e m a t i c i a n s s t a y e d i n J a p a n i n t h o s e d a y s . O n c e a m a t h e m a t i c i a n a c h i e v e d s o m e r e n o w n , h e o r s h e w o u l d t r y t o m o v e t o a u n i v e r s i t y i n A m e r i c a o r E u r o p e , w h e r e t h e m a t h e m a t i c a l c o m m u n i t y w a s m o r e e s t a b l i s h e d a n d w h e r e c o n n e c t i o n s w i t h p e o p l e d o i n g r e s e a r c h i n t h e s a m e f i e l d s w e r e p o s s i b l e . S u c h l i n k s w e r e i m p o r t a n t f o r
c o n d u c t i n g r e s e a r c h i n e s o t e r i c a r e a s a b o u t w h i c h n o t m u c h w a s k n o w n . T o t r y t o f o s t e r r e s e a r c h t i e s w i t h p e o p l e w i t h k n o w l e d g e i n t h e i r f i e l d o f i n t e r e s t , t h e t w o f r i e n d s h e l p e d o r g a n i z e t h e T o k y o - N i k k o S y m p o s i u m o n A l g e b r a i c N u m b e r T h e o r y i n S e p t e m b e r , 1955. S o m e s t a t e m e n t s m a d e a t t h i s s m a l l c o n f e r e n c e , w h i l e d e s t i n e d t o r e m a i n o b s c u r e f o r a l o n g t i m e , w o u l d e v e n t u a l l y l e a d t o m o m e n t o u s r e s u l t s — a n d a s a v a g e c o n t r o v e r s y — a l m o s t f o r t y y e a r s l a t e r .
Goro Shimura, circa 1965, when he first
developed his conjecture
A Hopeful Beginning
T h e t w o f r i e n d s h e l p e d f i l e t h e n e c e s s a r y f o r m s w i t h t h e a d m i n i s t r a t i o n , a r r a n g e d f o r t h e c o n f e r e n c e f a c i l i t i e s , a n d h e l p e d s e n d o u t i n v i t a t i o n s t o l o c a l a n d f o r e i g n m a t h e m a t i c i a n s t h e y h o p e d w o u l d a t t e n d t h e c o n f e r e n c e . A n d r e W e i l , w h o h a d l e f t F r a n c e i n t h e m e a n t i m e a n d w a s a p r o f e s s o r a t t h e U n i v e r s i t y o f C h i c a g o , w a s o n e o f t h o s e i n v i t e d t o a t t e n d . A t t h e I n t e r n a -t i o n a l C o n g r e s s o f M a t h e m a t i c i a n s f i v e y e a r s e a r l i e r , W e i l h a d b r o u g h t t o
t h e a t t e n t i o n o f t h e m a t h e m a t i c a l c o m m u n i t y a n
u n k n o w n c o n j e c t u r e b y a m a t h e m a t i c i a n b y t h e n a m e o f H a s s e a b o u t t h e " z e t a f u n c t i o n o f a v a r i e t y o v e r a n u m b e r f i e l d . " T h e o b s c u r e s t a t e m e n t h e l d s o m e i n t e r e s t t o r e s e a r c h e r s i n n u m b e r t h e o r y . A p p a r e n t l y , W e i l w a s c o l l e c t i n g t h e s e c o n j e c t u r a l i d e a s i n t h e t h e o r y o f n u m b e r s a n d i n c l u d e d t h i s o n e i n h i s Collected Papers, g i v i n g c r e d i t t o H a s s e .
H i s i n t e r e s t i n r e s u l t s i n t h i s a r e a m a d e h i m a t t r a c t i v e t o T a n i y a m a a n d S h i m u r a , a n d t h e y w e r e p l e a s e d w h e n h e a c c e p t e d t h e i n v i t a t i o n t o a t t e n d t h e i r c o n f e r e n c e . A n o t h e r f o r e i g n m a t h e m a t i c i a n t o c o m e t o T o k y o - N i k k o w a s a y o u n g e r F r e n c h m a t h e m a t i c i a n , J e a n - P i e r r e S e r r e . W h i l e h e m a y n o t h a v e b e e n a m e m b e r o f B o u r b a k i a t t h a t t i m e , s i n c e t h e s o c i e t y i n c l u d e d o n l y w e l l -k n o w n m a t h e m a t i c i a n s , h e w o u l d b e c o m e o n e w i t h i n t h e f o l l o w i n g d e c a d e s . S e r r e h a s b e e n d e s c r i b e d b y s o m e m a t h e m a t i c i a n s a s a m b i t i o u s a n d f i e r c e l y c o m p e t i t i v e . H e c a m e t o T o k y o - N i k k o t o l e a r n a s m u c h a s h e c o u l d . T h e J a p a n e s e k n e w s o m e t h i n g s a b o u t n u m b e r t h e o r y , a n d t h e y h a d m a n y r e s u l t s p u b l i s h e d o n l y i n J a p a n e s e a n d t h u s h i d d e n f r o m t h e r e s t o f t h e w o r l d . T h i s w a s a g r e a t o p p o r t u n i t y t o l e a r n o f t h e s e r e s u l t s , s i n c e t h e c o n f e r e n c e w a s t o b e
c o n d u c t e d i n E n g l i s h . H e w o u l d b e o n e o f f e w p e o p l e o u t s i d e J a p a n w i t h k n o w l -e d g e o f t h e m a t h e m a t i c s p r e s e n t e d t h e r e . W h e n t h e p r o c e e d i n g s o f t h e s y m p o s i u m w o u l d b e p u b l i s h e d , i t w o u l d b e i n J a p a n e s e o n l y . T w e n t y y e a r s l a t e r , S e r r e w o u l d d r a w a t t e n t i o n t o s o m e e v e n t s a t T o k y o - N i k k o , a n d t h e w o r l d w o u l d h e a r h i s v e r s i o n — n o t t h e o n e r e c o r d e d i n t h e J a p a n e s e p r o c e e d i n g s .
T h e p r o c e e d i n g s i n c l u d e d t h i r t y - s i x p r o b l e m s . P r o b l e m s 10, 11, 12, a n d 13 w e r e w r i t t e n b y Y u t a k a T a n i y a m a . S i m i l a r t o t h e i d e a s o f H a s s e , T a n i y a m a ' s p r o b l e m s c o n s t i t u t e d a c o n j e c t u r e
a b o u t z e t a f u n c t i o n s . H e s e e m e d t o c o n n e c t P o i n c a r e ' s a u t o m o r p h i c f u n c t i o n s o f t h e c o m p l e x p l a n e w i t h t h e z e t a f u n c t i o n o f a n e l l i p t i c c u r v e . I t w a s m y s t e r i o u s t h a t a n e l l i p t i c c u r v e s h o u l d s o m e h o w b e c o n n e c t e d w i t h s o m e t h i n g i n t h e c o m p l e x p l a n e .
"You Are Saying What...?"
T h e c o n j e c t u r e e m b o d i e d i n t h e f o u r p r o b l e m s w a s n e b u l o u s . T a n i y a m a d i d n o t f o r m u l a t e t h e p r o b l e m s i n a v e r y m e a n i n g f u l w a y , p o s s i b l y b e c a u s e h e w a s n ' t q u i t e s u r e w h a t t h e c o n n e c -t i o n w a s . B u t t h e b a s i c i d e a w a s t h e r e . I t w a s a n i n t u i t i o n , a g u t f e e l i n g t h a t t h e a u t o m o r p h i c f u n c t i o n s w i t h t h e i r m a n y s y m m e t r i e s o n t h e c o m p l e x p l a n e w e r e s o m e h o w c o n n e c t e d w i t h t h e e q u a t i o n s o f D i o p h a n t u s . I t c e r t a i n l y w a s n ' t o b v i o u s . H e w a s p o s i t i n g a h i d d e n c o n n e c t i o n b e t w e e n t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s .
A n d r e W e i l w a n t e d t o k n o w e x a c t l y w h a t T a n i y a m a h a d i n m i n d . A c c o r d i n g t o t h e w r i t t e n r e c o r d o f t h e c o n f e r e n c e , t h e p r o c e e d i n g s p u b l i s h e d i n J a p a n e s e , t h e f o l l o w i n g e x c h a n g e t o o k p l a c e b e t w e e n W e i l a n d T a n i y a m a : 1 6
Weil: Do you think all elliptic functions are
uniformized by modular functions?
Taniyama: Modular functions alone will not be
enough. I think other special types of automorphic
functions are necessary. Weil: Of course some of
them can probably be handled that way. But in the
general case, they look completely different and
mysterious...
T w o t h i n g s a r e e v i d e n t f r o m t h e c o n v e r s a t i o n . F i r s t , T a n i y a m a
w a s r e f e r r i n g t o " a u t o m o r p h i c f u n c t i o n s " r a t h e r t h a n " m o d u l a r f u n c t i o n s a l o n e " a s b e i n g a s s o c i a t e d w i t h t h e e l l i p t i c c u r v e s . A n d s e c o n d , W e i l d i d n o t b e l i e v e t h a t i n g e n e r a l t h e r e w a s s u c h a c o n n e c t i o n . L a t e r h e w o u l d b e m o r e s p e c i f i c a b o u t t h i s d i s b e l i e f , a l l o f w h i c h w o u l d m a k e i t a s t o u n d i n g t h a t his n a m e , o f a l l p e o p l e ' s , s h o u l d e n d u p b e i n g a s s o c i a t e d w i t h a c o n j e c t u r e h e n e i t h e r f o r m u l a t e d n o r e v e n b e l i e v e d w a s t r u e . B u t f a t e s o m e t i m e s t a k e s s t r a n g e , i m p l a u s i b l e t u r n s , a n d e v e n m o r e b i z a r r e o c c u r r e n c e s w e r e g o i n g t o t r a n s p i r e .
A l l t h i s w o u l d m a t t e r d e c a d e s l a t e r . T o k n o w e x a c t l y w h a t Y u t a k a T a n i y a m a m e a n t , t h o u g h t , a n d s a i d w o u l d b e a b o o n t o m o d e r n h i s t o r i a n s . B u t , u n f o r t u n a t e l y , t r a g e d y s t a l k e d T a n i y a m a , a s i t d i d s o m a n y o t h e r y o u n g m a t h e m a t i c a l g e n i u s e s .
W i t h i n a c o u p l e o f y e a r s , G o r o S h i m u r a l e f t T o k y o , f i r s t f o r P a r i s , t h e n f o r t h e I n s t i t u t e f o r A d v a n c e d S t u d y a n d P r i n c e t o n U n i v e r s i t y . T h e t w o f r i e n d s c o n t i n u e d t o c o m m u n i c a t e b y m a i l . I n S e p t e m b e r , 1958, G o r o S h i m u r a r e c e i v e d t h e l a s t l e t t e r w r i t t e n b y Y u t a k a T a n i y a m a . I n t h e m o r n i n g o f N o v e m b e r 17, 1958, f i v e d a y s a f t e r h i s t h i r t y - f i r s t b i r t h d a y , Y u t a k a T a n i y a m a w a s f o u n d d e a d i n
h i s a p a r t m e n t , a s u i c i d e n o t e o n h i s d e s k .
Shimura's Conjecture
A d e c a d e p a s s e d s i n c e t h e T o k y o - N i k k o c o n f e r e n c e a n d G o r o S h i m u r a , n o w a t P r i n c e t o n , c o n t i n u e d h i s r e s e a r c h o n n u m b e r t h e o r y , z e t a f u n c t i o n s , a n d e l l i p t i c c u r v e s . H e u n d e r s t o o d w h e r e h i s l a t e f r i e n d h a d b e e n w r o n g , a n d h i s o w n r e s e a r c h a n d q u e s t f o r h i d d e n h a r m o n i e s i n t h e r e a l m s o f m a t h e m a t i c s l e d h i m t o f o r m u l a t e a d i f f e r e n t , b o l d e r a n d m o r e p r e c i s e c o n j e c t u r e . H i s c o n j e c t u r e w a s t h a t e v e r y e l l i p t i c c u r v e o v e r t h e
r a t i o n a l n u m b e r s i s u n i f o r m i z e d b y a m o d u l a r f o r m . M o d u l a r f o r m s a r e m o r e s p e c i f i c e l e m e n t s o v e r t h e c o m p l e x p l a n e t h a n a r e t h e a u t o m o r p h i c f u n c t i o n s o f T a n i y a m a . A n d s p e c i f y i n g t h e d o m a i n a s t h e r a t i o n a l n u m b e r s , a n d o t h e r m o d i f i c a t i o n s , w e r e i m p o r t a n t c o r r e c t i o n s a s w e l l .S h i m u r a ' s c o n j e c t u r e c a n b e
e x p l a i n e d u s i n g a p i c t u r e :
I f w e " f o l d " t h e c o m p l e x p l a n e a s a t o r u s ( t h e d o u g h n u t i n t h e p i c t u r e ^ , t h e n t h i s s u r f a c e w i l l h o l d a l l s o l u t i o n s t o e l l i p t i c e q u a t i o n s o v e r t h e r a t i o n a l n u m b e r s , t h e s e i n t u r n a r i s i n g f r o m t h e e q u a t i o n s o f D i o p h a n t u s . W h a t w o u l d l a t e r b e i m p o r t a n t t o t h e s o l u t i o n o f F e r m a t ' s L a s t T h e o r e m i s t h a t i f a s o l u t i o n t o F e r m a t ' s e q u a t i o n x n + y n
= z n e x i s t e d , t h i s s o l u t i o n w o u l d a l s o h a v e t o l i e o n t h a t t o r u s . N o w , S h i m u r a c o n j e c t u r e d t h a t e v e r y e l l i p t i c c u r v e w i t h r a t i o n a l c o e f f i c i e n t s ( t h a t i s , a n e q u a t i o n w i t h c o e f f i c i e n t s o f
t h e f o r m alb w h e r e b o t h a a n d b
a r e i n t e g e r s ) h a s a " m a t e " o n t h e c o m p l e x h a l f - p l a n e o f P o i n c a r e , w i t h i t s n o n -E u c l i d e a n , h y p e r b o l i c g e o m e t r y . T h e p a r t i c u l a r m a t e o f e a c h r a t i o n a l e l l i p t i c c u r v e w a s a v e r y s p e c i f i c f u n c t i o n o n t h e
c o m p l e x h a l f - p l a n e , w h i c h w a s i n v a r i a n t u n d e r c o m p l i c a t e d t r a n s f o r m a t i o n s o f t h e p l a n e — t h e o n e s m e n t i o n e d e a r l i e r :
/ ( z ) >f{az+b/cz+d), t h e c o e f f i c i e n t s f o r m i n g a g r o u p
w i t h m a n y u n e x p e c t e d s y m m e t r i e s . A l l o f t h i s w a s v e r y c o m p l e x , v e r y t e c h n i c a l , a n d — a s m o s t m a t h e m a t i c i a n s w o u l d b e l i e v e f o r s e v e r a l d e c a d e s —i m p o s s i b l e t o p r o v e i n t h e f o r e s e e a b l e f u t u r e .W h a t S h i m u r a ' s c o n j e c t u r e
w a s s a y i n g w a s t h a t e v e r y e l l i p t i c c u r v e w a s t h e p a r t o f a n i c e b e r g l y i n g a b o v e t h e w a t e r l i n e . B e l o w t h e s u r f a c e l a y a w h o l e i n t r i c a t e s t r u c t u r e . T o p r o v e t h e c o n j e c t u r e , o n e w o u l d h a v e t o s h o w t h a t every i c e b e r g h a d a n u n d e r w a t e r p a r t . S o m e s p e c i a l g r o u p s o f i c e b e r g s w e r e k n o w n t o h a v e t h e u n d e r w a t e r p a r t , b u t s i n c e t h e r e w e r e i n f i n i t e l y m a n y i c e b e r g s , o n e c o u l d n ' t j u s t g o l o o k u n d e r e a c h o n e o f t h e m . A g e n e r a l p r o o f w a s n e c e s s a r y t o s h o w t h a t a n i c e b e r g c o u l d n ' t e x i s t w i t h o u t p a r t o f i t b e i n g u n d e r w a t e r . T h e f o r m u l a t i o n o f s u c h a p r o o f w a s c o n s i d e r e d e x c e e d i n g l y d i f f i c u l t .
