Yutaka TaniyamaEmoonah McClerklin

Yutaka Taniyama’s Life

Yutaka Taniyama was born the sixth of eight children to Sahei Taniyama and Kaku Taniyamain Kisai, Japan on November 12th, 1927. However, his name wasnt really Yutaka. Origi-nally born Toyo, his name was commonly misread until he decided to change it to Yutakapermanently. His father was a locally renowned country doctor, known for giving unsolicitedadvice. He had seven siblings in total, two older brothers, three older sisters, one youngerbrother, and one younger sister. He was a sickly child and even in his adult life Taniyamawas known to cough every ten to fifteen minutes.

Because he was so sickly, Taniyama missed two years of high school. Eventually, he graduatedand attended the University of Tokyo, majoring in mathematics. When Taniyama graduatedfrom 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 byMasao 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 researchstudent for the University. His salary was less than 15,000, which at the time was roughly42 U.S. dollars. He lived in a small, one bed room apartment. Each floor of his apartmentbuilding had one bathroom, and he had to walk to another building minutes away from hishome to take a bath. Taniyama wasn’t alone. Tokyo was in a part of its history whereeveryone 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 livedas many adult males fresh out of college do, rather lazily. For example, he wore the sameoutlandish, metallic, blue-green suit every day. He had acquired it from his father, who hadbrought the fabric because it was cheap, but couldn’t get anyone else in the family to wearit.

Taniyama rarely cooked and left his shoe laces untied. He wrote elaborate essays aboutdiscourse that didn’t concern him, and at times could be abrasive. However, Taniyama hada 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. Hewrote On n-division of abelian function fields, an essay similar to a senior thesis. In hisessay he combined knowledge from Hasse and Weil to prove the Mordell-Weil theorem. GoroShimura thought him to be the only person who knew the subject in depth. He also publishedJacobian varieties and number fields and L-functions of number fields and zeta functions ofabelian 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 themath 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 afew must have noticed I have been tired both physically and mentally. As to the causeof my suicide, I don’t quite understand it myself, but it is not the result of a particularincident, nor of a specific matter. Merely may I say, I am in the frame of mind that Ilost confidence in my future.”

Taniyama’s life was going well. He had just gotten engaged and had brought a new housewith his fianc, Misako Suzuki, who he affectionately referred to as M.S. Taniyama had chosenSuzuki despite his parents efforts to match him with other girls. His parents eventuallyapproved of their relationship and many thought Taniyama and Suzuki to be happy. At thesame time Taniyama’s work was becoming famous. It seemed like he had everything, yet heended his life. Stricken by grief, Taniyama’s wife killed herself as well two months after hissuicide. The suicides of Taniyama and his fiance are a great, perplexing, tragedy. Shimurahimself comments on Taniyamas death. He said,

“... he was the moral support of many of those who came into mathematical contactwith him, including of course myself. Probably he was never conscious of this role hewas playing. But I feel his noble generosity in this respect even more strongly nowthan when he was alive. And yet nobody was able to give him any support when hedesperately 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 rightdirection. His life touched many and even today he is remembered.

2

Yutaka’s mathematical works

Yutaka Taniyama is most famous for the Taniyama-Shimura conjecture which eventuallyproved Fermats Last Theorem. The theorem first started to take form at the 1955 Sympo-sium on Algebraic Number Theory hosted in Tokyo. 36 math problems were given to theparticiapnts 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 Taniyama-Shimura 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 Fourierseries obtained from Lc(s) by the inverse Mellin transformation must be an automorphicfrom of dimension –2 of a special type (see Hecke). If so, it is very plausible that this form isan elliptic differential of the field of associated automorphic functions. Now, going throughthese observations backward, is it possible to prove Hasse’s conjecture by finding a suitableautomorphic form from which Lc(s) can be obtained?

Problem 13. In connection with Problem 12, the following may be set as a problem: tocharacterize the field of elliptic modular functions of ’Stude’ N, and espexially to decomposethe Jacobian variety J of this function field into simple factors up to isogeny. Also, it is wellknown that if N=q, a prime, q ≡ 3 mod(4), then J contains elliptic curves with complexmultiplicaton. What can one say for general N ?

