Yutaka Taniyama
Emoonah McClerklin
Yutaka Taniyama’s Life
Yutaka Taniyama was born the sixth of eight children to Sahei Taniyama and Kaku Taniyama
in Kisai, Japan on November 12th, 1927. However, his name wasnt really Yutaka. Originally born Toyo, his name was commonly misread until he decided to change it to Yutaka
permanently. His father was a locally renowned country doctor, known for giving unsolicited
advice. He had seven siblings in total, two older brothers, three older sisters, one younger
brother, and one younger sister. He was a sickly child and even in his adult life Taniyama
was known to cough every ten to fifteen minutes.
Because he was so sickly, Taniyama missed two years of high school. Eventually, he graduated
and attended the University of Tokyo, majoring in mathematics. When Taniyama graduated
from the University of Tokyo he was older than many of his peers. At the university,
Taniyama began to develop his interest in algebraic number theory which was sparked by
Masao Sugwara. Taniyama also read a selection of books that would shape his career,
including Weil’s Foundations of Algebraic Geometry.
Taniyama graduated from college in March of 1953. He began working as a special research
student for the University. His salary was less than 15,000, which at the time was roughly
42 U.S. dollars. He lived in a small, one bed room apartment. Each floor of his apartment
building had one bathroom, and he had to walk to another building minutes away from his
home to take a bath. Taniyama wasn’t alone. Tokyo was in a part of its history where
everyone was poor. It was almost the norm to live how Taniyama did.
Anyone who knew Taniyama could contest to the fact that he was a strange man. He lived
as many adult males fresh out of college do, rather lazily. For example, he wore the same
outlandish, metallic, blue-green suit every day. He had acquired it from his father, who had
brought the fabric because it was cheap, but couldn’t get anyone else in the family to wear
it.
Taniyama rarely cooked and left his shoe laces untied. He wrote elaborate essays about
discourse that didn’t concern him, and at times could be abrasive. However, Taniyama had
a mathematical mind most did not possess, which allowed him to do great things.
While a graduate student Taniyama took a special interest in abelian number theory. He
wrote On n-division of abelian function fields, an essay similar to a senior thesis. In his
essay he combined knowledge from Hasse and Weil to prove the Mordell-Weil theorem. Goro
Shimura thought him to be the only person who knew the subject in depth. He also published
Jacobian varieties and number fields and L-functions of number fields and zeta functions of
abelian varieties.
Taniyama is famous for the journal Modern Number Theory which he wrote with Shimura.
Taniyama also started a conjecture that would eventually prove Fermat’s Last Theorem.
Before Taniyama could finish the conjecture and see just how his discovery effected the
math world, he killed himself. No one knows why Taniyama decided to take his life. In his
suicide note he admits that he was not sure of the reasoning himself. He says,
“Until yesterday I have had no definite intention of killing myself. But more than a
few must have noticed I have been tired both physically and mentally. As to the cause
of my suicide, I don’t quite understand it myself, but it is not the result of a particular
incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I
lost confidence in my future.”
Taniyama’s life was going well. He had just gotten engaged and had brought a new house
with his fianc, Misako Suzuki, who he affectionately referred to as M.S. Taniyama had chosen
Suzuki despite his parents efforts to match him with other girls. His parents eventually
approved of their relationship and many thought Taniyama and Suzuki to be happy. At the
same time Taniyama’s work was becoming famous. It seemed like he had everything, yet he
ended his life. Stricken by grief, Taniyama’s wife killed herself as well two months after his
suicide. The suicides of Taniyama and his fiance are a great, perplexing, tragedy. Shimura
himself comments on Taniyamas death. He said,
“... he was the moral support of many of those who came into mathematical contact
with him, including of course myself. Probably he was never conscious of this role he
was playing. But I feel his noble generosity in this respect even more strongly now
than when he was alive. And yet nobody was able to give him any support when he
desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief.”
Taniyama was a brilliant man, who made brilliant mistakes which led him to the right
direction. His life touched many and even today he is remembered.
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Yutaka’s mathematical works
Yutaka Taniyama is most famous for the Taniyama-Shimura conjecture which eventually
proved Fermats Last Theorem. The theorem first started to take form at the 1955 Symposium on Algebraic Number Theory hosted in Tokyo. 36 math problems were given to the
particiapnts to solve. Two of those problems posed by Taniyama concerned elliptic curves,
the topic of his conjecture. These two problems would eventually grow into the TaniyamaShimura conjecture. Below, are the problems translated from Japanese to English.