Intrigue and a Betrayal
A t a p a r t y a t t h e I n s t i t u t e f o r A d v a n c e d S t u d y a t P r i n c e t o n i n t h e e a r l y 1960s,
S h i m u r a a g a i n m e t J e a n -P i e r r e S e r r e . A c c o r d i n g t o
S h i m u r a , S e r r e a p p r o a c h e d h i m r a t h e r a r r o g a n t l y . " I d o n ' t t h i n k t h a t y o u r r e s u l t s o n m o d u l a r c u r v e s a r e a n y g o o d , " h e s a i d . " W h y , t h e y d o n ' t e v e n a p p l y t o a n a r b i t r a r y e l l i p t i c c u r v e . " I n r e s p o n s e , S h i m u r a s t a t e d h i s c o n j e c t u r e e x a c t l y : " S u c h a c u r v e s h o u l d always b e u n i f o r m i z e d b y a m o d u l a r c u r v e . " 1 7 S e r r e w e n t t o W e i l , w h o w a s n o t a t t h e p a r t y b u t w a s a m e m b e r o f t h e I n s t i -t u t e a n d t h e r e f o r e i n t h e i m m e d i a t e a r e a , a n d t o l d h i m o f h i s c o n v e r s a t i o n w i t h S h i m u r a . I n r e s p o n s e , A n d r e W e i l c a m e t o
Shimura. "Did you really say that?" he asked him, puzzled. "Yes,"
answered Shimura, "don't you think it plausible?" Ten years after
Taniyama's related conjecture, Andre Weil still did not believe
either conjecture. He answered: "I don't see any reason against it,
since one and the other of these sets are denumerable, but I don't
see any reason either for this hypothesis." What Weil said on this
occasion would later be described as "stupid," and "inane" by Serge
Lang of Yale University, who would circulate these comments
together with copies of two dozen letters he named collectively
"The Taniyama-Shimura File," among about fifty mathematicians
worldwide. What Weil meant in his response to Shimura was
tantamount to the following: If in a room you have seven men and
seven women and you conjecture that these are seven married
couples, then 1 see no reason against it, since the number of men
is the same as the number of women. But I don't see any reason for
your hypothesis, either. It could be that they are all single. What
made Lang describe the statement as stupid was that the counting
argument didn't really apply in any simple way here, because
"denumerable" means infinite and countable (such as the number
of all the positive integers: 1, 2, 3, 4,...) and matching such
infinite collections is no simple task. At any rate, it is clear that
Andre Weil did not believe Shimura's theory was necessarily true.
He would later admit that the conversation took place and, stupid,
inane, or otherwise, would quote it. But this would happen only in
1979, when he would write:18
Quelques annees plus tard, a Princeton, Shimura me demanda si
je trouvais plausible que toute courbe elliptique sur Q hit
contenue dans le jacobienne d'une courbe definie par une sous-
groupe de
congruence du groupe modulaire, je lui repondis, il
me semble, que je n'y voyais pas d'empechement,
puisque l'un et l'autre ensemble est denombrable,
mais je ne voyais rien non plus qui parlat en faveur
de cette hypothese.
[ " S o m e y e a r s l a t e r , a t P r i n c e t o n , S h i m u r a a s k e d m e i f I f o u n d i t p l a u s i b l e t h a t e v e r y e l l i p t i c c u r v e o v e r Q w a s c o n t a i n e d i n t h e j a c o b i a n o f a c u r v e d e f i n e d b y a c o n g r u e n c e s u b g r o u p o f a m o d u l a r g r o u p , - 1 r e s p o n d e d t o h i m , i t s e e m s t o m e , t h a t I d o n ' t s e e a n y t h i n g a g a i n s t i t , s i n c e o n e s e t a n d t h e o t h e r a r e d e n u m e r a b l e , b u t n e i t h e r d o 1 s e e a n y t h i n g t h a t s p e a k s i n f a v o r o f t h i s h y p o t h e s i s . " ]
B u t e v e n t h e n , W e i l w o u l d w r i t e " S h i m u r a a s k e d m e " ( m e demanda), r a t h e r t h a n " S h i m u r a t o l d m e , " w h e n r e f e r r i n g t o t h e s t a t e m e n t t h a t i s S h i m u r a ' s c o n j e c t u r e . W e i l p u b l i s h e d s o m e r e l a t e d p a p e r s , a n d w h i l e h e d i d n o t b e l i e v e S h i m u r a ' s t h e o r y , h i s o w n n a m e b e c a m e a s s o c i a t e d w i t h i t . T h e e r r o r w a s p e r p e t u a t e d w h e n m a t h e m a t i c i a n s m a d e r e f e r e n c e s i n t h e i r p a p e r s t o t h e w o r k s o f o t h e r s , a n d t h e m i s q u o t a t i o n i s p r e s e n t t o t h i s d a y w h e n w r i t e r s i g n o r a n t o f t h e h i s t o r y r e f e r t o t h e W e i l - T a n i y a m a c o n j e c t u r e i n s t e a d o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e . W e i l s e e m e d t o e n j o y h i s a s s o c i a t i o n w i t h a n
i m p o r t a n t t h e o r y w h i c h —w h i l e h e h i m s e l f d i d n o t b e l i e v e i n i t — m o s t m a t h e m a t i c i a n s t h o u g h t w o u l d b e p r o v e d s o m e d a y i n t h e d i s t a n t f u t u r e .
W i t h t h e p a s s i n g d e c a d e s , t h e r e w a s m o r e a n d m o r e r e a s o n f o r t h e c o n n e c t i o n t o e x i s t . I f a n d w h e n t h e c o n j e c t u r e w a s p r o v e d , i t w o u l d b e a s u b s t a n t i a l m a t h e m a t i c a l t h e o r y . W e i l w o r k e d a r o u n d t h e c o n j e c t u r e , n e v e r l e a v i n g m a t h e m a t i c a l
results he obtained too far away from the possible
connection between modular forms of the complex
plane and the elliptic curves of Diophantine
equations. And while he certainly knew better, he
held back references to Shimura and his crucial role
until almost two decades had passed. Then he gave
offhand praise to Shimura in casual conversation and
mentioned him— almost in passing—in a published
paper. Meanwhile, in France, Serre was actively
contributing to the false attribution. He made every
effort to associate the name of Andre Weil with the
conjecture, instead of that of Goro Shimura.
"An Exercise for the Interested Reader"
In 1967, Andre Weil wrote a paper in German, in
which he said:19
Ob sich diese Dinge immer, d.h. fur jede iiber Q
definierte Kurve C, so verhalten, scheint im
Moment noch problematisch zu sein und mag dem
interessierten Leser als Ubungsaufgabe empfohlen
werden. x
["Whether these things, that is for every curve C
defined over Q, so behave, at this moment is still
seen as problematic and will be recommended as an
exercise for the interested reader."] This paragraph
refers to elliptic curves over the rational numbers
(which mathematicians denote by Q), and "sich so
verhalten" here refers to being modular, that is, it
states Shimura's conjecture. But, here again, Weil
did not attribute the theory to its originator. He did
so only 12 years later, and even then as "Shimura
asked me..." as we have just seen. In this
p a p e r i n G e r m a n , a b o v e , W e i l c a l l s t h e c o n j e c t u r e " p r o b l e m a t i c . " A n d t h e n h e d o e s s o m e t h i n g s t r a n g e . H e s i m p l y a s s i g n s t h e c o n j e c t u r e a s a n exercise for the interested
reader ( " u n d m a g d e m i n t e r e s s i e r t e n L e s e r a l s U b u n g s a u f g a b e e m p f o h l e n w e r d e n " ) . T h i s e x e r c i s e f o r t h e " i n t e r e s t e d r e a d e r " w o u l d t a k e o n e o f t h e w o r l d ' s f i n e s t m a t h e m a t i c i a n s s e v e n y e a r s o f w o r k i n s o l i t u d e t o a t t e m p t t o p r o v e . W h e n a m a t h e m a t i c i a n a s s i g n s a h o m e w o r k p r o b l e m (Ubungsaufgabe),
u s u a l l y h e o r s h e k n o w s t h e p r o o f t h r o u g h a n d t h r o u g h , a n d b e l i e v e s — k n o w s f o r c e r -t a i n — t h a t t h e t h e o r e m i s t r u e , n o t " p r o b l e m a t i c " a s W e i l d e s c r i b e s i t .
T h e r e i s a n o l d s t o r y a b o u t a m a t h p r o f e s s o r w h o t e l l s h i s c l a s s " t h i s i s o b v i o u s , " w h e n r e f e r r i n g t o s o m e c o n c e p t . T h e c l a s s l o o k s c o n f u s e d s i n c e i t i s n o t a t a l l o b v i o u s , a n d f i n a l l y a b o l d s t u d e n t r a i s e s a h a n d a n d a s k s , " W h y ? " T h e p r o f e s s o r t h e n s t a r t s d r a w i n g a n d w r i t i n g o n t h e e d g e o f t h e b o a r d w i t h o n e h a n d , c o v e r i n g t h e w r i t i n g w i t h h i s o t h e r h a n d , a n d e r a s i n g e v e r y t h i n g a s h e i s d o n e . A f t e r a b o u t t e n m i n u t e s o f t h i s f u r t i v e s c r i b b l i n g , t h e p r o f e s s o r e r a s e s t h e b o a r d c o m p l e t e l y a n d a n n o u n c e s t o t h e b e f u d d l e d c l a s s : " Y e s , i t ' s
o b v i o u s . "
Tbe Lie
I n t h e 1970s, T a n i y a m a ' s p r o b l e m s f r o m t h e T o k y o -N i k k o m e e t i n g r e c e i v e d w i d e r d i s t r i b u t i o n . I n t h e m e a n t i m e , s i n c e W e i l h a d w r i t t e n a b o u t t h e c o n j e c t u r e h e d o u b t e d , m o d u l a r e l l i p t i c c u r v e s b e c a m e k n o w n a s " W e i l C u r v e s . " W h e n T a n i y a m a ' s p r o b l e m s b e c a m e b e t t e r k n o w n i n t h e W e s t , t h e c o n j e c t u r e a b o u t s u c h c u r v e s c a m e t o b e c a l l e d t h e " T a n i y a m a - W e i l c o n j e c t u r e . " S h i m u r a ' s n a m e w a s l e f t o u t c o m p l e t e l y . B u t
s i n c e T a n i y a m a ' s n a m e c a m e i n , W e i l s t a r t e d t o i n v e i g h a g a i n s t c o n j e c t u r e s a l t o g e t h e r . I n 1979, a m e r e f i v e y e a r s b e f o r e i t w a s p r o v e d b y G e r d F a l t i n g s , W e i l e v e n s p o k e a g a i n s t " t h e s o - c a l l e d ' M o r d e l l c o n j e c t u r e ' o n D i o p h a n t i n e e q u a t i o n s . " H e c o n t i n u e d , " I t w o u l d b e n i c e i f t h i s w e r e s o , a n d I w o u l d r a t h e r b e t f o r i t t h e n a g a i n s t i t . B u t i t i s n o m o r e t h a n w i s h f u l t h i n k i n g b e c a u s e t h e r e i s n o t a s h r e d o f e v i d e n c e f o r i t , a n d a l s o n o n e a g a i n s t i t . " B u t W e i l w a s w r o n g e v e n t h e n . A n u m b e r o f R u s s i a n m a t h e m a t i c i a n s , a m o n g t h e m S h a f a r e v i c h a n d P a r s h i n , w e r e a l r e a d y o b t a i n i n g r e s u l t s t h a t w o u l d s u p p l y e v i d e n c e f o r t h e M o r d e l l c o n j e c t u r e a s e a r l y a s t h e e a r l y 1970s. I n 1984, o f c o u r s e , G e r d F a l t i n g s w o u l d p r o v e t h e c o n j e c t u r e o u t r i g h t , m a k i n g F e r m a t ' s L a s t T h e o r e m " a l m o s t a l w a y s t r u e . "
W h i l e A n d r e W e i l w a s t u r n i n g a g a i n s t a l l c o n j e c t u r e s a t a t i m e w h e n h i s n a m e w a s n o l o n g e r b e i n g e x c l u s i v e l y a s s o c i a t e d w i t h t h e c o n j e c t u r e n o w c a l l e d T a n i y a m a - W e i l b y m a n y m a t h e m a t i c i a n s , S e r r e i n P a r i s w a s w o r k i n g t o k e e p S h i m u r a ' s n a m e d i s s o c i a t e d f r o m t h e c o n j e c t u r e . I n 1986,
a t a p a r t y a t t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y , a n d w i t h i n e a r s h o t o f a
n u m b e r o f p e o p l e , Jean- P i e r r e S e r r e t o l d S e r g e L a n g t h a t A n d r e W e i l h a d t o l d h i m o f a c o n v e r s a t i o n h e h a d h a d with
G o r o S h i m u r a . A c c o r d i n g t o S e r r e , this i s what W e i l t o l d h i m t o o k p l a c e :
Weil: Why did Taniyama think that all elliptic
curves are modular?
Shimura: You told him so yourself, and you have
forgotten.
A t this m o m e n t , L a n g , w h o u n k n o w i n g l y h a d b e e n u s i n g t h e t e r m s "Weil c u r v e " a n d " T a n i y a m a -Weil c o n j e c t u r e , " b e c a m e
s u s p i c i o u s . H e t o o k i t u p o n h i m s e l f t o f i n d t h e t r u t h . L a n g i m m e d i a t e l y w r o t e t o b o t h W e i l a n d S h i m u r a , t h e n t o S e r r e . S h i m u r a c a t e g o r i c a l l y d e n i e d t h a t s u c h a c o n v e r s a t i o n e v e r t o o k p l a c e , a n d g a v e a m p l e e v i d e n c e f o r t h i s c l a i m . W e i l d i d n o t r e p l y r i g h t a w a y . A n d S e r r e , i n h i s r e s p o n s e , c r i t i c i z e d L a n g ' s a t t e m p t t o f i n d t h e t r u t h . I n h i s B o u r b a k i S e m i n a r i n J u n e 1995,
S e r r e s t i l l r e f e r r e d t o t h e c o n j e c t u r e a s t h a t o f " T a n i y a m a - W e i l , " l e a v i n g o u t t h e n a m e o f i t s o r i g i n a t o r , w h o t r u s t e d h i m w i t h h i s c o n j e c t u r e 3 0 y e a r s e a r l i e r . W e i l r e s p o n d e d a f t e r a s e c -o n d a t t e m p t t o c o n t a c t h i m b y L a n g . H i s l e t t e r f o l l o w s . 2 0
3 December 1986.
Dear Lang,
I do not recall when and where your letter of
August 9 first reached me. When it did, I had (and
still have) far more serious matters to think about.
I cannot but resent strongly any suggestion that
I ever sought to diminish the credit due to
Taniyama and to Shimura. I am glad to see that
you admire them. So do 1.
Reports of conversations held long ago are open
to misunderstandings. You choose to regard them
as "history",- they are not. At best they are
anecdotes. Concerning the controversy which you
have found fit to raise, Shimura's letters seem to
me to put an end to it, once and for all.