The conjecture that came from these questions, which is now a theorem, connects typologyand number theory in a waymost never thought. The conjecture states that for every rationalelliptic 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 Theoremwasnt discovered until 1986. Gerhard Frey assumed that Fermat’s Last Theorem was false.By doing this he found a solution that fulfills

An + Bn = Cnforn > 2

3

When he checked the properties of the elliptic curves created by his solution he found thatit did not fit the Taniyama-Shimura conjecture. When Frey couldn’t prove his conjecturehe presented his partial results for the math community in hopes that someone could helphim. Ken Ribet eventually proved Frey’s conjecture in 1986, consequently proving that theTaniyama-Shimura Conjecture and Fermat’s Last Theorem are linked.

Though Ribet connected the conjecture to Fermats Last Theorem, the conjecture wasntfully proved until 1995. By the time it was proved, most scholars thought, like Fermats LastTheorem, 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 wasdone alongside Shimura, the two of them sharing and debating the parts of number theorythey know best to come up with their renowned conjecture.

Taniyama first corresponded with Goro Shimura about mathematics in 1954. They wereboth students at the university of Tokyo. This correspondence started when Shimura wroteTaniyama a letter, requesting that Taniyama return The Mathematische Annalen, Vol. 124to the Universitys library so that Shimura could check it out. As it turns out, both math-ematicians were planning to apply the reduction module p of algebraic varieties theory toelliptic curves.

At first, Shimura had a sort-of negative opinion about Taniyama. He thought that becauseTaniyama was in a lower grade, he wasn’t as experienced and smart as Shimura. Shimurawas soon proved wrong.

Working alongside Taniyama, Shimura realized that there was a sort of genius Taniyamahad that no one else did. Shimura credited most of his learning to Taniyama and otherundergrad students. According to Shimura, his professors taught him little yet Taniyamataught him many things. They spent a lot of time together during their graduate years, inwhich they fleshed out most of their work. In this time Shimura and Taniyama wrote ModernNumber Theory. After completing Modern Number Theory Taniyama started to flesh outthe Taniyama-Shimura Conjecture, the conjecture that would eventually enable the proofof 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 Frenchmathematician, most famous for his work on algebraic number theory and his role in theinfamous mathematical group Nicholas Bourbaki. In college Taniyama read many of Weilswork, including Foundations of Algebraic Geometry, as well as Weils books on algebraiccurves and algebraic varieties.

In 1955 Taniyama attended the Symposium on Algebraic Number Theory that was held inTokyo. During that symposium he met with Weil and was able to exchange ideas about thetopic. This meeting with Weil would eventually help shape the conjecture that helped make

4

Taniyama famous. Taniyama would use Weils work in a way no one thought to. This newway of thinking could be applied to Fermats Last Theorem.

Weil also had a hand in making sure Taniyama’s conjecture was known. When Shimurafinally 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 con-jectures importance, the conjecture was eventually related to Fermat’s Last Theorem. If theTaniyama-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 mathematicianscall 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 recentworld history. This was caused by two things. The first cause was the stock market crashof 1929. In the U.S. the stock market greatly expanded but stock value was inflated. InSeptember 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

5

Thursday. The following Friday investors tried to salvage the damage by buying out largeamounts of stock, but by Monday the market was in decline again. The next day the marketcrashed completely.

At this time the primary government in Japan was the Minseito, or The English DemocraticParty, a government party that had just recently taken power due to Japan’s economic declineafter WWI. The party appointed their president, Osachi Hamaguchi, as their prime minister.Osachi’s goal was to improve Japans Western relations, lessen governmental emphasis onmilitary power, and strengthen the fallen economy. Japan was hit by the stock market crashof 1929 and found themselves falling in to even more economic decline.

Though Japan was suffering financially, they suffered less than other nations caught in theeconomic blast of the stock market crash. In April of 1931 the Minseito attempted to returnJapan to its prewar financial state by using a deflationary policy to discriminate againstweak banks and firms via severe budget cuts. During their tenure, the Minseito party alsostrengthened their London Naval Agreement in order to reach their goal of improving Westernrelations.

In 1931 the Minseito was scrutinized for their peaceful policies. The party had been inter-fering in military affairs in order to inforce a peaceful agenda and to save money by cuttingdown on military spending. Many people considiered their actions to be treason. Despitetheir damaged economy and the Minseito’s attempts to cut down on military power, in theearly 1930s Japan’s Republican party, the Seiyukai party, pushed against legislaton designedto 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 amonarchy than a democracy. In 1930 Oschi was assassinated. The government began to falla part. In fall of 1931 Japan’s military acted independently and moved into Manchuria. In1932 there was an attempted assassinaton on Oschi’s successor.