Problem 12. Let C be an elliptic curve defined over an algebraic number field k, and Lc(s)
the L-function of C over k in the sense that:
is the zeta function of c over k. If Hasse’s conjecture is true for Lc(s), then the Fourier
series obtained from Lc(s) by the inverse Mellin transformation must be an automorphic
from of dimension –2 of a special type (see Hecke). If so, it is very plausible that this form is
an elliptic differential of the field of associated automorphic functions. Now, going through
these observations backward, is it possible to prove Hasse’s conjecture by finding a suitable
automorphic form from which Lc(s) can be obtained?
Problem 13. In connection with Problem 12, the following may be set as a problem: to
characterize the field of elliptic modular functions of ’Stude’ N, and espexially to decompose
the Jacobian variety J of this function field into simple factors up to isogeny. Also, it is well
known that if N=q, a prime, q ≡ 3 mod(4), then J contains elliptic curves with complex
multiplicaton. What can one say for general N ?
The conjecture that came from these questions, which is now a theorem, connects typology
and number theory in a waymost never thought. The conjecture states that for every rational
elliptic curve:
There exists non-constant modular functions f(z) and g(z) of the same level N such that:
In short, every rational elliptical curve is also modular.
Though the conjecture was created in the late 50s, its relation to Fermats Last Theorem
wasnt discovered until 1986. Gerhard Frey assumed that Fermat’s Last Theorem was false.
By doing this he found a solution that fulfills
An + B n = C n f orn > 2
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When he checked the properties of the elliptic curves created by his solution he found that
it did not fit the Taniyama-Shimura conjecture. When Frey couldn’t prove his conjecture
he presented his partial results for the math community in hopes that someone could help
him. Ken Ribet eventually proved Frey’s conjecture in 1986, consequently proving that the
Taniyama-Shimura Conjecture and Fermat’s Last Theorem are linked.
Though Ribet connected the conjecture to Fermats Last Theorem, the conjecture wasnt
fully proved until 1995. By the time it was proved, most scholars thought, like Fermats Last
Theorem, it was unsolvable. The theorem was eventually proved by Andre Wiles.
Collaboration with other scholars
Taniyama worked closely with Goro Shimura. The most famous of Taniyamas work was
done alongside Shimura, the two of them sharing and debating the parts of number theory
they know best to come up with their renowned conjecture.
Taniyama first corresponded with Goro Shimura about mathematics in 1954. They were
both students at the university of Tokyo. This correspondence started when Shimura wrote
Taniyama a letter, requesting that Taniyama return The Mathematische Annalen, Vol. 124
to the Universitys library so that Shimura could check it out. As it turns out, both mathematicians were planning to apply the reduction module p of algebraic varieties theory to
elliptic curves.
At first, Shimura had a sort-of negative opinion about Taniyama. He thought that because
Taniyama was in a lower grade, he wasn’t as experienced and smart as Shimura. Shimura
was soon proved wrong.
Working alongside Taniyama, Shimura realized that there was a sort of genius Taniyama
had that no one else did. Shimura credited most of his learning to Taniyama and other
undergrad students. According to Shimura, his professors taught him little yet Taniyama
taught him many things. They spent a lot of time together during their graduate years, in
which they fleshed out most of their work. In this time Shimura and Taniyama wrote Modern
Number Theory. After completing Modern Number Theory Taniyama started to flesh out
the Taniyama-Shimura Conjecture, the conjecture that would eventually enable the proof
of Fermats Last Theorem. However, Taniyama died before he could complete it. Shimura,
driven by a need to repay his friend, finished the conjecture.
Taniyama was also deeply affected by Andr Weil. Weil is an extremely influential French
mathematician, most famous for his work on algebraic number theory and his role in the
infamous mathematical group Nicholas Bourbaki. In college Taniyama read many of Weils
work, including Foundations of Algebraic Geometry, as well as Weils books on algebraic
curves and algebraic varieties.
In 1955 Taniyama attended the Symposium on Algebraic Number Theory that was held in
Tokyo. During that symposium he met with Weil and was able to exchange ideas about the
topic. This meeting with Weil would eventually help shape the conjecture that helped make
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Taniyama famous. Taniyama would use Weils work in a way no one thought to. This new
way of thinking could be applied to Fermats Last Theorem.