As to attaching names to concepts, theorems, or
(?) conjectures, I have often said: (a) that, when a
proper name gets attached to (say) a concept, this
should never be taken as a sign that the author in
question had anything to do with the concept;
more often than not, the opposite is true.
Pythagoras had
nothing to do with "his" theorem, nor Fuchs with the Fonctions
fuchsiennes, any more than Auguste Comte with rue Auguste-
Comte,- (b) proper names tend, quite properly, to get replaced
by more appropriate ones,- the Leray-Koszul sequence is now a
spectral sequence (and as Siegel once told Erdos, abelian is now
written with a small a).
Why shouldn't I have made "stupid" remarks sometimes, as
you are pleased to say? But indeed, 1 was "out of it" in 1979
when expressing some skepticism about Mordell's conjecture,
since at that time 1 was totally ignorant of the work of the
Russians (Parshin, etc.) in that direction. My excuse, if it is one,
is that I had had long conversations with Shafarevich in 1972,
and he never mentioned any of that work.
Sincerely,
A. Weil
AW:ig
RS. Should you wish to run this letter through your Xerox
machine, do feel free to do so. I wonder what the Xerox Co.
would do without you and the like of you.
Deep in the Black Forest, Fall i 984
W h i l e t h e c o n t r o v e r s y a b o u t w h o o r i g i n a t e d t h e S h i m u r a -T a n i y a m a c o n j e c t u r e w a s r a g i n g i n B e r k e l e y , N e w H a v e n , P r i n c e t o n , a n d a c r o s s t h e A t l a n t i c i n P a r i s , s o m e t h i n g t o t a l l y u n e x p e c t e d w a s h a p p e n i n g d e e p i n t h e B l a c k F o r e s t o f s o u t h w e s t G e r m a n y .
G e r h a r d F r e y r e c e i v e d h i s D i p l o m a f r o m t h e U n i v e r s i t y o f T u b i n g e n , a n d h i s P h . D . f r o m t h e U n i v e r s i t y o f H e i d e l b e r g ,
w h e r e h e s t u d i e d n u m b e r t h e o r y a n d w a s i n f l u e n c e d b y t h e w o r k s o f H a s s e a n d W e i l . F r e y w a s f a s c i n a t e d b y t h e i n t e r p l a y b e t w e e n t h e t h e o r y o f n u m b e r s a n d a l g e b r a i c g e o m e t r y , a n a r e a o f m a t h e m a t i c s w h i c h w a s d e v e l o p e d i n t h e l a s t f i f t y y e a r s . H e w a s a l s o i n t e r e s t e d i n a r i t h m e t i c g e o m e t r y . I t w a s t h e c o n n e c t i o n s b e t w e e n n u m b e r t h e o r y a n d t h e n e w e r f i e l d s o f a l g e b r a i c a n d a r i t h m e t i c g e o m e t r y t h a t w o u l d l e a d h i m t o m a k e a n u n e x p e c t e d m a t h e m a t i c a l s t a t e m e n t . I n t h e 1970s, F r e y d i d a l o t o f w o r k o n e l l i p t i c c u r v e s a n d D i o p h a n t i n e e q u a t i o n s , a n d i n 1978 h e r e a d t h e p a p e r " M o d u l a r c u r v e s a n d t h e E i s e n s t e i n i d e a l , " b y B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . F r e y w a s s t r o n g l y i n f l u e n c e d b y t h e p a p e r , a s w e r e m a n y n u m b e r t h e o r i s t s , a m o n g t h e m B e r k e l e y ' s K e n n e t h R i b e t a n d P r i n c e t o n ' s A n d r e w W i l e s . F r e y b e c a m e c o n v i n c e d b y M a z u r ' s p a p e r t h a t h e s h o u l d t h i n k v e r y s e r i o u s l y a b o u t a p p l i c a t i o n s o f m o d u l a r c u r v e s a n d G a l o i s r e p r e s e n t a t i o n s t o t h e t h e o r y o f e l l i p t i c c u r v e s . H e f o u n d t h a t t h i s l e d h i m
a l m o s t u n a v o i d a b l y t o D i o p h a n t i n e q u e s t i o n s c l o s e l y r e l a t e d t o e q u a t i o n s o f F e r m a t ' s t y p e . T h i s w a s a p o w e r f u l i n s i g h t , w h i c h F r e y t r i e d t o m a k e m o r e p r e c i s e .
I n 1981, G e r h a r d F r e y s p e n t a f e w w e e k s a t H a r v a r d U n i -v e r s i t y a n d h a d a n u m b e r o f d i s c u s s i o n s w i t h B a r r y M a z u r . T h e s e d i s c u s s i o n s w e r e c l e a r i n g t h i n g s i n h i s m i n d . T h e h e a v y f o g s u r r o u n d i n g t h e d i f f i c u l t c o n n e c t i o n s h e e n v i s i o n e d b e t w e e n F e r m a t - l i k e e q u a t i o n s a n d t h e r e l a t i o n b e t w e e n m o d u l a r f o r m s a n d e l l i p t i c c u r v e s w a s s l o w l y l i f t i n g . F r e y w e n t o n t o B e r k e l e y , w h e r e h e s p o k e w i t h K e n R i b e t , a b r i g h t n u m b e r t h e o r i s t w h o w a s a g r a d u a t e o f H a r v a r d a n d h a d w o r k e d w i t h M a z u r o n r e l a t e d i s s u e s . F r e y r e t u r n e d t o h i s n a t i v e G e r m a n y .
no
T h r e e y e a r s l a t e r , h e w a s i n v i t e d t o g i v e a l e c t u r e a t t h e O b e r - w o l f a c h c e n t e r d e e p i n t h e B l a c k F o r e s t .
O b e r w o l f a c h w a s d e s i g n e d a s a c o n f e r e n c e a n d w o r k s h o p c e n t e r i n m a t h e m a t i c s , s e t i n b e a u t i f u l a n d p e a c e f u l s u r r o u n d i n g s f a r f r o m c i t i e s a n d c r o w d s . E v e r y y e a r , t h e c e n t e r s p o n s o r s a b o u t f i f t y i n t e r n a t i o n a l m e e t i n g s o n d i f f e r e n t t o p i c s o f m a t h e -m a t i c s . L e c t u r e s , a n d e v e n j u s t a t t e n d a n c e a t t h e m e e t i n g s , a r e e x c l u s i v e l y b y i n v i t a t i o n . E v e r y e f f o r t i s m a d e t o a l l o w f o r t h e e a s y e x c h a n g e o f i d e a s a m o n g e x p e r t s f r o m d i f f e r e n t c o u n -t r i e s . I n 1984, G e r h a r d F r e y g a v e a t a l k a t a n u m b e r t h e o r y c o n f e r e n c e t h e r e . H e m a d e w h a t l o o k e d l i k e a c r a z y a s s e r t i o n . T h e m i m e o g r a p h e d s h e e t s f i l l e d w i t h m a t h e m a t i c a l f o r m u l a s h e p a s s e d a r o u n d a t t h e c o n f e r e n c e s e e m e d t o i m p l y t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w e r e i n d e e d t r u e , F e r m a t ' s L a s t T h e o r e m w o u l d b e p r o v e d . T h i s m a d e n o s e n s e a t a l l . W h e n K e n R i b e t f i r s t h e a r d o f F r e y ' s s t a t e m e n t , h e t h o u g h t i t w a s a j o k e . W h a t c o u l d m o d u l a r i t y o f e l l i p t i c c u r v e s p o s s i b l y h a v e t o d o w i t h F e r m a t ' s L a s t T h e o r e m ? h e a s k e d h i m s e l f . H e g a v e t h i s s t r a n g e a s s e r t i o n n o f u r t h e r t h o u g h t a n d w e n t a b o u t h i s u s u a l w o r k . B u t a n u m b e r o f p e o p l e i n P a r i s a n d
e l s e w h e r e w e r e i n t e r e s t e d i n F r e y ' s u n p r o v e n , a n d s o m e w h a t i n c o m p l e t e s t a t e m e n t . J e a n - P i e r r e S e r r e w r o t e a p r i v a t e l e t t e r t o a m a t h e m a t i c i a n b y t h e n a m e o f J . - F . M e s t r e . T h i s l e t t e r b e c a m e p u b l i c , a n d S e r r e s u b s e q u e n t l y p u b l i s h e d a p a p e r r e p e a t i n g h i s o w n c o n j e c t u r e s f r o m t h e l e t t e r t o M e s t r e . 2 1
Ribet's Theorem
K e n R i b e t , w h o f i r s t t h o u g h t t h i s s t a t e m e n t w a s a j o k e , s t a r t e d t h i n k i n g a b o u t S e r r e ' s c o n j e c t u r e s , a n d i n f a c t r e c o g n i z e d i n
t h e m s o m e t h i n g h e h a d a l r e a d y f o r m u l a t e d f o r h i m s e l f w h e n h e f o u n d t i m e t o t h i n k a b o u t F r e y ' s " j o k e . " T h e s e w e r e c e r t a i n c l a r i f i c a t i o n s o f G e r h a r d F r e y ' s s t a t e m e n t s , w h i c h , i f p r o v e n , w o u l d e s t a b l i s h t h e f o l l o w i n g i m p l i c a t i o n :
Shimura-Taniyama conjecture > Fermat's Last Theorem
T h e w a y t h e F r e y i d e a w o r k e d w a s i n g e n i o u s . F r e y r e a s o n e d a s f o l l o w s : S u p p o s e t h a t F e r m a t ' s L a s t T h e o r e m i s not t r u e . T h e n , f o r s o m e p o w e r n t h a t i s g r e a t e r t h a n 1 t h e r e is a s o l u t i o n t o F e r m a t ' s e q u a t i o n : x n + yn = z",
w h e r e x , y, a n d z a r e i n t e g e r s . T h i s p a r t i c u l a r s o l u t i o n , a , b , a n d c, w o u l d t h e n g i v e r i s e t o a s p e c i f i c e l l i p t i c c u r v e . N o w F r e y w r o t e ' d o w n t h e g e n e r a l e q u a t i o n o f t h i s c u r v e t h a t w o u l d r e s u l t f r o m t h e s o l u t i o n o f F e r m a t ' s e q u a t i o n . H i s c o n j e c t u r e p r e s e n t e d a t O b e r w o l f a c h s t a t e d t h a t t h i s v e r y c u r v e , n o w c a l l e d t h e F r e y c u r v e , w a s a v e r y s t r a n g e a n i m a l . I t w a s s o s t r a n g e , i n f a c t , t h a t i t c o u l d n ' t p o s s i b l y e x i s t . A n d , m o s t i m p o r t a n t , t h e e l l i p t i c c u r v e t h a t w o u l d a r i s e i f F e r m a t ' s L a s t T h e o r e m w e r e f a l s e w a s d e f i n i t e l y not m o d u l a r . S o , i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s i n d e e d t r u e ,
t h e n a l l e l l i p t i c c u r v e s m u s t b e m o d u l a r . T h e r e f o r e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r c o u l d n ' t p o s s i b l y e x i s t . A n d i t w o u l d f o l l o w t h a t F r e y ' s c u r v e , a n e l l i p t i c c u r v e t h a t w a s n o t m o d u l a r ( i n a d d i t i o n t o a l l i t s o t h e r s t r a n g e c h a r a c t e r i s t i c s ) c o u l d n o t e x i s t . T h e r e f o r e , t h e s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t e i t h e r . W i t h o u t t h e e x i s t e n c e o f s o l u t i o n s t o t h e F e r m a t e q u a t i o n , F e r m a t ' s L a s t T h e o r e m ( w h i c h s t a t e s t h a t t h e r e a r e no s o l u t i o n s t o t h e e q u a t i o n f o r a n y n>2),
w o u l d b e p r o v e d . T h i s w a s a c o m p l i c a t e d s e q u e n c e o f i m p l i c a t i o n s , b u t i t f o l l o w e d b e a u t i f u l l y t h e l o g i c o f m a t h e m a t i c a l
Ken Ribet presenting his important
theorems.
Andrew Wiles being
interviewed.
Barry Mazur of Harvard
University— the "granddaddy" of
them all, a mathematician whose
geometrical insights inspired
everyone who contributed to the
final proof of Fermat's Last
Theorem.
Gerhard Frey, who had the
"crazy idea" that an elliptic
curve resulting from a
solution of Fermat's
equation simply could not
exist.
p r o o f . T h e l o g i c w a s : A i m p l i e s B ; t h e r e f o r e , i f B i s not t r u e , t h e n A c a n n o t b e t r u e e i t h e r . H o w e v e r , t h e F r e y s t a t e m e n t w a s , i t s e l f , a conjecture. I t w a s a c o n j e c t u r e w h i c h s a i d t h a t i f a n o t h e r c o n j e c t u r e ( S h i m u r a -T a n i y a m a ) w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d b e e s t a b l i s h e d . T h e p a i r o f s u b s e q u e n t c o n j e c t u r e s i n S e r r e ' s l e t t e r t o M e s t r e f u r t h e r a l l o w e d K e n R i b e t t o t h i n k a b o u t t h e F r e y c o n j e c t u r e i n c l e a r t e r m s .
K e n R i b e t h a d n e v e r b e f o r e b e e n i n t e r e s t e d i n F e r m a t ' s L a s t T h e o r e m . H e h a d s t a r t e d o u t a s a c h e m i s t r y m a j o r a t B r o w n U n i v e r s i t y . U n d e r t h e i n f l u e n c e a n d t u t e l a g e o f K e n n e t h F . I r e -l a n d , R i b e t w a s s t e e r e d t o m a t h e m a t i c s a n d g o t i n t e r e s t e d i n z e t a f u n c t i o n s , e x p o n e n t i a l s u m s , a n d n u m b e r t h e o r y . H e h a d d i s m i s s e d F e r m a t ' s L a s t T h e o r e m a s " o n e o f t h o s e p r o b l e m s a b o u t w h i c h n o t h i n g f u r t h e r o f r e a l i m p o r t a n c e c o u l d b e s a i d . " T h i s w a s a v i e w h e l d b y m a n y m a t h e m a t i c i a n s , b e c a u s e p r o b l e m s i n n u m b e r t h e o r y t e n d t o b e i s o l a t e d ,
w i t h n o u n i f y i n g s c h e m e o r u n d e r l y i n g g e n e r a l p r i n c i p l e b e h i n d t h e m . W h a t i s i n t e r e s t i n g a b o u t F e r m a t ' s L a s t T h e o r e m , h o w e v e r , i s t h a t i t s p a n s m a t h e m a t i c a l h i s t o r y f r o m t h e d a w n o f c i v i l i z a t i o n t o o u r o w n t i m e . A n d t h e t h e o r e m ' s u l t i m a t e s o l u t i o n a l s o s p a n s t h e b r e a d t h o f m a t h e m a t i c s , i n v o l v i n g f i e l d s o t h e r t h a n n u m b e r t h e o r y : a l g e b r a , a n a l y s i s , g e o m e t r y , a n d t o p o l o g y — v i r t u a l l y a l l o f m a t h e m a t i c s .
