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. Origi- nally 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. 2 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 Sympo- sium 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 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 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 + Bn = Cnforn > 2 3 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 math- ematicians 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.