Weil also had a hand in making sure Taniyama’s conjecture was known. When Shimura
finally completed the conjecture, its relation to Fermat’s Last Theorem wasnt evident.
Through the work of other mathematicians and Weil, who spread the word of the conjectures importance, the conjecture was eventually related to Fermat’s Last Theorem. If the
Taniyama-Shimura conjecture could be proven, then Fermat’s Last Theorem could be proven.
Because of this and Taniyamas use of Weils work to aid his discovery, some mathematicians
call the conjecture the Taniyama-Shimura-Weil conjecture.
Historical events that marked Taniyama’s life.
In the early 1930s, Japan was going through one of the biggest economic crisis in recent
world history. This was caused by two things. The first cause was the stock market crash
of 1929. In the U.S. the stock market greatly expanded but stock value was inflated. In
September and October of 1929 stock prices began to decline. Investors started to panic.
On October 24th a record amount of shares were traded. This historic day is called Black
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Thursday. The following Friday investors tried to salvage the damage by buying out large
amounts of stock, but by Monday the market was in decline again. The next day the market
crashed completely.
At this time the primary government in Japan was the Minseito, or The English Democratic
Party, a government party that had just recently taken power due to Japan’s economic decline
after WWI. The party appointed their president, Osachi Hamaguchi, as their prime minister.
Osachi’s goal was to improve Japans Western relations, lessen governmental emphasis on
military power, and strengthen the fallen economy. Japan was hit by the stock market crash
of 1929 and found themselves falling in to even more economic decline.
Though Japan was suffering financially, they suffered less than other nations caught in the
economic blast of the stock market crash. In April of 1931 the Minseito attempted to return
Japan to its prewar financial state by using a deflationary policy to discriminate against
weak banks and firms via severe budget cuts. During their tenure, the Minseito party also
strengthened their London Naval Agreement in order to reach their goal of improving Western
relations.
In 1931 the Minseito was scrutinized for their peaceful policies. The party had been interfering in military affairs in order to inforce a peaceful agenda and to save money by cutting
down on military spending. Many people considiered their actions to be treason. Despite
their damaged economy and the Minseito’s attempts to cut down on military power, in the
early 1930s Japan’s Republican party, the Seiyukai party, pushed against legislaton designed
to aid the economy. Instead, they wanted to take part in global colonialism. Like Germany,
the damaged economy caused a rise in ultra-nationailsm.
Ultra-nationalists spread xenophobic, asian-centric ideals that were more representative of a
monarchy than a democracy. In 1930 Oschi was assassinated. The government began to fall
a part. In fall of 1931 Japan’s military acted independently and moved into Manchuria. In
1932 there was an attempted assassinaton on Oschi’s successor.
After Oschi’s assassination the Seiyukai party worked quickly to undo all the progress the
Minseito party had done. In 1932 officers tried to assassinate his successor. From 1932
through 1936 Japan was ruled by admirals. During this time the Greater East Asian CoProsperity Sphere emerged. The ’Greater East Asian Co-Prosperity Sphere” was an ideal
rooted in ultra-nationalism. The plan was to unify east asian countries against the west
by uniting them under the Japanese governement. The plan really meant that east asians
should allow themselves to be conquered by Japan, because it was a better alternative than
being conquered by the West.
In 1937 Japan tried to invade China under the excuse that China attacked first. The invasion
turned into a war, the Japenese army conquering cities as quickly as possible. By December
of 1937 Japan had occupied Shanghai and Nanking, two of China’s most influential cities.
While there, they committedd horrendous war acts.“The Rape of Nanking” refers to the
brutal killing of 300,000 civilians while Japan held the city. By 1940 Japan signed the
Tipartite Pact that strengthened the alliance built between them, Germany and Italy in 1936.
In 1941, America tried to force Japanese troops out of China by delivering an ultimatum.
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When Japan refused to comply, the United States issued an oil embargo on Japan. This act
lead Japan to iniate war on America with the attack on Pearl Harbor.
Significant historical events around the world during Taniyama’s life
Taniyama was born in the midst of one of the greatest political conflicts the world has ever
seen. Born in 1927, Yutaka was only four years old when the Minseito party took over
Japanese government, and nine years old when his country started to pursue the idea of the
Greater East Asian Co Prosperity Sphere, a feat that would have great consequences. He
was born in an era of great poverty, not only for Japan but for other parts of the world as
well.