R i b e t w e n t o n t o p u r s u e a P h . D . i n m a t h e m a t i c s a t H a r v a r d U n i v e r s i t y . T h e r e , f i r s t i n d i r e c t l y a n d t o w a r d h i s g r a d u a t i o n m o r e d i r e c t l y , h e f e l l u n d e r t h e i n f l u e n c e o f t h e g r e a t n u m b e r t h e o r i s t a n d g e o m e t e r B a r r y M a z u r , w h o s e v i s i o n i n s p i r e d e v e r y m a t h e m a t i c i a n i n v o l v e d e v e n i n t h e s m a l l e s t w a y i n e f f o r t s t o p r o v e F e r m a t ' s L a s t T h e o r e m . M a z u r ' s p a p e r o n t h e
I 14
Eisenstein ideal acted as an abstraction of the theory
of ideals developed in the last century by Ernst
Kummer into the modern fields of mathematics,
algebraic geometry and new approaches to the
theory of numbers through geometry.22
Ken Ribet eventually became a professor of
mathematics at the University of California at
Berkeley and did research in number theory. In 1985,
he heard about Frey's "crazy" notion that if a solution
of Fermat's equation existed, that is, if Fermat's Last
Theorem were false, then this solution would give
rise to a very weird curve. This Frey Curve would be
associated with an elliptic curve that could not be
modular. The pair of associated conjectures in
Serre's letter to Mestre made him interested in trying
to prove Frey's conjecture. While he wasn't really
interested in Fermat's Last Theorem, Ribet
recognized that this had become a hot problem, and
it happened to be in an area he knew well. During the
week of August 18-24, 1985, Ribet was at a meeting
on arithmetic algebraic geometry in Areata,
California. He began thinking about Frey's statement,
and the problem remained on his mind for the next
year. When he was freed of his teaching obligations
at Berkeley early in the summer of 1986, Ribet flew
to Germany where he was to do research at the
world-famous center for mathematics, the Max
Planck Institute. Just as he arrived at the Institute,
Ribet made his great breakthrough. He was now
almost able to prove Frey's conjecture.
But he was not quite there. When he returned to
Berkeley, Ribet ran into Barry Mazur, who was
visiting from Harvard. "Barry, let's go for a cup of
coffee," Ribet suggested. The two retreated to a
popular cafe by the University of California campus.
While sipping a cappuccino, Ribet confided to
M a z u r : " I ' m t r y i n g t o g e n e r a l i z e w h a t I ' v e d o n e , s o t h a t I ' l l b e a b l e t o p r o v e t h e F r e y c o n j e c t u r e . I j u s t d o n ' t s e e m t o g e t t h i s o n e t h i n g t o g e n e r a l i z e i t . . . " M a z u r l o o k e d a t w h a t h e w a s s h o w i n g h i m . " B u t y o u ' v e d o n e i t a l r e a d y K e n , " h e s a i d , " a l l y o u n e e d t o d o i s a d d o n s o m e e x t r a g a m m a z e r o o f N s t r u c t u r e , a n d r u n t h r o u g h y o u r a r g u m e n t , a n d y o u ' r e t h e r e ! " R i b e t l o o k e d a t M a z u r , h e l o o k e d b a c k a t h i s c a p p u c c i n o , t h e n b a c k a t M a z u r w i t h d i s b e l i e f . " M y G o d , y o u ' r e a b s o l u t e l y r i g h t ! " h e s a i d . L a t e r h e w e n t b a c k t o h i s o f f i c e t o f i n i s h o f f t h e p r o o f . " K e n ' s i d e a w a s b r i l l i a n t , " M a z u r e x c l a i m e d w h e n d e s c r i b i n g K e n R i b e t ' s i n g e n i o u s p r o o f a f t e r i t w a s p u b l i s h e d a n d b e c a m e k n o w n i n t h e w o r l d o f m a t h e m a t i c i a n s .
R i b e t f o r m u l a t e d a n d p r o v e d a t h e o r e m w h i c h e s t a b l i s h e d a s f a c t t h a t i f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e , t h e n F e r m a t ' s L a s t T h e o r e m w o u l d f a l l o u t o f i t a s a d i r e c t c o n s e q u e n c e . T h e m a n w h o o n l y a y e a r e a r l i e r t h o u g h t t h a t F r e y ' s s u g g e s t i o n w a s a j o k e n o w p r o v e d t h a t t h e " j o k e " w a s a c t u a l l y a m a t h e m a t i c a l r e a l i t y . T h e d o o r t o t h e a t t a c k o n F e r m a t ' s p r o b l e m u s i n g t h e m o d e r n m e t h o d s o f a r i t h m e t i c a l g e b r a i c g e o m e t r y w a s n o w w i d e o p e n . A l l t h e w o r l d n e e d e d
n o w w a s s o m e o n e w h o w o u l d p r o v e t h e s e e m i n g l y -i m p o s s i b l e S h i m u r a - T a n i y a m a c o n j e c t u r e . T h e n F e r m a t ' s L a s t T h e o r e m w o u l d a u t o m a t i c a l l y b e t r u e .
A Child's Dream
T h e p e r s o n w h o w a n t e d d o j u s t t h a t w a s A n d r e w W i l e s . W h e n h e w a s t e n y e a r s o l d , A n d r e w W i l e s w e n t t o t h e p u b l i c l i b r a r y i n h i s t o w n i n E n g l a n d a n d l o o k e d a t a b o o k o n m a t h e m a t i c s . I n t h a t b o o k h e r e a d a b o u t F e r m a t ' s L a s t T h e o r e m . T h e t h e o r e m , a s d e s c r i b e d i n t h e b o o k , s e e m e d s o s i m p l e , t h a t a n y
c h i l d c o u l d u n d e r s t a n d i t . I n W i l e s ' o w n w o r d s : " I t s a i d t h a t y o u w i l l n e v e r f i n d n u m b e r s , x, y , a n d z , s o t h a t x3 + y 3 = z 3 . N o m a t t e r h o w h a r d y o u t r i e d , y o u w i l l n e v e r , e v e r f i n d s u c h n u m b e r s . A n d i t s a i d t h a t t h e s a m e w a s t r u e f o r x4 + y 4 = z 4 , a n d f o r x5
+ y5 = z 5 , a n d s o o n . . . I t s e e m e d s o s i m p l e . A n d i t s a i d t h a t n o b o d y h a s e v e r f o u n d a p r o o f o f t h i s f o r o v e r t h r e e h u n d r e d y e a r s . I w a n t e d t o p r o v e i t . . . "
I n t h e 1970s, A n d r e w W i l e s w e n t t o t h e u n i v e r s i t y . W h e n h e f i n i s h e d h i s d e g r e e h e w a s a d m i t t e d a s a r e s e a r c h s t u d e n t i n m a t h e m a t i c s t o C a m b r i d g e . H i s a d v i s e r w a s P r o f e s s o r J o h n C o a t e s . W i l e s h a d t o d r o p h i s c h i l d h o o d d r e a m o f p r o v i n g F e r m a t ' s L a s t T h e o r e m . R e s e a r c h o n t h e p r o b l e m w o u l d h a v e t u r n e d i n t o s u c h a w a s t e o f t i m e t h a t n o g r a d u a t e s t u d e n t c o u l d a f f o r d i t . B e s i d e s , w h a t d o c t o r a l a d v i s e r w o u l d h a v e a c c e p t e d a s t u d e n t w o r k i n g o n s u c h a n a n c i e n t p u z z l e , o n e t h a t h a d k e p t t h e w o r l d ' s b r i g h t e s t m i n d s f r o m a s o l u t i o n f o r t h r e e c e n t u r i e s ? I n t h e 1970s,
F e r m a t w a s n o t i n f a s h i o n . W h a t w a s " i n " a t t h e t i m e , t h e r e a l h o t t o p i c f o r r e s e a r c h i n n u m b e r t h e o r y , w a s e l l i p t i c c u r v e s . S o A n d r e w W i l e s s p e n t h i s t i m e d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d i n a n a r e a c a l l e d I w a s a w a t h e o r y . H e c o m p l e t e d h i s d o c t o r a l d i s s e r t a t i o n , a n d w h e n h e w a s a w a r d e d h i s P h . D . , h e
g o t a p o s i t i o n i n m a t h e m a t i c s a t P r i n c e t o n U n i v e r s i t y a n d m o v e d t o t h e U n i t e d S t a t e s . T h e r e , h e c o n t i n u e d d o i n g r e s e a r c h o n e l l i p t i c c u r v e s a n d I w a s a w a t h e o r y .
An Old Flame Rekindled
I t w a s a w a r m s u m m e r e v e n i n g , a n d A n d r e w W i l e s w a s s i p p i n g i c e d t e a a t a f r i e n d ' s h o u s e . S u d d e n l y , i n t h e m i d d l e o f t h e c o n -v e r s a t i o n , h i s f r i e n d s a i d : " B y t h e w a y , d i d y o u h e a r t h a t K e n
R i b e t j u s t p r o v e d t h e E p s i l o n C o n j e c t u r e ? " T h e E p s i l o n C o n j e c t u r e i s w h a t n u m b e r t h e o r i s t s w e r e i n f o r m a l l y c a l l i n g F r e y ' s c o n j e c t u r e , a s m o d i f i e d b y S e r r e , a b o u t t h e c o n n e c t i o n b e t w e e n F e r m a t ' s L a s t T h e o r e m a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e . W i l e s w a s e l e c t r i f i e d . A t t h a t m o m e n t , h e k n e w t h a t h i s l i f e w a s c h a n g i n g . T h e c h i l d h o o d d r e a m h e h a d o f p r o v i n g F e r m a t ' s L a s t T h e o r e m — a d r e a m h e h a d h a d t o g i v e u p t o u n d e r t a k e m o r e f e a s i b l e r e s e a r c h — c a m e a l i v e a g a i n w i t h i n c r e d i b l e f o r c e . H e w e n t h o m e a n d s t a r t e d t h i n k i n g a b o u t h o w h e w o u l d p r o v e t h e S h i m u r a - T a n i y a m a c o n j e c t u r e .
" F o r t h e f i r s t f e w y e a r s , " h e l a t e r c o n f i d e d , "I k n e w I
h a d n o c o m p e t i t i o n , s i n c e I
k n e w t h a t n o b o d y — m e i n c l u d e d — h a d a n y i d e a w h e r e t o s t a r t . " H e d e c i d e d t o w o r k i n c o m p l e t e s e c r e c y , a n d i n i s o l a t i o n . " T o o m a n y s p e c t a t o r s w o u l d s p o i l t h e c o n c e n t r a t i o n , a n d I
d i s c o v e r e d e a r l y o n t h a t j u s t a m e n t i o n o f F e r m a t i m m e d i a t e l y g e n e r a t e s t o o m u c h i n t e r e s t . " O f c o u r s e , g i f t e d , a b l e m a t h e m a t i c i a n s a b o u n d , e s p e c i a l l y a t a p l a c e l i k e P r i n c e t o n , a n d t h e d a n g e r o f s o m e o n e c o m p l e t i n g y o u r w o r k f o r y o u— a n d e v e n d o i n g i t b e t t e r — i s v e r y r e a l .
W h a t e v e r t h e r e a s o n , W i l e s l o c k e d h i m s e l f u p i n h i s a t t i c o f f i c e a n d w e n t t o w o r k . H e a b a n d o n e d a l l o t h e r
r e s e a r c h p r o j e c t s t o d e v o t e h i s t i m e c o m p l e t e l y t o F e r m a t . W i l e s w o u l d u s e a l l t h e p o w e r o f t h e m o d e r n m a c h i n e r y o f a l g e b r a , g e o m e t r y , a n a l y s i s , a n d t h e o t h e r a r e a s o f m a t h e m a t i c s . H e w o u l d a l s o m a k e u s e o f t h e i m p o r t a n t m a t h e m a t i c a l r e s u l t s o f h i s c o n t e m p o -r a r i e s , a n d o f h i s h i s t o r i c a l p r e d e c e s s o r s . H e w o u l d m a k e u s e o f R i b e t ' s c l e v e r m e t h o d s o f p r o o f , a n d h i s r e s u l t s , - h e w o u l d u s e t h e t h e o r i e s o f B a r r y M a z u r , a n d t h e i d e a s o f S h i m u r a , F r e y , S e r r e , A n d r e W e i l , a n d t h o s e o f m a n y , m a n y o t h e r m a t h e m a t i c i a n s .
W i l e s ' g r e a t n e s s , G e r h a r d F r e y w o u l d l a t e r s a y , w a s t h a t h e b e l i e v e d i n w h a t h e w a s d o i n g a t a t i m e w h e n v i r t u a l l y e v e r y m a t h e m a t i c i a n i n t h e w o r l d b e l i e v e d t h a t t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c o u l d n o t b e p r o v e n i n t h e t w e n t i e t h c e n t u r y .
To p r o v e t h e S h i m u r a -T a n i y a m a C o n j e c t u r e , A n d r e w W i l e s k n e w h e h a d t o p r o v e t h a t e v e r y e l l i p t i c c u r v e i s m o d u l a r . H e h a d t o s h o w t h a t e v e r y e l l i p t i c c u r v e , w h o s e s o l u t i o n s l i e o n a d o u g h n u t , w a s r e a l l y a m o d u l a r f o r m i n d i s g u i s e . T h e d o u g h n u t w a s , i n a s e n s e , a l s o t h i s s p a c e o f i n t r i c a t e l y s y m m e t r i c f u n c t i o n s o n t h e c o m p l e x p l a n e c a l l e d m o d u l a r f o r m s . N o b o d y h a d a n y i d e a h o w t o s h o w s u c h a w e i r d c o n n e c t i o n b e t w e e n t h e s e t w o s e e m i n g l y v e r y d i f f e r e n t e n t i t i e s .
W i l e s r e a l i z e d t h a t t h e b e s t i d e a w a s t o t r y t o count
t h e n u m b e r o f e l l i p t i c c u r v e s , a n d t o c o u n t t h e n u m b e r o f modular e l l i p t i c c u r v e s , a n d t h e n t o s h o w t h a t t h e i r " n u m b e r " w a s t h e s a m e . T h i s c o n s t r u c t i o n w o u l d p r o v e t h a t t h e e l l i p t i c c u r v e s a n d t h e m o d u l a r e l l i p t i c c u r v e s w e r e o n e a n d t h e s a m e , a n d h e n c e e v e r y e l l i p t i c c u r v e i s i n d e e d m o d u l a r , a s t h e S h i m u r a -T a n i y a m a c o n j e c t u r e c l a i m s .
W i l e s r e a l i z e d t w o t h i n g s . O n e w a s t h a t h e d i d n ' t h a v e t o p r o v e t h e e n t i r e S h i m u r a -T a n i y a m a c o n j e c t u r e , b u t
o n l y a s p e c i a l c a s e : semistable
e l l i p t i c c u r v e s w i t h r a t i o n a l n u m b e r s a s t h e c o e f f i c i e n t s . P r o v i n g t h a t t h e c o n j e c t u r e w a s t r u e f o r t h i s s m a l l e r c l a s s o f e l l i p t i c c u r v e s w o u l d b e e n o u g h t o e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . T h e o t h e r t h i n g W i l e s k n e w w a s t h a t " c o u n t i n g " w o u l d n o t w o r k h e r e b e c a u s e h e w a s d e a l i n g w i t h infinite s e t s . T h e s e t o f s e m i s t a b l e e l l i p t i c c u r v e s w a s i n f i n i t e . A n y r a t i o n a l n u m b e r alb, w h e r e a a n d b a r e i n t e g e r s , w o u l d
g i v e y o u a n o t h e r e l l i p t i c c u r v e ( w e s a y a n e l l i p t i c c u r v e over t h e r a t i o n a l s ) . S i n c e t h e r e a r e i n f i n i t e l y m a n y s u c h n u m b e r s — a a n d b c a n b e a n y o f t h e i n f i n i t e l y m a n y n u m b e r s 1,2,3,4..., t o i n f i n i t y , t h e r e a r e i n f i n i t e l y m a n y e l l i p t i c c u r v e s . S o c o u n t i n g a s w e k n o w i t w o u l d n ' t w o r k .
