Itaka, the chief editor of Sugaku, and Nakajima, an editor of Sugaku visited Professor Iwasawa's house at 2 PM on March 18th, 1993. It is located in a quiet residential area in Meguro, Tokyo. We also asked Fujisaki to visit with us. We talked with Professor Iwasawa about many topics.

Iwasawa: First of all, I will recall some of my memories. I was born in 1917 (the 6-th year of Taisho in the Japanese calendar) and entered the University of Tokyo in 1937. In those days, the curriculum of the University was for three years, and I graduated in 1940. After that, I entered the graduate school of the University of Tokyo and became an assistant. In December, 1941 the Pacific War broke out, and it became more and more difficult to study calmly. In March, 1945 our mathematics department was evacuated to Suwa (Nagano prefecture), with the help of Professor Kodaira's father. On August 15th the war finally ended, and we could come back to Hongo, Tokyo. But at that time I was suffering with a serious disease. It was pleurisy. I had been in bed every day with a high fever. In those days, we could not get enough medicine, and even enough food, so many people around me were seriously worried about me. Fortunately, I found a good doctor, and almost recovered by the autumn of 1946. I was advised to take a rest for a while and came back to the university in April,1947.

I was interested in group theory from the time when I was a student, and studied a book by Zassenhaus (Gruppentheorie) at our seminar with the help of Professors Suetsuna and Iyanaga. This was a very good book. At the same time, Topological groups by Pontryagin was translated from Russian into English, and published by Princeton University Press. After reading this book, I became interested in topological groups and started working on locally compact groups. I wrote a relatively long paper (long for me), and showed it to Professor Iyanaga, who sent it to Chevalley. Chevalley kindly wrote to me that I did very well, and also pointed out that some arguments were vague. I was very happy. This paper ¹ was published in the Annals of Mathematics in 1949. I think this was also possible with Chevalley's assistance.

In 1950, the first ICM (International Congress of Mathematicians) after the war was held at Harvard University and MIT in Cambridge, Massachusetts. I was invited to give a talk about the topic of my paper at the session on topological groups. In those days, only 5 years after the war, daily life itself was generally hard to live. I wondered whether I had suitable clothes to attend such a conference, even though I was invited.

At almost the same time that I received the letter of invitation from the ICM, I also received a letter of invitation from Professor Morse at the Institute for Advanced Study. Professor Morse knew that I would be attending the ICM to give a talk, and invited me to be a temporary member of IAS after the ICM, and enclosed an application form with his letter. Kakutani stayed at IAS two years before, and Kodaira was there at that time. Tadashi Nakayama also had stayed there before the war. I was told that IAS was a very good place for research. So I had hoped to visit there. Of course, I returned the application form.

In the letter from Professor Morse, he told me the amount of money for one year as a member of IAS. I received the amount in advance, and prepared to go abroad. But I had no preparation in English conversation. I was worried about giving a lecture in English there.

So I decided to go to the United States a few months before the ICM. Though the ICM began in the end of August, I left Japan in June by ship, and landed in San Francisco. I met Kodaira there. He came to San Francisco with Tomonaga who was returning to Japan. Both Kodaira and Tomonaga came to the IAS one year before, but Tomonaga stayed only for one year. It was fortunate that I could meet Kodaira there. We took an airplane together from San Francisco to Chicago. But the route was unusual. For example, we went to El Paso, Texas on the way to Chicago. I met Weil in Chicago.

I had to work hard for Weil. More frankly, Weil pushed me hard. I had been in Chicago for more than one month. In those days, Harvard and MIT were centers of mathematics, and Chicago was constructing a new center of mathematics. Stone, Chern and MacLane were in Chicago. Among younger researchers, Kaplansky, Segal and, of course, Weil were there. In addition to the usual seminars, Weil had a seminar on the history of mathematics. I had never attended such a seminar in Japan. I attended the seminar. Chern also attended the seminar. They were studying some French geometers, such as Monge and Poncelet. Weil told each graduate student to read a paper by Poncelet from his Collected Papers, and to explain it at the seminar. Chern also explained some papers. I knew Weil was a great mathematician, but was also surprised at his deep study of the history of mathematics. I felt that Weil was really great. This was just after Weil finished writing his book ² on the Riemann hypothesis for algebraic curves over finite fields.

There is a four or six lane road for cars along Lake Michigan. Weil could easily cross the road, slipping through the high-speed cars and getting to the shore. He often told us to come after him. But it was very dangerous for us, and we could not cross the road as Weil did.

Weil told me that fiber spaces would become important in mathematics, and suggested some books for me to read. Weil also kindly attended the rehearsal for my talk at the ICM.