Meanwhile, other parts of the world were also taking part in colonialism in the name of ethnic
purity. In 1933, When Yutaka was 6 years old, Adolph Hitler took office as Chancellor of
Germany. Like most countries, Germany had fallen into a period of economic poverty. Unlike
Japan, however, the economic downfall of the 1930s hit Germany rather hard. Not only was
Germany in a bleak economic period, they had just lost World War I and suffered with a
weak government. The people of Germany became restless and scared, and believed that
something within their country needed to change. Out of this fear rose Adolph Hitler and
the Nazi Party.
Once Adolph Hitler became Chancellor in 33, he worked swiftly to lead Germany to a
dictatorship. He declared Germany was in a state of emergency, convincing his cabinet
to give the government the power to invade his citizen’s privacy. The government was
now allowed to listen to telephone calls, read emails, and search property without warrant.
Hitlers rhetoric convinced the terrified Germans to strap on uniforms and join the army,
thousands of young men choosing to join Hitler’s oppressive regime. As the 30s went on
Hitler centered his campaign around anti-sematic rhetoric, blaming the Jews for Germanys
economic downfall and promoting the idea of a superior, Aryan race.
In 1936 Germany allied with Italy because of the countries interest in militarized imperialism
and the destruction of soviet communism. A month later Germany allied with Japan under
the same grounds. In the years following, Germany and Japan both did things that would
lead to World War II. Germany continue to invade European countries, ignoring Britain and
Frances warnings to stop. Japan continued to pursue the Greater East Asian Co Prosperity
Sphere. In 1937 Japan invaded China, staring the war in the East. In 1939 Germany invaded
Poland initiating the war in Europe. Italy joined the war when it was clear France was going
to fall.
In the summer of 1940 Germany took it’s first loss in the Battle of Britain. The German
air force was no match for Brittain’s, mainly becuase Germany’s airplanes were not adept
enough. While Germany recovered from its loss Italy invaded Greece and North Africa.
However, Italy failed at occupying Greece and Germany had to rally behind them. In 1941,
America joined the war after the attack on Pearl Harbor. Meanwhile, Germany attempted
to invade the Soviet Union. This would prove to be the end of the war for Germany. The
german troops weren’t prepared for Russia’s harsh winters, which allowed them to be over
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come easily. The Russians chased the Germans out of Russia, and out of all of the other
countries they occupied, up until 1945. On D-day (June 1944) France and Britain started
to push against German forces from the west. Germany was now surrounded by both sides
and surrendered quickly after russia invaded the capital Berlin.
Meanwhile, in the summer of 1942 the United States and Japan began to participate in
heavy naval battles. The U.S. and Japan fought over the Solomon Islands until 1943 when
Japan was forced out. The U.S. then chased them across the Pacific, taking back islands
Japan had formerly conquered. Fighting continued into early 1945. By that time most of the
Japanese colonies had been liberated and Japan was trapped in their homeland – much like
Germany had been. In the summer of 1945 the United States dropped two atomic bomvs
on Hiroshima and Nagasaki. Japan was forced to surrender a few days later.
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Connections between history and the development of mathematics
The major world conflict that occurred during Taniyamas life was World War II. The rapid
invention of new weapons and forms of communication during World War II had a great
effect on math. During World War II there was a large demand for cryptographers. Countries
needed a way to decipher their enemys messages if they wanted to win the war. Japan, as
well as other countries, began to train cryptographers for the war. The most famous example
of this is the British cryptographer, Alan Turig.
Other than the importance of cryptography increasing, World War II did not have a large
effect on math. The increasing need for cryptographers did not have any large effect on
Taniyamas life.
References
1. http://self.gutenberg.org/articles/Rikken Minseito
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2. http://www.history.com/topics/1929-stock-market-crash
3. http://wgordon.web.wesleyan.edu/papers/coprospr.htm
4. http://shayfam.com/David/flt/flt8.htm
5. http://mathworld.wolfram.com/Taniyama-ShimuraConjecture.html
6. http://www.storyofmathematics.com/20th weil.html
7. http://www-history.mcs.st-and.ac.uk/Biographies/Taniyama.html
8. http://www.biographybase.com/biography/Taniyama Yutaka.html
9. https://www.jstor.org/stable/2320947?seq=1page scan tab contents
10. http://www.history.com/topics/world-war-ii/hirohito
11. http://blms.oxfordjournals.org/content/21/2/186.full.pdf
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