Breaking Down a Big Problem into Smaller Ones W i l e s t h o u g h t h e m i g h t t r y t o w o r k o n s m a l l e r p r o b l e m s , o n e a t a t i m e . M a y b e h e c o u l d l o o k a t s e t s o f e l l i p t i c f u n c t i o n s a n d s e e w h a t h e c o u l d d o a b o u t t h e m . T h i s w a s a g o o d a p p r o a c h s i n c e i t b r o k e d o w n t h e t a s k s o t h a t , s t e p b y s t e p , h e c o u l d u n d e r s t a n d e a c h s e t . F i r s t o f a l l , s o m e e l l i p t i c c u r v e s w e r e a l r e a d y k n o w n t o b e m o d u l a r . T h e s e w e r e v e r y i m p o r t a n t r e s u l t s , d e v e l o p e d b y m a n y o t h e r n u m b e r t h e o r i s t s . B u t s o o n A n d r e w W i l e s r e a l i z e d t h a t l o o k i n g o n l y a t e l l i p t i c c u r v e s a n d t r y i n g t o c o u n t t h e m o f f a g a i n s t m o d u l a r f o r m s m i g h t n o t b e a g o o d a p p r o a c h — h e w a s d e a l i n g w i t h t w o i n f i n i t e s e t s . I n f a c t , h e w a s n o c l o s e r t o a s o l u t i o n t h a n w a s t h e s k e p t i c a l A n d r e W e i l w h e n h e s a i d : " I s e e n o r e a s o n a g a i n s t t h e c o n j e c t u r e s i n c e o n e a n d t h e o t h e r o f t h e t w o s e t s a r e d e n u m e r a b l e [ i n f i -n i t e b u t o f t h e o r d e r o f i n f i n i t y o f t h e i n t e g e r s a n d r a t i o n a l n u m b e r s , n o t t h e h i g h e r o r d e r o f i n f i n i t y o f t h e i r r a t i o n a l n u m b e r s a n d t h e c o n t i n u u m ] , b u t I d o n ' t s e e a n y r e a s o n f o r i t ,
e i t h e r . . . " A f t e r t w o y e a r s o f g e t t i n g n o w h e r e , W i l e s t r i e d a n e w a p p r o a c h . H e t h o u g h t h e m i g h t transform t h e e l l i p t i c c u r v e s i n t o G a l o i s r e p r e s e n t a t i o n s , a n d t h e n c o u n t t h e s e G a l o i s r e p -r e s e n t a t i o n s a g a i n s t t h e m o d u l a r f o r m s .
T h e i d e a w a s a n e x c e l l e n t o n e , a l t h o u g h i t w a s n o t o r i g i n a l . T h e p r i n c i p l e b e h i n d t h i s m o v e i s i n t e r e s t i n g . N u m b e r t h e o r i s t s
a r e c o n c e r n e d w i t h f i n d i n g s o l u t i o n s o f e q u a t i o n s , s u c h a s t h e F e r m a t e q u a t i o n . T h e m a t h e m a t i c a l t h e o r y o f f i e l d s o f n u m b e r s s e t s t h i s p r o b l e m i n t h e c o n t e x t o f f i e l d e x t e n s i o n s . F i e l d s a r e l a r g e , i n f i n i t e c o l l e c t i o n s w h i c h a r e d i f f i c u l t t o a n a l y z e . T h e r e f o r e , w h a t n u m b e r t h e o r i s t s h a v e o f t e n d o n e i s t o u s e t h e t h e o r i e s o f E v a r i s t e G a l o i s , c a l l e d G a l o i s t h e o r y , i n o r d e r t o t r a n s l a t e t h e s e p r o b l e m s f r o m t h e c o m p l i c a t e d f i e l d s t o w h a t a r e k n o w n a s g r o u p s . O f t e n a g r o u p i s g e n e r a t e d b y a f i n i t e ( r a t h e r t h a n a n i n f i n i t e ) s e t o f e l e m e n t s . U s i n g G a l o i s t h e o r y t h u s a l l o w s t h e n u m b e r t h e o r i s t t o m o v e f r o m a n i n f i n i t e c o l l e c t i o n t o o n e t h a t i s r e p r e s e n t e d b y a f i n i t e s e t . T h i s t r a n s l a t i o n o f a p r o b l e m c o n s t i t u t e s a n i m m e n s e s t e p f o r w a r d , s i n c e a f i n i t e s e t o f e l e m e n t s i s s o m u c h e a s i e r t o h a n d l e t h a n a n i n f i n i t e s e t . C o u n t i n g m a k e s s e n s e o n l y f o r a f i n i t e n u m b e r o f e l e m e n t s . T h e a p p r o a c h s e e m e d t o w o r k f o r s o m e s e t s o f e l l i p t i c c u r v e s . T h i s w a s a g o o d b r e a k t h r o u g h . B u t a f t e r a n o t h e r y e a r , W i l e s w a s s t u c k a g a i n .
The Flack Paper
W h a t A n d r e w W i l e s w a s t r y i n g t o d o n o w w a s t o c o u n t s e t s o f G a l o i s r e p r e s e n t a t i o n s , c o r r e s p o n d i n g t o t h e ( s e m i -s t a b l e ) e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s ,
a n d t o s h o w t h a t t h e y a r e t h e s a m e . I n d o i n g s o , h e w a s u s i n g h i s a r e a o f e x p e r -t i s e , i n w h i c h h e h a d d o n e h i s d i s s e r t a t i o n , c a l l e d H o r i z o n t a l I w a s a w a T h e o r y . W i l e s w a s t r y i n g t o u s e t h i s t h e o r y t o a t t a i n t h e C l a s s N u m b e r F o r m u l a , a r e s u l t w h i c h h e n e e d e d f o r t h e " c o u n t i n g . " B u t h e r e h e w a s u p a g a i n s t a b r i c k w a l l . N o t h i n g h e c o u l d d o b r o u g h t h i m n e a r e r t o t h e a n s w e r .
I n t h e s u m m e r o f 1991, W i l e s w a s a t a c o n f e r e n c e i n B o s t o n , w h e r e h e m e t h i s f o r m e r d o c t o r a l a d v i s e r a t C a m b r i d g e , J o h n
C o a t e s . P r o f e s s o r C o a t e s t o l d W i l e s t h a t o n e o f C o a t e s ' s t u d e n t s , M a t t h i a s F l a c h , u s i n g e a r l i e r w o r k b y a R u s s i a n m a t h e m a t i c i a n n a m e d K o l y v a g i n , h a d d e v i s e d a n E u l e r S y s t e m ( n a m e d a f t e r L e o n h a r d E u l e r ) i n a n a t t e m p t t o p r o v e t h e C l a s s N u m b e r F o r -m u l a . T h i s w a s e x a c t l y w h a t W i l e s n e e d e d f o r h i s p r o o f o f t h e S h i m u r a - T a n i y a m a c o n j e c t u r e — i f , i n d e e d , h e c o u l d e x t e n d F l a c h ' s p a r t i a l r e s u l t s t o t h e f u l l C l a s s N u m b e r F o r m u l a . W i l e s w a s e l a t e d o n h e a r i n g f r o m C o a t e s a b o u t t h i s w o r k o f F l a c h . " T h i s w a s t a i l o r -m a d e " f o r h i s o w n p r o b l e m , W i l e s s a i d , a s i f M a t t h i a s F l a c h h a d d o n e a l l t h i s w o r k j u s t f o r h i m . A n d W i l e s i m m e d i a t e l y a b a n d o n e d a l l h i s H o r i z o n t a l I w a s a w a T h e o r y w o r k a n d i m m e r s e d h i m s e l f d a y a n d n i g h t i n t h e w o r k o f K o l y v a g i n a n d F l a c h . I f t h e i r " E u l e r S y s t e m " r e a l l y w o r k e d , W i l e s w o u l d h o p e f u l l y a c h i e v e t h e C l a s s N u m b e r r e s u l t a n d t h e S h i m u r a - T a n i y a m a C o n j e c t u r e w o u l d b e p r o v e d f o r s e m i s t a b l e e l l i p t i c c u r v e s —e n o u g h t o p r o v e F e r m a t ' s L a s t T h e o r e m .
T h i s w a s h a r d w o r k , h o w e v e r , a n d i t w a s o u t s i d e t h e I w a s a w a r e a l m w h i c h W i l e s k n e w s o w e l l . I n c r e a s i n g l y , W i l e s s t a r t e d t o f e e l t h e n e e d t o f i n d s o m e b o d y t o t a l k t o . H e w a n t e d s o m e o n e w h o c o u l d c h e c k h i s p r o g r e s s i n t h e s e u n c h a r t e d w a t e r s , b u t
s o m e o n e w h o w o u l d n o t r e v e a l a t h i n g t o a n y o n e e l s e .
A Good Friend
W i l e s f i n a l l y h a d t o m a k e a d e c i s i o n : s h o u l d h e c o n t i n u e t o k e e p e v e r y t h i n g s e c r e t a s h e h a d d o n e f o r s o l o n g , o r s h o u l d h e b r e a k d o w n a n d t a l k t o s o m e o n e w i t h g o o d k n o w l e d g e o f n u m b e r t h e o r y ? H e f i n a l l y d e c i d e d h e c o u l d p r o b a b l y n o t d o v e r y w e l l b y k e e p i n g s e c r e c y f o r e v e r . A s h e h i m s e l f s a i d , o n e
c o u l d w o r k o n a p r o b l e m f o r a n e n t i r e l i f e t i m e a n d n o t s e e a n y r e s u l t s . T h e n e e d t o c o m p a r e n o t e s w i t h a n o t h e r p e r s o n f i n a l l y o u t w e i g h e d t h a t i n t e n s e n e e d t o k e e p i t a l l t o h i m s e l f . B u t n o w t h e q u e s t i o n w a s : w h o ? W h o m c o u l d h e t r u s t t o k e e p h i s s e c r e t ?
I n J a n u a r y o f 1993, a f t e r s i x y e a r s o f w o r k i n g a l o n e , W i l e s m a d e h i s c o n t a c t . H e c a l l e d i n P r o f e s s o r N i c k K a t z , o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t . K a t z w a s a n e x p e r t o n m a n y o f t h e t h e o r i e s t h a t w e n t i n t o a t t e m p t s t o p r o v e t h e C l a s s N u m b e r F o r m u l a . B u t m o r e i m p o r t a n t l y , K a t z w a s c o m p l e t e l y t r u s t w o r t h y . H e w o u l d n e v e r r e v e a l w h a t A n d r e w W i l e s w a s u p t o . T h i s a s s e s s m e n t o n W i l e s ' p a r t t u r n e d o u t t o b e c o r r e c t . N i c k K a t z k e p t h i s m o u t h s h u t t h r o u g h o u t h i s w o r k w i t h W i l e s o v e r t h e m a n y r e m a i n i n g m o n t h s o f t h e p r o j e c t . T h e i r m u t u a l c o l l e a g u e s a t t h e t i g h t l y -k n i t m a t h e m a t i c a l c o m m u n i t y a t P r i n c e t o n n e v e r s u s p e c t e d a t h i n g , e v e n a f t e r w e e k s o f s e e i n g t h e t w o s p e n d h o u r s h u d d l e d t o g e t h e r o v e r c o f f e e a t t h e f a r e n d o f t h e C o m m o n s R o o m .
B u t A n d r e w W i l e s s t i l l w o r r i e d t h a t s o m e o n e m i g h t s u s p e c t w h a t h e w a s w o r k i n g o n . H e c o u l d n ' t t a k e a c h a n c e . S o h e c a m e u p w i t h a s c h e m e t o h i d e t h e f a c t t h a t h e w a s w o r k i n g v e r y i n t e n s e l y o n " s o m e t h i n g " w i t h N i c k K a t z . W i l e s w o u l d
o f f e r a n e w g r a d u a t e c o u r s e i n m a t h e m a t i c s f o r t h e s p r i n g o f 1993, a c o u r s e w h i c h N i c k K a t z w o u l d a t t e n d a s o n e o f t h e s t u d e n t s , a n d t h i s w o u l d a l l o w t h e t w o o f t h e m t o w o r k t o g e t h e r w i t h o u t o t h e r s s u s p e c t i n g w h a t t h e y w e r e d o i n g . O r a t l e a s t t h i s i s w h a t W i l e s h a s s a i d . T h e g r a d u a t e s t u d e n t s c o u l d n o t s u s p e c t t h a t b e h i n d t h e s e l e c t u r e s w a s a r o a d t o F e r m a t ' s L a s t T h e o r e m , a n d W i l e s w o u l d b e a b l e t o p i c k t h e i r b r a i n s f o r a n y p o s s i b l e h o l e s i n h i s t h e o r y , w i t h t h e h e l p o f h i s g o o d f r i e n d K a t z .
T h e c o u r s e w a s a n n o u n c e d . I t w a s c a l l e d " C a l c u l a t i o n s w i t h E l l i p t i c C u r v e s , " w h i c h w a s i n n o c e n t e n o u g h s o n o o n e c o u l d s u s p e c t a n y t h i n g . A n d a t t h e s t a r t o f t h e c o u r s e , P r o f e s s o r W i l e s s a i d t h a t t h e p u r p o s e o f t h e l e c t u r e s w a s t o s t u d y s o m e r e c e n t w o r k o f M a t t h i a s F l a c h o n t h e C l a s s N u m b e r F o r m u l a . T h e r e w a s n o m e n t i o n o f F e r m a t , n o m e n t i o n o f S h i m u r a o r T a n i y a m a , a n d n o o n e c o u l d s u s p e c t t h a t t h e C l a s s N u m b e r F o r m u l a t h e y w e r e g o i n g t o s t u d y w o u l d b e t h e k e y s t o n e t o p r o v i n g F e r m a t ' s L a s t T h e o r e m . A n d n o o n e h a d a n y i d e a t h a t t h e t r u e p u r p o s e o f t h e l e c t u r e s w a s n o t t o t e a c h g r a d u a t e s t u -d e n t s m a t h e m a t i c s b u t t o a l l o w W i l e s a n d K a t z t o w o r k t o g e t h e r o n t h i s p r o b l e m w i t h o u t s u s p i c i o n b y a n y o f t h e i r c o l l e a g u e s , w h i l e a t t h e s a m e t i m e g e t t i n g u n s u s p e c t i n g g r a d u a t e s t u d e n t s t o c h e c k t h e w o r k f o r t h e m .
B u t w i t h i n a f e w w e e k s a l l t h e g r a d u a t e s t u d e n t s d r i f t e d a w a y . T h e y c o u l d n ' t k e e p u p w i t h a c o u r s e t h a t w a s n ' t r e a l l y g o i n g a n y w h e r e . T h e o n l y " s t u d e n t " w h o s e e m e d t o k n o w a n y t h i n g a n d t o p a r t i c i p a t e i n c l a s s w a s t h e o t h e r m a t h p r o f e s s o r w h o w a s s i t t i n g t h e r e w i t h t h e m . S o a f t e r a w h i l e , N i c k K a t z w a s t h e o n l y o n e i n t h e a u d i e n c e . B u t W i l e s j u s t w e n t o n u s i n g t h e " c l a s s " t o w r i t e h i s l o n g p r o o f o f t h e
C l a s s N u m b e r T h e o r e m o n t h e b o a r d , c o n t i n u i n g t o t h e n e x t s t e p e v e r y c l a s s m e e t i n g , w i t h N i c k K a t z v e r i f y i n g e a c h s t e p .