After Oschi’s assassination the Seiyukai party worked quickly to undo all the progress theMinseito party had done. In 1932 officers tried to assassinate his successor. From 1932through 1936 Japan was ruled by admirals. During this time the Greater East Asian Co-Prosperity Sphere emerged. The ’Greater East Asian Co-Prosperity Sphere” was an idealrooted in ultra-nationalism. The plan was to unify east asian countries against the westby uniting them under the Japanese governement. The plan really meant that east asiansshould allow themselves to be conquered by Japan, because it was a better alternative thanbeing conquered by the West.

In 1937 Japan tried to invade China under the excuse that China attacked first. The invasionturned into a war, the Japenese army conquering cities as quickly as possible. By Decemberof 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 thebrutal killing of 300,000 civilians while Japan held the city. By 1940 Japan signed theTipartite 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.

6

When Japan refused to comply, the United States issued an oil embargo on Japan. This actlead 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 everseen. Born in 1927, Yutaka was only four years old when the Minseito party took overJapanese government, and nine years old when his country started to pursue the idea of theGreater East Asian Co Prosperity Sphere, a feat that would have great consequences. Hewas born in an era of great poverty, not only for Japan but for other parts of the world aswell.

Meanwhile, other parts of the world were also taking part in colonialism in the name of ethnicpurity. In 1933, When Yutaka was 6 years old, Adolph Hitler took office as Chancellor ofGermany. Like most countries, Germany had fallen into a period of economic poverty. UnlikeJapan, however, the economic downfall of the 1930s hit Germany rather hard. Not only wasGermany in a bleak economic period, they had just lost World War I and suffered with aweak government. The people of Germany became restless and scared, and believed thatsomething within their country needed to change. Out of this fear rose Adolph Hitler andthe Nazi Party.

Once Adolph Hitler became Chancellor in 33, he worked swiftly to lead Germany to adictatorship. He declared Germany was in a state of emergency, convincing his cabinetto give the government the power to invade his citizen’s privacy. The government wasnow 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 onHitler centered his campaign around anti-sematic rhetoric, blaming the Jews for Germanyseconomic downfall and promoting the idea of a superior, Aryan race.

In 1936 Germany allied with Italy because of the countries interest in militarized imperialismand the destruction of soviet communism. A month later Germany allied with Japan underthe same grounds. In the years following, Germany and Japan both did things that wouldlead to World War II. Germany continue to invade European countries, ignoring Britain andFrances warnings to stop. Japan continued to pursue the Greater East Asian Co ProsperitySphere. In 1937 Japan invaded China, staring the war in the East. In 1939 Germany invadedPoland initiating the war in Europe. Italy joined the war when it was clear France was goingto fall.

In the summer of 1940 Germany took it’s first loss in the Battle of Britain. The Germanair force was no match for Brittain’s, mainly becuase Germany’s airplanes were not adeptenough. 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 attemptedto invade the Soviet Union. This would prove to be the end of the war for Germany. Thegerman troops weren’t prepared for Russia’s harsh winters, which allowed them to be over

7

come easily. The Russians chased the Germans out of Russia, and out of all of the othercountries they occupied, up until 1945. On D-day (June 1944) France and Britain startedto push against German forces from the west. Germany was now surrounded by both sidesand surrendered quickly after russia invaded the capital Berlin.

Meanwhile, in the summer of 1942 the United States and Japan began to participate inheavy naval battles. The U.S. and Japan fought over the Solomon Islands until 1943 whenJapan was forced out. The U.S. then chased them across the Pacific, taking back islandsJapan had formerly conquered. Fighting continued into early 1945. By that time most of theJapanese colonies had been liberated and Japan was trapped in their homeland – much likeGermany had been. In the summer of 1945 the United States dropped two atomic bomvson Hiroshima and Nagasaki. Japan was forced to surrender a few days later.

8

Connections between history and the development of mathematics

The major world conflict that occurred during Taniyamas life was World War II. The rapidinvention of new weapons and forms of communication during World War II had a greateffect on math. During World War II there was a large demand for cryptographers. Countriesneeded a way to decipher their enemys messages if they wanted to win the war. Japan, aswell as other countries, began to train cryptographers for the war. The most famous exampleof this is the British cryptographer, Alan Turig.

Other than the importance of cryptography increasing, World War II did not have a largeeffect on math. The increasing need for cryptographers did not have any large effect onTaniyamas life.

References

1. http://self.gutenberg.org/articles/Rikken Minseito

9

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

10