After Chicago, I went to Princeton with Kodaira, chose my accommodation, and went to Cambridge. At the end of August, the ICM began. I felt like a man from the countryside visiting a city for the first time. Too many people were talking noisily between the lectures, and I could not understand their talk at all. I could not understand the English of the lectures well either. I lectured about topological groups as an invited speaker, but I guess that the audience could not understand my English either. I also applied for a short talk of ten minutes, and explained how Hecke's L-functions can be expressed as integrals on idele groups ³ . Just after my talk, Artin came to me, and told me that one of his students was doing essentially the same thing. It was Tate.

Of course, I knew Artin's name when I was in Japan from his papers. Professors Suetsuna and Iyanaga had also told me about Artin when I was in Japan. He made a good impression on me. He was tall, broad, unpretentious, and frank. He seemed to say exactly what he thought. He was also very kind to me. I was very happy that Artin mentioned my work in his special lecture at the ICM. I met Tate at his lecture. Lang might have been there too.

After the ICM, I went back to Princeton. Kodaira had been in Princeton, but he was not there that year, because he was teaching at Johns Hopkins for a year. Kentaro Yano also came to IAS at that time. Yano had lived in Paris before the war, and knew well about the life style in Europe. Though the life style in Europe is slightly different from that in the States I could learn many things from him about the Western life style, which was fortunate for me. At that time, Veblen, Morse, Weyl, Siegel, and Goedel were in the Mathematics Department at IAS, and, of course, Einstein was there. Among younger researchers, Montgomery and Selberg were there. Everyone was a remarkable person, and I liked Princeton very much. There was no obligation for members of the Institute. I liked the informal and family-like atmosphere of the Institute, and the size, which was not so big. I felt quite comfortable at the Institute.

I had been at IAS for two years. My work concerned topological groups, and so it was related with that of Montgomery. But he was busy with Hilbert's 5-th problem, and studying with Zippin. I talked with Montgomery, but I usually spent my time on a seminar of Artin at Princeton University, which was quite stimulating.

I was interested in number theory from when I was a student. After talking with Weil in Chicago and Artin at the ICM, I was tempted to study number theory more.

You can read about the atmosphere at IAS in those days in Kodaira's book Diary of an Idle Mathematician (in Japanese, published by Iwanami). It contains what I forgot.

I will talk about Einstein. Einstein's house was on Mercer Street, which is in between the town and the Institute. Usually he walked to the Institute. But on rainy days he took the free shuttle of the Institute, and I had a chance to talk with him because I lived in the town and used the shuttle every day. I talked with him about a lot of topics. What I remember the most is about the atomic bomb. It concerned Einstein very much, and he asked me what the Japanese thought about the atomic bombs dropped on Nagasaki and Hiroshima.

In the spring of 1952, I was preparing to go back to Japan. In those days, it was very difficult for us to go abroad. I wanted to see Europe. My plan was not for Mathematics, but just for real sightseeing. I reserved a ticket on a ship from Marseilles to Yokohama, and went to New York to get a visa to visit Europe. But in the middle of April, Professor Martin, who was the chairman of MIT, asked me if I could stay in the United States and come to MIT. I had already decided to go back to Japan. So I was confused and asked Montgomery, who was very dependable and, like a parent, gave me advice. I also asked the young mathematician Kadison, who later became one of the leading experts on operator algebras. He was very kind to me. I recently read his article in the Bulletin (of the AMS) in which he reviewed a book by Shoichiro Sakai. It was a very carefully written and kind review, and reminded me of those days in Princeton. I miss those days. Kadison was younger than me, but thought about my problem for a long time. His advice was that I should stay in the United States to study for some time. I finally decided to stay in the United States a little more, and went to MIT. My original plan was to stay for one or two years. But the result was to stay until 1967.

Professor Weiner must have been the most famous at MIT in those days. Martin (function theory in several variables) , Levinson (function theory, he has good results on the Riemann hypothesis), Ambrose, Whitehead and Hurewicz (one of the founders of homotopy theory) were also there. MIT had a strong group in topology. So it was said that MIT means Massachusetts Institute of Topology. There were a lot of stories about Weiner. When Weiner bought a new house and moved, he went back to his old house. The door was locked, and he told a girl with white clothes near the house that he was the owner of the house, but could not enter inside. Then she said that her mother had told her to come here to wait for her father, and to go home together with him. This means that he forgot not only his new house, but also his daughter. Kakutani told me not to talk about mathematics to Wiener in his car, because he explained his ideas by using both hands even when he was driving, and so it was very dangerous.

In 1967 I moved to Princeton, and stayed there until 1986. I came back to Japan in 1987. So I had been in Princeton for 19 years. This is what I remember. Let's take a rest.

Editors: Tell us something about the time when you wrote your book Theory of Algebraic Functions ?

Iwasawa: That book was one of the new series published by Iwanami. I was asked to write it by Professor Iyanaga. I remember that I wrote it before I went abroad, but the footnotes were written in the United States.

Editors: It was published in 1952?