T h e l e c t u r e s r e v e a l e d n o e r r o r s . I t s e e m e d t h a t t h e C l a s s N u m b e r F o r m u l a w a s w o r k i n g , a n d W i l e s w a s o n h i s w a y t o t h e s o l u t i o n o f t h e F e r m a t p r o b l e m . A n d s o i n t h e l a t e s p r i n g o f 1993, a s t h e c o u r s e w a s c o m i n g t o a n e n d , A n d r e w W i l e s w a s a l m o s t f i n i s h e d . S t i l l , h e w a s w r e s t l i n g w i t h j u s t o n e f i n a l o b s t a c l e . H e w a s a b l e t o p r o v e t h a t m o s t o f t h e e l l i p t i c c u r v e s
w e r e m o d u l a r , b u t a f e w o f t h e m r e m a i n e d u n p r o v a b l e . H e t h o u g h t h e c o u l d o v e r c o m e t h e s e d i f f i c u l t i e s , a n d h e w a s g e n e r a l l y o p t i m i s t i c . W i l e s f e l t i t w a s t i m e t o t a l k t o o n e m o r e p e r s o n , t o t r y t o g a i n a l i t t l e m o r e i n s i g h t i n t o t h i s l a s t d i f f i c u l t y f a c i n g h i m . S o h e c a l l e d i n a n o t h e r o n e o f h i s c o l l e a g u e s a t t h e P r i n c e t o n m a t h e m a t i c s d e p a r t m e n t , P r o f e s s o r P e t e r S a r n a k , a n d s w o r e h i m t o s e c r e c y a s w e l l . " 1 t h i n k I a m a b o u t t o p r o v e F e r m a t ' s L a s t T h e o r e m , " h e t o l d t h e s t u n n e d S a r n a k .
" T h i s w a s i n c r e d i b l e , " S a r n a k l a t e r r e c a l l e d . " 1 w a s f l a b b e r g a s t e d , e l a t e d , d i s t u r b e d — I m e a n . . . I r e m e m b e r f i n d i n g i t d i f f i c u l t t o s l e e p t h a t n i g h t . " S o n o w t h e r e w e r e t w o c o l l e a g u e s t r y i n g t o h e l p W i l e s f i n i s h h i s p r o o f . W h i l e n o b o d y s u s p e c t e d w h a t i t w a s t h a t t h e y w e r e d o i n g , p e o p l e w e r e n o t i c i n g s o m e t h i n g . A n d w h i l e h e m a i n t a i n e d t h a t n o o n e e v e r f o u n d a n y -t h i n g t h r o u g h h i m , S a r n a k l a t e r a d m i t t e d t h a t h e d r o p p e d " m a y b e a f e w h i n t s . . . "
The Last Piece oj the Puzzle
I n M a y , 1993, A n d r e w W i l e s w a s s i t t i n g a l o n e a t h i s d e s k . H e w a s g e t t i n g s o m e w h a t f r u s t r a t e d . I t s e e m e d t h a t t h o s e f e w e l l i p t i c c u r v e s t h a t g o t a w a y f r o m h i m w e r e n o t c o m i n g a n y n e a r e r . H e
s i m p l y c o u l d n ' t p r o v e t h a t t h e y w e r e m o d u l a r . A n d h e n e e d e d t h e m , t o o , i f h e w e r e t o p r o v e t h a t all ( s e m i s t a b l e ) e l l i p t i c c u r v e s w e r e m o d u l a r s o t h a t F e r m a t ' s L a s t T h e o r e m w o u l d f o l l o w . D o i n g i t f o r m o s t o f t h e s e m i s t a b l e e l l i p t i c c u r v e s w a s a g r e a t m a t h e m a t i c a l r e s u l t o n i t s o w n r i g h t , b u t n o t e n o u g h t o r e a c h h i s g o a l . To r e s t a l i t t l e h o r n h i s i n t e n s e s e a r c h l e a d i n g n o w h e r e , W i l e s p i c k e d u p a n o l d p a p e r o f t h e g r e a t m a s t e r , B a r r y M a z u r o f H a r v a r d U n i v e r s i t y . M a z u r h a d m a d e s o m e
g r o u n d b r e a k i n g d i s c o v e r i e s i n n u m b e r t h e o r y — r e s u l t s t h a t h a d i n s p i r e d m a n y o f t h e e x p e r t s i n t h e f i e l d , i n c l u d i n g R i b e t a n d F r e y , w h o s e w o r k p a v e d t h e w a y f o r W i l e s ' e f f o r t . M a z u r ' s p a p e r t h a t W i l e s w a s n o w r e r e a d i n g w a s a n e x t e n s i o n o f t h e t h e o r y o f i d e a l s , s t a r t i n g w i t h K u m m e r a n d D e d e k i n d , a n d c o n t i n u i n g w i t h y e t a t h i r d n i n e t e e n t h c e n t u r y m a t h e m a t i c i a n —F e r d i n a n d G o t t h o l d E i s e n s t e i n (1823-1852). A l t h o u g h h e d i e d y o u n g , E i s e n s t e i n m a d e g r e a t s t r i d e s i n n u m b e r t h e o r y . G a u s s , i n f a c t , i s q u o t e d a s h a v i n g s a i d : ' T h e r e h a v e b e e n o n l y t h r e e e p o c h - m a k i n g m a t h e m a t i c i a n s , A r c h i m e d e s , N e w t o n , a n d E i s e n s t e i n . "
M a z u r ' s p a p e r o n t h e E i s e n s t e i n I d e a l h a d o n e l i n e i n i t t h a t n o w c a u g h t W i l e s ' a t t e n t i o n . M a z u r w a s s a y i n g t h a t i t w a s p o s s i b l e t o switch
f r o m o n e s e t o f e l l i p t i c c u r v e s t o a n o t h e r . T h e s w i t c h h a d t o d o w i t h p r i m e n u m b e r s . W h a t M a z u r w a s s a y i n g w a s t h a t i f o n e w a s d e a l i n g w i t h e l l i p t i c c u r v e s t h a t w e r e b a s e d o n t h e p r i m e n u m b e r t h r e e , i t w a s p o s s i b l e t o t r a n s f o r m t h e m s o t h a t o n e c o u l d s t u d y t h e m u s i n g t h e p r i m e n u m b e r f i v e i n s t e a d . T h i s 3- t o -5 s w i t c h w a s e x a c t l y w h a t W i l e s w o u l d n e e d . H e w a s s t u c k w i t h n o t b e i n g a b l e t o p r o v e t h a t c e r t a i n c l a s s e s o f e l l i p t i c c u r v e s b a s e d o n t h e
p r i m e n u m b e r t h r e e w e r e m o d u l a r . A n d h e r e M a z u r w a s s a y i n g t h a t h e c o u l d s w i t c h t h e m t o c u r v e s b a s e d o n t h e p r i m e n u m b e r f i v e . B u t t h e s e c u r v e s b a s e d o n f i v e W i l e s h a d a l r e a d y p r o v e d w e r e m o d u l a r . S o t h e 3- t o -5 s w i t c h w a s t h e f i n a l t r i c k . I t t o o k t h e d i f f i c u l t e l l i p t i c c u r v e s b a s e d o n t h r e e , t r a n s f o r m e d t h e m i n t o b a s e f i v e a n d t h e s e w e r e k n o w n t o b e m o d u l a r . O n c e a g a i n , s o m e o t h e r m a t h e m a t i c i a n ' s b r i l l i a n t i d e a h e l p e d W i l e s o v e r c o m e a s e e m i n g l y i n s u r m o u n t a b l e h u r d l e . A n d r e w W i l e s w a s f i n a l l y d o n e .
H i s t i m i n g w a s p e r f e c t , t o o . I n t h e n e x t m o n t h , J u n e , h i s
f o r m e r a d v i s e r J o h n C o a t e s w o u l d b e h o s t i n g a n u m b e r t h e o r y c o n f e r e n c e i n C a m b r i d g e . A l l t h e b i g n a m e s i n n u m b e r t h e o r y w o u l d b e t h e r e . A n d C a m b r i d g e w a s W i l e s ' o l d h o m e t o w n a n d w h e r e h e w e n t t o g r a d u a t e s c h o o l . W o u l d n ' t i t b e p e r f e c t t o p r e s e n t h i s p r o o f o f F e r m a t ' s L a s t T h e o r e m t h e r e ? W i l e s w a s n o w r a c i n g a g a i n s t t h e c l o c k . H e h a d t o p u t t o g e t h e r h i s e n t i r e p r o o f t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w a s t r u e f o r s e m i s t a b l e e l l i p t i c c u r v e s . T h i s m e a n t t h a t t h e F r e y c u r v e c o u l d n o t e x i s t . A n d i f t h e F r e y c u r v e c o u l d n o t " e x i s t , t h a t m e a n t t h a t s o l u t i o n s t o F e r m a t ' s e q u a t i o n c o u l d n o t e x i s t f o r n>2 a n d t h e r e f o r e F e r m a t ' s L a s t T h e o r e m w a s p r o v e n . T h e w r i t e - u p t o o k A n d r e w W i l e s 200 p a g e s . H e w a s d o n e j u s t i n t i m e t o c a t c h h i s p l a n e f o r E n g l a n d . A n d a t t h e e n d o f h i s l a s t l e c t u r e t h e r e , h e e m e r g e d v i c t o r i o u s t o t h e s w e e p i n g a p p l a u s e , t h e f l a s h i n g c a m e r a s , a n d t h e r e p o r t e r s .
The Aftermath
N o w i t w a s t i m e f o r t h e p e e r - r e v i e w . U s u a l l y , a m a t h e m a t i c a l r e s u l t — o r a n y a c a d e m i c f i n d i n g , f o r t h a t m a t t e r — i s s u b m i t t e d t o a " r e f e r e e d j o u r n a l . " S u c h r e f e r e e d j o u r n a l s a r e t h e s t a n d a r d v e h i c l e f o r s c h o l a r s t o s u b m i t t h e i r w o r k f o r p o s s i b l e p u b l i c a t i o n . T h e
j o u r n a l ' s c h a r g e i s t h e n t o s e n d t h e p r o p o s e d p a p e r t o o t h e r e x p e r t s i n t h e a p p r o p r i a t e a r e a o f s t u d y w h o r e v i e w t h e p a p e r ' s c o n t e n t a n d d e t e r m i n e w h e t h e r i t i s c o r r e c t , a n d w h e t h e r t h e p a p e r m a k e s a c o n t r i b u t i o n w o r t h y o f p u b l i c a t i o n . R e f e r e e d j o u r n a l p u b l i c a t i o n s a r e t h e b r e a d a n d b u t t e r o f a c a d e m i a . T e n u r e a n d p r o m o t i o n , a n d o f t e n s a l a r y l e v e l s a n d r a i s e s , a r e a l l d e p e n d e n t o n a r e s e a r c h e r ' s o u t p u t o f r e f e r e e d j o u r n a l a r t i c l e s .
B u t A n d r e w W i l e s c h o s e a d i f f e r e n t a p p r o a c h . I n s t e a d o f s u b m i t t i n g h i s p r o o f t o a p r o f e s s i o n a l m a t h e m a t i c s j o u r n a l — a s a l m o s t a n y o n e e l s e w o u l d h a v e d o n e — h e p r e s e n t e d i t a t a c o n f e r e n c e . T h e r e a s o n w a s p r o b a b l y t w o f o l d . T h r o u g h o u t t h e y e a r s o f w o r k o n t h e p r o o f , W i l e s w a s o b s e s s e d w i t h s e c r e c y . I f h e s u b m i t t e d t h e p r o o f t o a j o u r n a l , t h e p r o o f w o u l d h a v e b e e n s e n t t o a n u m b e r o f r e f e r e e s c h o s e n b y t h e j o u r n a l a n d o n e o f t h e m , o r t h e e d i t o r s , m i g h t h a v e s a i d s o m e t h i n g t o t h e w o r l d a t l a r g e . W i l e s p r o b a b l y w a s a l s o w o r r i e d t h a t s o m e o n e w h o r e a d t h e p r o p o s e d p r o o f m i g h t s o m e h o w s t e a l i t a n d s e n d i t o u t u n d e r h i s o r h e r o w n n a m e . T h i s , u n f o r t u n a t e l y , d o e s h a p p e n i n a c a d e m i a . T h e o t h e r r e a s o n , l i n k e d w i t h t h e f i r s t , w a s t h a t W i l e s w a n t e d t o m a i n t a i n t h e b u i l d u p o f s u s p e n s e a s h e p r e s e n t e d h i s p r o o f a t C a m b r i d g e .
B u t e v e n s o , h a v i n g p r e s e n t e d t h e r e s u l t s a t a c o n f e r e n c e , t h e w o r k w o u l d s t i l l h a v e t o b e r e f e r e e d . T h e s t e p s w o u l d s t i l l h a v e t o b e p e e r - r e v i e w e d , t h a t i s , o t h e r e x p e r t s i n n u m b e r t h e o r y w o u l d h a v e t o g o t h r o u g h W i l e s ' p r o o f , l i n e b y l i n e , t o a s c e r t a i n t h a t h e h a d i n d e e d e s t a b l i s h e d w h a t h e s e t o u t t o p r o v e .
The Deep Gulf Materializes
W i l e s ' 200- p a g e p a p e r w a s
s e n t t o a n u m b e r o f l e a d i n g e x p e r t s i n n u m b e r t h e o r y . S o m e o f t h e m q u i c k l y e x p r e s s e d c o n c e r n s , b u t g e n e r a l l y m a t h e m a t i c i a n s t h o u g h t t h e p r o o f w a s p r o b a b l y c o r r e c t . T h e y w o u l d w a i t t o h e a r t h e e x p e r t s ' v e r d i c t , h o w e v e r . " O h y e s ! " s a i d K e n R i b e t w h e n I a s k e d h i m i f h e b e l i e v e d W i l e s ' p r o o f . " I w a s u n a b l e t o s e e w h a t s o m e p e o p l e w e r e s a y -i n g s o o n a f t e r t h e y r e a d t h e p r o o f — n a m e l y t h a t t h e r e w a s n o E u l e r S y s t e m h e r e . "
O n e o f t h e e x p e r t s c h o s e n t o g o o v e r W i l e s ' p r o o f w a s h i s P r i n c e t o n f r i e n d , N i c k K a t z . P r o f e s s o r K a t z s p e n t t w o w h o l e m o n t h s , J u l y a n d A u g u s t o f 1993, d o i n g n o t h i n g b u t s t u d y i n g t h e e n t i r e p r o o f . E v e r y d a y , h e w o u l d s i t a t h i s d e s k a n d s l o w l y r e a d e v e r y l i n e , e v e r y m a t h e m a t i c a l s y m b o l , e v e r y l o g i c a l i m p l i c a t i o n , t o m a k e s u r e t h a t i t m a d e p e r f e c t s e n s e a n d t h a t i t w o u l d i n d e e d b e a c c e p t a b l e t o a n y m a t h e m a t i c i a n w h o m i g h t r e a d t h e p r o o f . O n c e o r t w i c e a d a y , K a t z w o u l d s e n d a n e - m a i l m e s s a g e t o A n d r e w W i l e s , w h o s t a y e d a w a y f r o m P r i n c e t o n t h a t s u m m e r , a s k i n g h i m : " W h a t d o y o u m e a n i n t h i s l i n e o n t h i s p a g e ? " o r " I d o n ' t s e e h o w t h i s i m p l i c a t i o n f o l l o w s f r o m t h e o n e a b o v e , " e t c . I n r e s p o n s e , W i l e s w o u l d e - m a i l b a c k , a n d i f t h e p r o b l e m r e q u i r e d m o r e d e t a i l s , h e w o u l d f a x t h e a n s w e r t o K a t z .
