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**http://www-history.mcs.st-andrews.ac.uk/😀**

**Article by:** *J J O’Connor* and *E F Robertson*

** ****Kenkichi Iwasawa** attended elementary school in the town of his birth but went to Tokyo for his high school studies which were at the Musashi High School. In 1937 he entered Tokyo University where he was taught by Shokichi Shokichi Iyanaga and Zyoiti Suetuna. At this time Tokyo University had become a centre for the study of algebraic number theory as a result of Teiji Takagi’s remarkable contributions. Takagi had retired in 1936, the year before Iwasawa began his studies, but his students Iyanaga and Suetuna were bringing to the university many ideas which they had developed during studies with the leading experts in Europe.

Iwasawa graduated in 1940 and remained at Tokyo University to undertake graduate studies. He also was employed as an Assistant in the Mathematics Department. Although the great tradition in number theory at Tokyo inspired him to an interest in that topic, some of his early research contributions were to group theory. The Second World War disrupted life in Japan and essentially ended Suetuna’s research career. Iyanaga did not fare much better. He wrote:-

… towards the end of the War, Tokyo and other Japanese cities were often bombarded and we had to find refuge in the countryside. Everyone was mobilised in one way or another for the War.

Clearly Iwasawa found this a most difficult period in which to try to complete work for his doctorate. However, despite the difficulties he succeeded brilliantly, and was awarded the degree of Doctor of Science in 1945. It was not without a high cost, however, for after being awarded his doctorate he became seriously ill with pleurisy and this prevented him returning to the University of Tokyo until April 1947.

For a glimpse of the research that Iwasawa undertook at this time we look briefly at the paper *On some types of topological groups* which he published in the *Annals of Mathematics* in 1949. Iwasawa’s results are related to Hibert’s fifth problem which asks whether any locally Euclidean topological groups is necessarily a Lie group. In his 1949 paper Iwasawa gives what is now known as the ‘Iwasawa decomposition’ of a real semisimple Lie group. He gave many results concerning Lie groups, proving in particular that if a locally compact group G has a closed normal subgroup N such that N and G/N are Lie groups then G is a Lie group.

In 1950 Iwasawa was invited to give an address at the International Congress of Mathematicians in Cambridge, Massachusetts. He then received an invitation to the Institute for Advanced Study at Princeton and he spent two years there from 1950 until 1952. Artin was at the Institute during Iwasawa’s two years there and he was one of the main factors in changing the direction of Iwasawa’s research to algebraic number theory. In 1952 Iwasawa published *Theory of algebraic functions* in Japanese. The book begins with an historical survey of the theory of algebraic functions of one variable, from analytical, algebraic geometrical, and algebro-arithmetical view points. Iwasawa then studies valuations, fields of algebraic functions giving definitions of prime divisors, ideles, valuation vectors and genus. A proof of the Riemann-Roch theorem is given, and the theory of Riemann surfaces and their topology is studied.

It was Iwasawa’s intention to return to Japan in 1952 after his visit to the Institute for Advanced Study but when he received the offer of a post of assistant professor at the Massachusetts Institute of Technology he decided to accept it. Coates, in [2], describes the fundamental ideas which Iwasawa introduced that have had such a fundamental impact on the development of mathematics in the second half of the 20^{th} century. Iwasawa introduced:-

… a general method in arithmetical algebraic geometry, known today as Iwasawa theory, whose central goal is to seek analogues for algebraic varieties defined over number field of the techniques which have been so successfully applied to varieties defined over finite fields by H Hasse, A Weil, B Dwork, A Grothendieck, P Deligne, and others. … The dominant theme of his work in number theory is his revolutionary idea that deep and previously inaccessible information about the arithmetic of a finite extension F of Q can be obtained by studying coarser questions about the arithmetic of certain infinite Galois towers of number fields lying above F.

Iwasawa first lectured on his revolutionary ideas at the meeting of the American Mathematical Society in Seattle, Washington in 1956. The ideas were taken up immediately by Serre who saw their great potential and gave lectures to the Seminaire Bourbaki in Paris on Iwasawa theory. Iwasawa himself produced a series of deep papers throughout the 1960s which pushed his ideas much further. R Greenberg, who became a student of Iwasawa’s in 1967 wrote:-

By the time that I became his student, Professor Iwasawa had developed his ideas considerably. The theory had become richer, and at the same time, more mysterious. Even though only a few mathematicians had studied the theory thoroughly at that time, there was a general feeling that the theory was very promising. When I look back at the developments that have taken place in the last three decades, that promise has been fulfilled even beyond expectations.

In 1967 Iwasawa left MIT when he was offered the Henry Burchard Fine Chair of Mathematics at Princeton and it was not long after he arrived there that he took on Greenberg as a research student. We learn a lot about Iwasawa if we look at Greenberg’s description of how Iwasawa supervised his studies:-

It was the tradition at Princeton to have tea every afternoon in fine Hall. This provided one of the best opportunities for graduate students to informally discuss mathematics with their professors. Professor Iwasawa usually came to the afternoon teas. It was then that he often suggested problems for me to think about and every few weeks he would ask me if I had made any progress on some of these problems. I recall that these problems seemed quite hard, but sometimes I was able to report some real progress, and then we would go to his office so that he could hear what I had done. He would help me push some of my ideas forward, but it was quite clear that he wanted me to accomplish as much as I could on my own. I often had the feeling that he was purposely not revealing everything that he knew about a specific problem.

In the late 1960s Iwasawa made a conjecture for algebraic number fields which, in some sense, was the analogue of the relationship which Weil had found between the zeta function and the divisor class group of an algebraic function field. This conjecture became known as “the main conjecture on cyclotomic fields” and it remained one of the most outstanding conjectures in algebraic number theory until it was solved by Mazur and Wiles in 1984 using modular curves.

Iwasawa remained as Henry Burchard Fine Professor of mathematics at Princeton until he retired in 1986. Then he returned to Tokyo where he spent his final years. He published *Local class field theory* in the year that he retired:-

This carefully written monograph presents a self-contained and concise account of the modern formal group-theoretic approach to local class field theory.

Iwasawa was much honoured for his achievements. He received the Asahi Prize (1959), the Prize of the Academy of Japan (1962), the Cole Prize from the American Mathematical Society (1962), and the Fujiwara Prize (1979).

The importance of his work is summed up by Coates [2]:-

… today it is no exaggeration to say that Iwasawa’s ideas have played a pivotal role in many of the finest achievements of modern arithmetical algebraic geometry on such questions as the conjecture of B Birch and H Swinnerton-Dyer on elliptic curve; the conjecture of B Birch, J Tate, and S Lichtenbaum on the orders of the K-groups of the rings of integers of number fields; and the work of A Wiles on the modularity of elliptic curves and Fermat’s Last Theorem.