Fujisaki: Topological Analysis, by Professor Yoshida, was published as a book of the series at the same time as Theory of Algebraic Functions. I thought many books would be published in succession. I ordered in advance the whole series at the COOP, and prepaid the price for one book, because doing so made the price discounted. But the next book had not been published for a long time, and the COOP forgot my order. So I lost my money.

Editors: Tell us about your work on L-functions, which you talked about at the ICM.

Iwasawa: I had an interest in number theory from the time when I was a student. I studied it by myself. When Chevalley invented the idele group, its topology was different from the modern topology. It was not Hausdorff, and the closure of the identity was the connected component. Artin also said that this topology was a "falsche Topologie". If we adopt the modern topology and study Fourier analysis, then we obtain the functional equations of L-functions. This is essentially the same as Tate's work. The functional equation is interesting, but I prefer the fact that the finiteness of the volume of a certain quotient of the idele group (which is equivalent to the finiteness of the ideal class group together with Dirichlet's unit theorem) can be deduced from the convergence of the zeta function.

Editors: What is the motivation of Iwasawa theory, especially the theory of Z_p-extensions?

Iwasawa: In about 1950, cohomology theory was brought to class field theory by Nakayama and Hochschild. This work has great significance. After that, Artin gave a series of lectures on class field theory in Princeton from 1950 to 1952. In the first year, he talked about the local theory, and the content was published in his book Algebraic Numbers and Algebraic Functions. In the second year, Artin discussed the global theory. This was later developed and published as the famous book with Tate, Class Field Theory. Even now, I think I was very fortunate that I could attend his lectures. The content of his lectures was of course excellent, but, more than that, it was very good for me to learn his way of thinking and his method of doing mathematics.

In those days, many good mathematicians studied cohomology. So I tried to study a different topic and chose the theory of cyclotomic fields. I thought that cyclotomic fields were important because they are also related to elementary number theory. When I stayed in Princeton from 1950 to 1952, I wrote a paper on the characterization of the adele ring (which was called the ring of valuation vectors at that time). I learned about p-adic L-functions ⁴ after I moved to Princeton in 1967, where Dwork and Washnitzer were studying them⁵. I think the direction of my study was sometimes decided by accidental reasons.

Editors: We thought the motivation of Iwasawa theory was an analogy with Jacobian varieties for number fields. Isn't this correct?

Iwasawa: No. I didn't write it in that way. I couldn't see everything from the start. After I defined the lambda and mu invariants, I noticed the analogy because the structure of ideal class groups and the structure of the torsion points of Jacobian varieties are similar if the mu invariants vanish. For an algebraic curve, the mu invariant is always zero, but it is not always so for a number field. This seems related to the fact that an algebraic curve is projective, but a number field is not (the influence of archimedean primes) .

Editors: If your starting point was not to look for an analogy with Jacobian varieties, why did you notice that there is a good theory when we study all the fields of pⁿ-th roots of unity systematically?

Iwasawa: Anyone would naturally notice it if he studied this kind of object (the theory of cyclotomic fields).

Editors: Is it really true?

We considered the p-part A of the ideal class group of the field of p-th roots of unity. For the study of A, we need class field theory and also Kummer theory. If every element of A has order p, there is no problem. But there is a possibility that A has an element of order p². Then, we have to study the field of p²-th roots of unity to apply Kummer theory. In the same way, we need p³-th roots of unity, etc. Naturally, we need fields with all p-power roots of unity.

Editors: Tell us about your students when you were in the United States.

Iwasawa: When I was at MIT, Mattson (computer science), Schue (Lie algebras), Hamara (from Finland) and Knee (integral representations) were my students. In Princeton, Greenberg, Washington and Ferrero were my students. I had a seminar with de Shalit in my last year in Princeton. Besides them, at MIT, I was the advisor of W. Browder and B. Mazur for their undergraduate thesis.

Mrs. Iwasawa kindly baked very tasty cakes for us. They were really tasty and sweet. So we ate many cakes and drank many cups of tea. As a result, we forgot the questions which we prepared. After this interview, we found several questions we should have asked, but we will stop here.

(from Sugaku, Vol. 45, No.4, 366-372, October 1993)


Footnotes (which were included in the text itself as notes from the editors.):

1. This is the paper On some types of topological groups which treats the topics related to Hilbert's 5-th problem. The "Iwasawa decomposition" was introduced in this paper.

2.Courbes algebriques et varietes abeliennes.

3. Now called "the method of Iwasawa and Tate". In the proceedings of the ICM, one can find an article A note on L-functions by Iwasawa.

4. which plays an important role in Iwasawa's main conjecture

5. Fujisaki later heard from Iwasawa that this part was not correct. Iwasawa said that he found his notebook for his lectures in Princeton, which were given the same year that he moved there. The content of the notebook was about p-adic L-functions. So this means that Iwasawa knew about them before he moved to Princeton.



Translated by Masato Kurihara with assistance from Ralph Greenberg.