O n e d a y , w h e n K a t z w a s a b o u t t w o t h i r d s o f t h e w a y t h r o u g h W i l e s ' l o n g m a n u s c r i p t , h e c a m e a c r o s s a p r o b l e m . I t s e e m e d i n n o c e n t e n o u g h a t f i r s t , o n e o f m a n y W i l e s h a d a n s w e r e d e a r l i e r t o K a t z ' s c o m p l e t e s a t i s f a c t i o n . B u t n o t t h i s t i m e . I n r e s p o n s e t o K a t z ' s q u e s t i o n s , W i l e s e -m a i l e d b a c k a n a n s w e r . B u t K a t z h a d t o e - m a i l b a c k : " I s t i l l d o n ' t u n d e r s t a n d i t , A n d r e w . " S o t h i s t i m e W i l e s s e n t a f a x t r y i n g t o m a k e t h e l o g i c a l c o n n e c t i o n .
A g a i n K a t z w a s n o t s a t i s f i e d . S o m e t h i n g w a s s i m p l y n o t r i g h t . T h i s w a s s u p p o s e d t o b e e x a c t l y o n e o f t h e a r g u m e n t s t h a t W i l e s a n d K a t z w e n t o v e r c a r e f u l l y i n t h e s p r i n g w h e n W i l e s w a s t e a c h i n g h i s " c o u r s e . " A n y d i f f i c u l t i e s s h o u l d h a v e a l r e a d y b e e n i r o n e d o u t . B u t a p p a r e n t l y t h e h o l e i n W i l e s ' l o g i c e l u d e d t h e m b o t h . P o s s i b l y i f t h e g r a d u a t e s t u -d e n t s h a d s t a y e d o n , o n e o f t h e m m i g h t h a v e a l e r t e d t h e t w o t o t h e p r o b l e m .
B y t h e t i m e K a t z f o u n d t h e e r r o r , o t h e r m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d w e r e a w a r e o f t h e e x a c t s a m e p r o b l e m w i t h W i l e s ' p r o o f . T h e r e s i m p l y w a s n o E u l e r S y s t e m h e r e , a n d t h e r e w a s n o t h i n g d o i n g . A n d w i t h o u t t h e E u l e r S y s t e m — s u p -p o s e d l y a g e n e r a l i z a t i o n o f t h e e a r l i e r w o r k o f F l a c h a n d K o l y - v a g i n — t h e r e w a s n o C l a s s N u m b e r F o r m u l a . W i t h o u t t h e C l a s s N u m b e r F o r m u l a i t w a s i m p o s s i b l e t o " c o u n t " t h e G a l o i s r e p r e s e n t a t i o n s o f t h e e l l i p t i c c u r v e s a g a i n s t t h e m o d u l a r f o r m s , a n d S h i m u r a -T a n i y a m a w a s n o t e s t a b l i s h e d . A n d w i t h o u t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e p r o v e d a s c o r r e c t , t h e r e w a s n o p r o o f o f F e r m a t ' s L a s t T h e o r e m . I n s h o r t , t h e h o l e i n t h e E u l e r S y s t e m m a d e e v e r y t h i n g c o l l a p s e l i k e a h o u s e o f c a r d s .
The Agony
A n d r e w W i l e s r e t u r n e d t o P r i n c e t o n i n t h e f a l l o f 1993.
H e w a s e m b a r r a s s e d , h e w a s u p s e t , h e w a s a n g r y , f r u s t r a t e d , h u m i l i a t e d . H e h a d p r o m i s e d t h e w o r l d a p r o o f o f F e r m a t ' s L a s t T h e o r e m — b u t h e c o u l d n ' t d e l i v e r . I n m a t h e m a t i c s , a s i n a l m o s t a n y o t h e r f i e l d , t h e r e a r e n o r e a l " s e c o n d p r i z e s " o r " a l s o r a n " a w a r d s . T h e c r e s t f a l l e n W i l e s w a s b a c k i n h i s a t t i c t r y i n g t o f i x t h e p r o o f . " A t t h i s p o i n t , h e w a s h i d i n g a s e c r e t f r o m
t h e w o r l d , " r e c a l l e d N i c k K a t z , " a n d I t h i n k h e m u s t h a v e f e l t p r e t t y u n c o m f o r t a b l e a b o u t i t . " O t h e r c o l l e a g u e s h a d t r i e d t o h e l p W i l e s , i n c l u d i n g h i s f o r m e r s t u d e n t R i c h a r d T a y l o r w h o w a s t e a c h i n g a t C a m b r i d g e b u t j o i n e d W i l e s a t P r i n c e t o n t o h e l p h i m t r y t o f i x t h e p r o o f .
" T h e f i r s t s e v e n y e a r s , w o r k i n g a l l a l o n e , I e n j o y e d e v e r y m i n u t e o f i t , " W i l e s r e c a l l e d , " n o m a t t e r h o w h a r d o r s e e m i n g l y i m p o s s i b l e a h u r d l e I f a c e d . B u t n o w , d o i n g m a t h e m a t i c s
i n t h i s o v e r - e x p o s e d w a y w a s c e r t a i n l y n o t m y s t y l e . I h a v e n o w i s h t o e v e r r e p e a t t h i s e x p e r i e n c e . " A n d t h e b a d e x p e r i e n c e l a s t e d a n d l a s t e d . R i c h a r d T a y l o r , h i s s a b b a t i c a l l e a v e o v e r , r e t u r n e d t o C a m b r i d g e a n d s t i l l W i l e s s a w n o e n d i n s i g h t . H i s c o l l e a g u e s l o o k e d a t h i m w i t h a m i x t u r e o f a n t i c i p a t i o n , h o p e , a n d p i t y , a n d h i s s u f f e r i n g w a s c l e a r t o e v e r y o n e a r o u n d h i m . P e o p l e w a n t e d t o k n o w . T h e y w a n t e d t o h e a r g o o d n e w s , b u t n o n e o f h i s c o l l e a g u e s d a r e d a s k h i m h o w h e w a s d o i n g w i t h t h e p r o o f . O u t s i d e h i s d e p a r t m e n t , t h e r e s t o f t h e w o r l d w a s c u r i o u s , t o o . S o m e t i m e i n t h e n i g h t o f D e c e m b e r 4, 1993,
A n d r e w W i l e s p o s t e d a n e -m a i l m e s s a g e t o t h e c o m p u t e r n e w s g r o u p S c i . m a t h , t o w h i c h s e v e r a l n u m b e r t h e o r i s t s a n d o t h e r m a t h e m a t i c i a n s b e l o n g e d :
In view of the speculation on the status of my
work on the Taniyama-Shimura conjecture and
Fermat's Last Theorem 1 will give a brief account
of the situation. During the review process a
number of problems emerged, most of which have
been resolved, but one in particular 1 have not yet
settled... I believe that 1 will be able to finish this
in the near future using the ideas explained in my
Cambridge lectures. The fact that a lot of work
remains to be done on the manuscript makes it
unsuitable for release as a preprint. In my course
in Princeton beginning in February I will give a full
account of this work.
Andrew Wiles
The Post-Mortem
B u t A n d r e w W i l e s w a s
p r e m a t u r e l y o p t i m i s t i c . A n d w h a t e v e r c o u r s e h e m a y h a v e p l a n n e d t o o f f e r a t P r i n c e t o n w o u l d n o t
y i e l d a n y s o l u t i o n . W h e n m o r e t h a n a y e a r p a s s e d s i n c e h i s s h o r t - l i v e d t r i u m p h i n C a m b r i d g e , A n d r e w W i l e s w a s a b o u t t o g i v e u p a l l h o p e a n d t o f o r g e t h i s c r i p p l e d p r o o f .
O n M o n d a y m o r n i n g , S e p t e m b e r 19, 1994, W i l e s w a s s i t t i n g a t h i s d e s k a t P r i n c e t o n U n i v e r s i t y , p i l e s o f p a p e r s t r e w n a l l a r o u n d h i m . H e d e c i d e d h e w o u l d t a k e o n e l a s t l o o k a t h i s p r o o f b e f o r e c h u c k i n g i t a l l a n d a b a n d o n i n g a l l h o p e t o p r o v e F e r m a t ' s L a s t T h e o r e m . H e w a n t e d t o s e e e x a c t l y w h a t i t w a s t h a t w a s p r e v e n t i n g h i m f r o m c o n s t r u c t i n g t h e E u l e r S y s t e m . H e w a n t e d t o k n o w —j u s t f o r h i s o w n s a t i s f a c t i o n— w h y h e h a d f a i l e d . W h y w a s t h e r e n o E u l e r S y s t e m ? — h e w a n t e d t o b e a b l e t o p i n p o i n t p r e c i s e l y w h i c h t e c h n i c a l f a c t w a s m a k i n g t h e w h o l e t h i n g f a i l . I f h e w a s g o i n g t o g i v e u p , h e f e l t , t h e n a t l e a s t h e w a s o w e d a n a n s w e r t o w h y h e h a d b e e n w r o n g .
W i l e s s t u d i e d t h e p a p e r s i n f r o n t o f h i m , c o n c e n t r a t i n g v e r y h a r d f o r a b o u t t w e n t y m i n u t e s . A n d t h e n h e s a w e x a c t l y w h y h e w a s u n a b l e t o m a k e t h e s y s t e m w o r k . F i n a l l y , h e u n d e r s t o o d w h a t w a s w r o n g . " I t w a s t h e m o s t i m p o r t a n t m o m e n t i n m y e n t i r e w o r k i n g l i f e , " h e l a t e r d e s c r i b e d t h e f e e l i n g . " S u d d e n l y , t o t a l l y u n e x p e c t e d l y , I h a d t h i s i n c r e d i b l e r e v e l a t i o n .
N o t h i n g I ' l l e v e r d o a g a i n w i l l . . . " a t t h a t m o m e n t t e a r s w e l l e d u p a n d W i l e s w a s c h o k i n g w i t h e m o t i o n . W h a t W i l e s r e a l i z e d a t t h a t f a t e f u l m o m e n t w a s " s o i n d e s c r i b a b l y b e a u t i f u l , i t w a s s o s i m p l e a n d s o e l e g a n t . . . a n d I j u s t s t a r e d i n d i s b e l i e f . " W i l e s r e a l i z e d t h a t e x a c t l y w h a t w a s m a k i n g t h e E u l e r S y s t e m f a i l i s w h a t w o u l d m a k e t h e H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h h e h a d a b a n d o n e d t h r e e y e a r s e a r l i e r work. W i l e s s t a r e d a t h i s p a p e r f o r a l o n g t i m e . H e m u s t b e d r e a m i n g , h e t h o u g h t , t h i s w a s j u s t t o o g o o d t o b e t r u e . B u t l a t e r h e s a i d i t w a s s i m p l y t o o g o o d t o b e
false. T h e d i s c o v e r y w a s s o p o w e r f u l , s o b e a u t i f u l , t h a t i t bad t o b e t r u e .
W i l e s w a l k e d a r o u n d t h e d e p a r t m e n t f o r s e v e r a l h o u r s . H e d i d n ' t k n o w w h e t h e r h e w a s a w a k e o r d r e a m i n g . E v e r y o n c e i n a w h i l e , h e w o u l d r e t u r n t o h i s d e s k t o s e e i f h i s f a n t a s t i c f i n d i n g w a s s t i l l t h e r e — a n d i t w a s . H e w e n t h o m e . H e h a d t o s l e e p o n i t — m a y b e i n t h e m o r n i n g h e w o u l d f i n d s o m e f l a w i n t h i s n e w a r g u m e n t . A y e a r o f p r e s s u r e f r o m t h e e n t i r e w o r l d , a y e a r o f o n e f r u s t r a t e d a t t e m p t a f t e r a n o t h e r h a d s h a k e n W i l e s ' c o n f i d e n c e . H e c a m e b a c k t o h i s d e s k i n t h e m o r n i n g , a n d t h e i n c r e d i b l e g e m h e h a d f o u n d t h e d a y b e f o r e w a s s t i l l t h e r e , w a i t i n g f o r h i m .
W i l e s w r o t e u p h i s p r o o f u s i n g t h e c o r r e c t e d H o r i z o n t a l I w a s a w a T h e o r y a p p r o a c h . F i n a l l y , e v e r y t h i n g f e l l p e r f e c t l y i n t o p l a c e . T h e a p p r o a c h h e h a d u s e d t h r e e y e a r s e a r l i e r w a s t h e c o r r e c t o n e . A n d t h a t k n o w l e d g e c a m e t o h i m f r o m t h e f a i l i n g o f t h e F l a c h a n d K o l y v a g i n r o u t e h e h a d c h o s e n i n m i d s t r e a m . T h e m a n u s c r i p t w a s r e a d y t o b e s h i p p e d o u t . E l a t e d , A n d r e w
W i l e s l o g g e d i n t o h i s c o m p u t e r a c c o u n t . H e s e n t e -m a i l m e s s a g e s a c r o s s t h e I n t e r n e t t o a s c o r e o f m a t h e m a t i c i a n s a r o u n d t h e w o r l d : " E x p e c t a F e d e r a l E x p r e s s p a c k a g e i n t h e n e x t f e w d a y s , " t h e m e s s a g e s r e a d .
A s h e h a d p r o m i s e d h i s f r i e n d R i c h a r d T a y l o r , w h o h a d c o m e f r o m E n g l a n d e s p e c i a l l y t o h e l p h i m c o r r e c t h i s p r o o f , t h e n e w p a p e r c o r r e c t i n g t h e I w a s a w a t h e o r y b o r e b o t h o f t h e i r n a m e s e v e n t h o u g h W i l e s o b t a i n e d t h e a c t u a l r e s u l t a f t e r T a y l o r ' s d e p a r t u r e . W i t h i n t h e n e x t f e w w e e k s , t h e m a t h e m a t i c i a n s w h o r e c e i v e d W i l e s ' c o r r e c t i o n t o h i s C a m b r i d g e p a p e r s w e n t o v e r a l l t h e d e t a i l s . T h e y c o u l d f i n d n o t h i n g w r o n g .
I 33
W i l e s n o w u s e d t h e c o n v e n t i o n a l a p p r o a c h t o t h e p r e s e n t a t i o n o f m a t h e m a t i c a l r e s u l t s . I n s t e a d o f d o i n g w h a t h e h a d d o n e i n C a m b r i d g e a y e a r a n d a h a l f e a r l i e r , h e s e n t t h e p a p e r s t o a p r o f e s s i o n a l j o u r n a l , t h e Annals oj Mathematics, w h e r e t h e y c o u l d b e p e e r - r e v i e w e d b y o t h e r m a t h e m a t i c i a n s . T h e r e v i e w p r o c e s s t o o k a f e w m o n t h s , b u t n o f l a w s w e r e f o u n d t h i s t i m e . T h e M a y , 1995, i s s u e o f t h e j o u r n a l c o n t a i n e d W i l e s ' o r i g i n a l C a m b r i d g e p a p e r a n d t h e c o r r e c t i o n b y T a y l o r a n d W i l e s . F e r m a t ' s L a s t T h e o r e m w a s f i n a l l y l a i d t o r e s t .
Did Fermat Possess the Prooj?
