Memories of Professor Iwasawa

I still remember very clearly the first time that I met Professor Iwasawa.
It was in 1967 when he had just become a faculty member at Princeton University.
I was a second year graduate student and had decided that I was ready to
take the General Examination. Students were not told in advance which
faculty members were to be on their examining committee. I had hoped that
Professor Iwasawa would be on my committee and, on the day of the examination,
when I was told that the committee was waiting for me in his office, I
knew that my hope would be fulfilled. It was at that time that I
first met him.

About a week after my General Examination, I came to Professor Iwasawa's
office again to ask him if I could become his Ph.D. student. I felt rather
nervous because, although I did fairly well in most of the examination,
my answer to one of his questions about Algebraic Number Theory was
rather poor. But nevertheless he did say yes, and I then began to
study what was referred to at that time as the theory of Gamma-extensions.
Iwasawa had introduced some of the basic ideas of that theory just eleven
years earlier. His first lecture about the subject was at the summer meeting
of the American Mathematical Society in 1956, which took place in Seattle,
Washington. Just a few years after that lecture, the French mathematician
Jean-Pierre Serre gave an exposition of Iwasawa's theory at the Seminaire
Bourbaki in Paris. 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 past three decades, that promise has been
fulfilled even beyond expectations.

Since the beginning of the 20th century, one of the continuing themes which
motivated Algebraic Number Theory was the analogy between algebraic number
fields and algebraic function fields. This theme was behind the work
of many great number theorists. I believe that one reason that Iwasawa's
theory of Gamma-extensions seemed so promising was that it suggested a
much deeper aspect of this analogy. It was already clear in some
of Iwasawa's first papers on the topic that this analogy was very much
in his mind, and by the late 1960s, he was able to formulate a very
remarkable conjecture which pushed this analogy much further. In the theory
of algebraic function fields, one of the famous and important theorems,
which was discovered by Andre Weil in the 1940s, states that there is a
very precise relationship between the zeta function and the divisor class
group of an algebraic function field. It was undoubtedly hard to
imagine how to even formulate an analogous result for algebraic number
fields, but Iwasawa's theory provided exactly the right framework to make
that possible. Iwasawa's Conjecture asserts that there should be a precise
relationship between two analogous objects associated with a large class
of algebraic number fields: the "p-adic zeta function" which had
recently been invented by Tomio Kubota and Heinrich Leopoldt and
the "Iwasawa Module" which was constructed from ideal class groups. In
the 1970s, this conjecture came to be called "Iwasawa's Main Conjecture."

It was a 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. Professor Iwasawa also advised me
to avoid the natural tendency of trying to read everything that was written
about related topics before tackling some of my research problems.
His philosophy was that one might then have a much better chance of coming
up with a really novel and fresh approach. I tried to follow his
advice to some extent, but I must admit that I did read Iwasawa's own papers
quite thoroughly.

Professor
Iwasawa's courses at Princeton were very popular among the graduate students.
Visitors at the Institute for Advanced Study would also attend. His lectures
were very beautiful, clear, and perfectly organized. But, remarkably,
he lectured entirely without lecture notes. I remember only one exception
to that, when once he briefly glanced at a small paper which he pulled
from his pocket, put it back, and then continued his lecture. During the
year 1968-69, Professor Iwasawa gave an introductory course on the theory
of Gamma-extensions. I took very careful notes from that courses which,
for years afterwards, were quite in demand and circulated widely.

In the early 1970s, various mathematicians began to realize that Iwasawa's
ideas had some important implications for other parts of number theory.
Barry Mazur was inspired by these ideas to develop an analogous theory
for elliptic curves, including a version of Iwasawa's Main Conjecture in
that context. On the one hand, Mazur constructed an Iwasawa Module
which reflected algebraic properties of the elliptic curve. And on the
other hand, Mazur, together with Swinnerton-Dyer, discovered that they
could construct a natural p-adic L-function which reflected properties
of the classical L-function for the elliptic curve defined by Hasse
and Weil decades earlier. Mazur realized that this version of the
Main Conjecture provided an approach to studying the famous Conjecture
of Birch and Swinnerton-Dyer concerning elliptic curves. At about
the same time, John Tate realized that some of Iwasawa's results and techniques
could be applied to some basic questions in Algebraic K-Theory for number
fields. These ideas were then pursued systematically by John Coates
and Steven Lichtenbaum. But one of the most dramatic developments
was the Theorem of Coates and Wiles proved in 1977. This was the first
truly general result concerning the Birch and Swinnerton-Dyer Conjecture.
The essence of their proof was to adapt a very deep theorem of Iwasawa
concerning the Gamma-extension constructed from roots of unity to another
Gamma-extension which could be constructed from certain elliptic curves.
It was an exciting demonstration of the power of Iwasawa's Theory.

These developments of course attracted many mathematicians to start working
on different aspects of Iwasawa Theory and important breakthroughs occurred
every few years. Bruce Ferrero and Larry Washington proved one of Iwasawa's
earliest conjectures in 1978. Andrew Wiles and Barry Mazur succeeded
in proving Iwasawa's Main Conjecture in 1983. But even as these results
were proved, it came to be realized that the Main Conjecture and
some of the other conjectures could be formulated in a vastly more general
context. For me personally, it has been a great privilege to watch
how fruitful Iwasawa's original ideas have been, how they have blossomed
over the years, and to have contributed along with other mathematicians
to their development. Certainly, there are now several hundred research
papers in the mathematical literature on different aspects of Iwasawa Theory.
It has become one of the fundamental branches of modern number theory.

I visited Professor Iwasawa in Princeton rather often after
completing my graduate studies. I have very fond memories of those
visits. Professor and Mrs. Iwasawa would welcome me at their front door.
Then Professor Iwasawa and myself would sit in the living room and chat
about a great variety of things for a couple of hours. Mrs. Iwasawa
would bring tea and various sweets, sometimes traditional Japanese pastries,
sometimes American-style deserts. Often, their beautiful dog
Zeta would come into the room, but just to stay briefly.
I missed those visits after their return to Japan, but a number of years
later I did have the opportunity to visit them in Japan. Those
visits are cherished memories for me.

--*Ralph Greenberg*

Professor Kenkichi Iwasawa. (I'm in the background with my bike.)

Grounds of the Institute for Advanced Study, Princeton, 1972

Photo taken by Mrs. Aiko Iwasawa

Interview with Iwasawa published in Sugaku in 1983