A n d r e w W i l e s d e s c r i b e d h i s p r o o f a s " a t w e n t i e t h c e n t u r y p r o o f . " I n d e e d , W i l e s u s e d t h e w o r k o f m a n y t w e n t i e t h - c e n t u r y m a t h e m a t i c i a n s . H e a l s o u s e d t h e w o r k o f e a r l i e r m a t h e m a t i c i a n s . A l l t h e m y r i a d e l e m e n t s o f W i l e s ' c o n s t r u c t i o n s c a m e f r o m t h e w o r k o f o t h e r s , m a n y o t h e r s . S o t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m w a s r e a l l y t h e a c h i e v e m e n t o f a l a r g e n u m b e r o f m a t h e m a t i c i a n s l i v i n g i n t h e t w e n t i e t h c e n t u r y — a n d a l l t h e p r e c e d i n g o n e s t o t h e t i m e o f F e r m a t h i m s e l f . A c c o r d i n g t o W i l e s , F e r m a t c o u l d n o t p o s s i b l y h a v e h a d t h i s p r o o f i n m i n d w h e n h i s w r o t e h i s f a m o u s n o t e i n t h e m a r g i n . T h i s m u c h i s t r u e , o f c o u r s e , b e c a u s e t h e S h i m u r a -
T a n i y a m a c o n j e c t u r e d i d n o t e x i s t u n t i l t h e t w e n t i e t h c e n t u r y . B u t c o u l d F e r m a t h a v e h a d a n o t h e r p r o o f i n m i n d ?
T h e a n s w e r i s p r o b a b l y n o t . B u t t h i s i s n o t a c e r t a i n t y . We w i l l n e v e r k n o w . O n t h e o t h e r h a n d , F e r m a t l i v e d a n o t h e r 28 y e a r s a f t e r h e w r o t e h i s t h e o r e m i n t h e m a r g i n . A n d h e n e v e r s a i d a n y t h i n g m o r e a b o u t i t . P o s s i b l y h e k n e w h e c o u l d n ' t p r o v e t h e t h e o r e m . O r h e m a y h a v e e r r o n e o u s l y t h o u g h t t h a t
h i s m e t h o d o f i n f i n i t e d e s c e n t u s e d i n p r o v i n g t h e s i m p l e c a s e « = 3 c o u l d a p p l y t o a g e n e r a l s o l u t i o n . O r m a y b e h e s i m p l y f o r g o t a b o u t t h e t h e o r e m a n d w e n t o n t o d o o t h e r t h i n g s .
P r o v i n g t h e t h e o r e m t h e w a y i t w a s f i n a l l y d o n e i n t h e 1990s r e q u i r e d a l o t m o r e m a t h e m a t i c s t h a n F e r m a t h i m s e l f c o u l d h a v e k n o w n . T h e p r o f o u n d n a t u r e o f t h e t h e o r e m i s t h a t n o t o n l y d o e s i t s h i s t o r y s p a n t h e l e n g t h o f h u m a n c i v i l i z a t i o n , b u t t h e f i n a l s o l u t i o n o f t h e p r o b l e m c a m e a b o u t b y h a r n e s s i n g — a n d i n a s e n s e u n i f y i n g — t h e e n t i r e b r e a d t h o f m a t h e m a t i c s . I t w a s t h i s u n i f i c a t i o n o f s e e m i n g l y d i s p a r a t e a r e a s o f m a t h e m a t i c s t h a t f i n a l l y n a i l e d t h e t h e o r e m . A n d d e s p i t e t h e f a c t t h a t A n d r e w W i l e s w a s t h e p e r s o n w h o d i d t h e i m p o r t a n t f i n a l w o r k o n t h e t h e o r e m b y p r o v i n g a f o r m o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e n e e d e d t o p r o v e F e r m a t ' s t h e o r e m , t h e e n t i r e e n t e r p r i s e w a s t h e w o r k o f m a n y p e o p l e . A n d i t i s a l l t h e i r c o n t r i b u -t i o n s , t a k e n t o g e t h e r , w h i c h b r o u g h t a b o u t t h e f i n a l s o l u t i o n . W i t h o u t t h e w o r k o f E r n s t K u m m e r t h e r e w o u l d h a v e b e e n n o t h e o r y o f i d e a l s , a n d w i t h o u t i d e a l s t h e w o r k o f B a r r y M a z u r w o u l d n o t h a v e e x i s t e d . W i t h o u t M a z u r t h e r e w o u l d h a v e b e e n n o c o n j e c t u r e b y F r e y , a n d w i t h o u t t h e c r u c i a l c o n j e c t u r e a n d i t s s y n t h e s i s
b y S e r r e t h e r e w o u l d n o t h a v e b e e n t h e p r o o f b y R i b e t t h a t t h e S h i m u r a - T a n i y a m a c o n j e c t u r e w o u l d e s t a b l i s h F e r m a t ' s L a s t T h e o r e m . A n d i t s e e m s t h a t n o p r o o f o f F e r m a t ' s L a s t T h e o r e m w o u l d b e p o s s i b l e w i t h o u t t h a t c o n j e c t u r e p u t f o r w a r d b y Y u t a k a T a n i y a m a a t T o k y o -N i k k o i n 1955 a n d t h e n r e f i n e d a n d m a d e s p e c i f i c b y G o r o S h i m u r a . O r w o u l d i t ?
F e r m a t , o f c o u r s e , c o u l d n o t h a v e f o r m u l a t e d s u c h a n o v e r a r c h i n g c o n j e c t u r e t h a t w o u l d u n i f y t w o v e r y d i f f e r e n t b r a n c h e s o f m a t h e m a t i c s . O r c o u l d h e h a v e d o n e s o ? N o t h i n g
i s c e r t a i n . We o n l y k n o w t h a t t h e t h e o r e m h a s f i n a l l y b e e n e s t a b l i s h e d a n d t h a t t h e p r o o f h a s b e e n c h e c k e d a n d v e r i f i e d t o i t s f i n e s t d e t a i l s b y s c o r e s o f m a t h e m a t i c i a n s t h r o u g h o u t t h e w o r l d . B u t j u s t b e c a u s e a p r o o f e x i s t s a n d i t i s a v e r y c o m p l i c a t e d , a d v a n c e d o n e d o e s n o t m e a n t h a t a s i m p l e r p r o o f i s n o t p o s s i b l e . R i b e t , i n f a c t , p o i n t s o u t i n o n e o f h i s p a p e r s a d i r e c t i o n w h e r e a p r o o f o f F e r m a t ' s t h e o r e m m i g h t b e p o s s i b l e w i t h o u t a p r o o f o f t h e S h i m u r a -T a n i y a m a c o n j e c t u r e . A n d p e r h a p s F e r m a t d i d k n o w a l o t o f p o w e r f u l " m o d e r n " m a t h e m a t i c s , n o w l o s t ( a c t u a l l y , t h e c o p y o f B a c h e t ' s D i o p h a n t u s i n w h i c h h e s u p p o s e d l y w r o t e h i s m a r g i n - s t a t e m e n t h a s n e v e r b e e n f o u n d ) . S o w h e t h e r o r n o t F e r m a t d i d p o s s e s s a " t r u l y m a r v e l o u s p r o o f " o f h i s t h e o r e m , o n e t h a t c o u l d n o t f i t i n t h e m a r g i n o f h i s b o o k , w i l l f o r e v e r r e m a i n h i s s e c r e t .
Gerd Faltings. He had a
totally different approach
to Fermat's Last Theorem.
When Wiles failed in his
first attempt in 1993,
many feared that Faltings
would now beat him to the
true proof.
Andrew Wiles at the crucial
moment of his third lecture
at Cambridge, June 1993,
when it was clear to
everyone Fermat was just
around the corner.
Ken Ribet at the famous
cafe where he finished the
proof that Shimura-
Taniyama would imply that
Fermat's Last Theorem had
to be true.
Endnotes
1. E.T. Bell, Men oj Mathematics, New York: Simon and Schuster, 1937, p. 56 .
I . Barry Mazur, "Number Theory as Gadfly," American Mathematical
Monthly, Vol. 98 , 1991, p. 593.
3 . Plimpton 322 and its implications about the advanced level of
Babylonian mathematics were brought to the attention of the scientific com-
munity by Otto Neugebauer in 1934. An account in English can be found in
his book The Exact Sciences in Antiquity (Princeton University Press, 1957 ) .
23.Actually, Cantor went much farther. He hypothesized.that the order of
infinity of the irrational numbers immediately follows that of the rationals.
That is, he believed that there is no order of infinity that is both higher
than that of the rational numbers and lower than that of the irrational
numbers. This statement became known as the Continuum Hypothesis,
and the work of Kurt Godel and Paul Cohen in the twentieth century
established that it is impossible to prove this hypothesis within the rest of
mathematics. The Continuum Hypothesis stands alone (with some
equivalent restatements) opposite the rest of mathematics, their
respective truths independent of each other. This remains one of the most
bizarre truths in the foundations of mathematics.
24.D. Wells, Curious and Interesting Numbers, London: Penguin Books, 1987,
p. 81 .
25.C. Boyer, A History of Mathematics, New York: Wiley, 1968 , p. 9 .
26.Reprinted in B. Mazur, op. cit.
27.Ian Stewart, Nature's Numbers, New York: Basic Books, 1995, p. 140.
9 . Michael Mahoney, The Mathematical Career oj Pierre de Fermat, 2d ed.,
Princeton University Press, 1994, p. 4 .
10 . Harold M. Edwards, Fermat's Last Theorem, New York: Springer-Verlag,
1977, pp. 61-73 .
I I . Much of what is publicly known about the secret society comes from
Paul R. Halmos, "Nicolas Bourbaki," Scientific American, 196, May 1957, pp.
88 -97 .
12 . Andre Weil, Oeuvres, Vols. Mil, Paris: Springer-Verlag, 1979.
13 . Adapted from Kenneth A. Ribet and Brian Hayes, "Fermat's Last
Theorem and Modern Arithmetic," American Scientist, Vol. 82 , March-April
1994, pp. 144-156 .
28.A good introduction to this topic is the book by Joseph H . Silverman and
John Tate, Rational Points on Elliptic Curves, New York: Springer-Verlag,
1992.
29.Most of the information on Yutaka Taniyama's life is from Goro Shimura,
"Yutaka Taniyama and His Time: Very Personal Recollections," Bulletin oj
the London Mathematical Society, Vol. 21 , 1989, pp. 184-196 .
30.Reprinted in the Japanese journal Sugaku, May 1956, pp. 227 -231 .
31.Shimura thus stated to Serre his actual conjecture, sharing it for the
first time, and implicitly trusting that Serre would acknowledge him as its
originator.
32.Andre Weil, Oeuvres, op. cit., Vol. Ill, p. 450.
33.Andre Weil, "Liber die Bestimmung Dirichletscher Reihen durch
Funktionalgleichungen," Math. Annalen, Vol. 168 , 1967, pp. 165-172.
34.Weil's letter to Lang, along with much of the chronology of events
described here, including private conversations and letters, are
reproduced in Serge Lang, "Some History of the Shimura-Taniyama
Conjecture," Notices oj the American Mathematical Society, November 1995,
pp. 1301-1307. It is to Lang's credit that his article and the "Taniyama-
Shimura File" he has been circulating among mathematicians for ten
years now are finally helping bring Goro Shimura the recognition he
rightly deserves.
35.Jean-Pierre Serre, "Lettre a J.-F. Mestre," reprinted in Current Trends in
Arithmetical Algebraic Geometry, Providence: American Mathematical
Society, 1987, pp. 263 -268 .
22 . Barry Mazur, "Modular Curves and the Eisenstein Ideal," Paris,
France: The Mathematical Publications ojI.H.E.S., Vol. 47 , 1977, pp. 33-186.
36.Barry Mazur, op. cit.
37.The first and more important of the two papers, Andrew Wiles, "Modular
Elliptic Curves and Fermat's Last Theorem," Annals of Mathematics, Vol.
142, 1995, pp. 443-551, begins with Fermat's actual margin-
statement of his theorem in Latin: Cubum autem in duos cubos, aut
quadratocjuadratum in duos quadratoquadratos, et generaliter nullam in infinitum
ultra quadratum potestatem in duos ejus-dem nominis fas est dividere: cujus rei
demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.
Pierre de Fermat. The journal sold out even before the publication date, and
for the first time imposed a charge of $14 per individual issue.
Author's Note
I n p r e p a r i n g t h i s b o o k I d r e w m u c h o f t h e h i s t o r i c a l b a c k -g r o u n d f r o m a v a r i e t y o f s o u r c e s . M y f a v o r i t e , a n d t h e m o s t c o m p l e t e a n d o r i g i n a l s o u r c e , i s E . T . B e l l ' s b o o k , Men oj Mathematics
( a l t h o u g h I d i s l i k e t h e s e x i s t t i t l e , w h i c h i s a l s o m i s l e a d i n g s i n c e t w o o f t h e m a t h e m a t i c i a n s a r e w o m e n , - t h e b o o k w a s w r i t t e n i n 1937).
A p p a r e n t l y o t h e r h i s t o r i a n s o f m a t h e m a t i c s h a v e d r a w n t h e i r i n f o r m a t i o n f r o m B e l l , s o I w i l l n o t m e n t i o n t h e m b y n a m e h e r e . A l l m y i m p o r t a n t s o u r c e s a r e r e f e r e n c e d i n t h e e n d n o t e s . A d d i t i o n a l l y , I f o u n d t h e a r t i c l e s o f J a c q u e l y n S a v a n i o f P r i n c e t o n U n i v e r s i t y (Princeton
Weekly Bulletin, S e p t e m b e r 6, 1993)
u s e f u l , a n d I t h a n k h e r f o r s e n d i n g m e a c o p y o f a p r o g r a m a i r e d o n t h e B B C a b o u t F e r m a t ' s L a s t T h e o r e m .
I a m i n d e b t e d t o C . J . M o z z o c h i f o r a n u m b e r o f p h o t o g r a p h s o f m a t h e m a t i c i a n s i n v o l v e d i n t h e p r o o f o f F e r m a t ' s L a s t T h e o r e m . V e r y w a r m t h a n k s t o P r o f e s s o r K e n n e t h A . R i b e t o f t h e U n i v e r s i t y o f C a l i f o r n i a a t B e r k e l e y f o r i n f o r m a t i v e i n t e r v i e w s a n d m u c h i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k l e a d i n g t o t h e p r o o f o f F e r m a t ' s t h e o r e m . M y d e e p g r a t i t u d e t o P r o f e s s o r G o r o S h i m u r a o f P r i n c e t o n U n i v e r s i t y f o r h i s g e n e r o s i t y w i t h h i s t i m e i n g i v i n g m e a c c e s s t o s o m u c h
i m p o r t a n t i n f o r m a t i o n a b o u t h i s w o r k , a n d h i s c o n j e c t u r e w i t h o u t w h i c h t h e r e w o u l d b e n o p r o o f o f F e r m a t ' s t h e o r e m . I a m g r a t e f u l t o P r o f e s s o r G e r d F a l t i n g s o f t h e M a x P l a n c k I n s t i t u t e i n B o n n a n d t o P r o f e s s o r G e r h a r d F r e y o f t h e U n i v e r s i t y o f E s s e n , G e r m a n y , f o r p r o v o c a t i v e i n t e r v i e w s a n d i n s i g h t f u l o p i n i o n s . T h a n k s a l s o t o P r o f e s s o r B a r r y M a z u r o f
H a r v a r d U n i v e r s i t y f o r e x p l a i n i n g t o m e i m p o r t a n t c o n c e p t s i n t h e g e o m e t r y o f n u m b e r t h e o r y . A n y e r r o r s t h a t r e m a i n a r e c e r t a i n l y m i n e .
I t h a n k m y p u b l i s h e r , J o h n O a k e s , f o r h i s e n c o u r a g e m e n t a n d s u p p o r t . T h a n k s a l s o t o J i l l E l l y n R i l e y a n d K a t h r y n B e l d e n o f F o u r W a l l s E i g h t W i n d o w s . A n d f i n a l l y , m y d e e p g r a t i t u d e t o m y w i f e , D e b r a .