Kenkichi Iwasawa (1917–1998)

mem-iwasawa.qxp 9/23/99 4:02 PM Page 1221 Kenkichi Iwasawa (1917–1998) John Coates Kenkichi Iwasawa, whose ideas have deeply influ- ematics and remained at Princeton until his re- enced the course of algebraic number theory in the tirement in 1986. In 1987 he and his wife returned second half of the twentieth century, died in Tokyo to live in Tokyo. on October 26, 1998. He spent much of his math- He was awarded the Asahi Prize in 1959, the ematical career in the United States, but he re- Prize of the Japan Academy in 1962, the American mained quintessentially Japanese and incarnated Mathematical Society Cole Prize in 1962, and the many of the finest qualities of the traditional Japan- Fujiwara Prize in 1979. His principal mathemati- ese scholar. Iwasawa was born on September 11, cal legacy is a general method in arithmetical al- 1917, in Shinshuku-mura near Kiryu in Gumma pre- gebraic geometry, known today as Iwasawa theory, fecture. After elementary school he was educated whose central goal is to seek analogues for alge- in Tokyo, where he first attended Musashi High braic varieties defined over number fields of the School and then did his undergraduate studies at techniques which have been so successfully applied Tokyo University from 1937 to 1940. In 1940 he to varieties defined over finite fields by H. Hasse, entered the graduate school of Tokyo University A. Weil, B. Dwork, A. Grothendieck, P. Deligne, and and became an Assistant in the Department of others. Mathematics, being awarded the degree of Doctor of Science in 1945. However, the war years were Cyclotomic Fields difficult for him. He became seriously ill with Until about 1950, most of Iwasawa’s papers were pleurisy in 1945 and was only well enough to re- on questions of group theory, and we shall briefly turn to his post at the university in April 1947. He discuss this aspect of his work later. However, he was appointed Assistant Professor at Tokyo Uni- himself stated that he was interested in number versity from 1949 to 1955. theory from his student days, and all of his pub- He travelled to the United States in 1950 to give lished papers from the early 1950s onwards are de- an invited lecture at the International Congress of voted to algebraic number theory. The dominant Mathematicians in Cambridge, Massachusetts, and theme of his work in number theory is his revo- then spent the two academic years 1950–52 at The lutionary idea that deep and previously inaccessi- Institute for Advanced Study in Princeton. While ble information about the arithmetic of a finite ex- preparing to return to Japan in the spring of 1952, tension F of Q can be obtained by studying coarser he received the offer of a post at the Massachusetts questions about the arithmetic of certain infinite Institute of Technology and ended up staying there Galois towers of number fields lying above F. This until 1967. In 1967 he moved to Princeton Uni- idea, whose power lies in subtly mixing p-adic an- versity as Henry Burchard Fine Professor of Math- alytic methods with Galois cohomology, has sub- sequently been applied to a much wider circle of John Coates is Sadleirian Professor of Pure Mathematics at the University of Cambridge. His e-mail address is problems in arithmetic algebraic geometry. But [email protected]. the origin and archetypical example of Iwasawa’s NOVEMBER 1999 NOTICES OF THE AMS 1221 mem-iwasawa.qxp 9/23/99 4:02 PM Page 1222 theory is in the analytic and an algebraicV interpretation. The ana- classical theory of lytic interpretation of (G) is as the algebra of cyclotomic fields measures on G withR values in Zp, allowingV us to when F is the field define the integral G fd for all in (G) and all generated over Q continuous functions f from G to Qp, the field of by the p-th roots of p-adic numbers. The algebraic interpretation is unity, where p is a that we can naturally extend the continuous action prime number, and of G on any compact Zp-moduleV X to an action of the infinite tower the whole Iwasawa algebra (G). above F is given by To apply these notions to cyclotomic fields, we the fields generated write p∞ for the group of all p-power roots of by all p-power unity; we define F∞ to be the maximal real subfield roots of unity. We of Q(p∞ ), that is, shall now discuss it in some detail, em- F∞ = Q(p∞ ) ∩ R; Photograph courtesy of John Coates. phasizing the new Left to right: Kenkichi Iwasawa with John ideas introduced and we take G to be the Galois group of F∞ over Coates, John Tate, and Ichiro Satake at the Q by Iwasawa. The principal analytic ingredient for Iwasawa’s 1976 Symposium on Algebraic Number We say that an theory of cyclotomic fields is the p-adic analogue Theory in Kyoto. odd prime number of the Riemann zeta function, whose existence p is irregular if p was already known from the work of Kummer and divides the class number of the field Q(p), where T. Kubota, and H. Leopoldt, but for which Iwasawa p denotes the group of p-th roots of unity. The gave a new construction in his 1969 paper [1] by first few irregular primes are given by an ingenious use of the classical Stickelberger el- 37, 59, 67, 101,... This notion was introduced by ements. Kummer in his work on Fermat’s Last Theorem. The We sayV that an element ϕ of the ring of frac- fact that it is at all feasible to determine whether tions of V(G) is a pseudo-measure if ( 1)ϕ be- a prime p is irregular is because of a mysterious longs to (G) for all in G (intuitively, one should and unexpected connexion between irregularity think of such a ϕ as possibly having a simple pole and the values of the Riemann zeta function (s). at the trivial character of G). Let GQ be the Galois Q For k =2, 4, 6,..., we have (1 k)=Bk/k , group of a fixed algebraic closure of , and write → Z where the Bk are the Bernoulli numbers defined by : GQ p for the homomorphism giving the ac- ( ) the expansion tion of GQ on p∞ , that is, ()= for all X∞ in GQ and in p∞ . By Galois theory G is a quo- k t n tient of GQ , and the characters , where k is any = Bnt /n!. et 1 n=0 even integer, factor through G. Modulo a slight change of language and normalisation, Iwasawa Kummer proved the remarkable result that p is ir- proved that there exists a unique pseudo-measure regular if and only if p divides the numerator of p on G such that Z at least one of the rational numbers k k1 dp =(1 p )(1 k) (1) G (1), ,(4 p). (k =2, 4, 6,...); Iwasawa was the first to realize, in a series of this makes sense, since one can integrate any non- papers published in the 1960s, that the key to a trivial p-adic homomorphism of G against any deeper understanding of this result was to study pseudo-measure. a natural arithmetic representation of the Galois The principal algebraic and arithmetic ingredi- group of the cyclotomic field generated by all the ent is a compact G-module X∞, which Iwasawa had p-power roots of unity over Q. To explain his idea, introduced in his papers in the 1950s. Let M∞ be we begin by defining the Iwasawa algebra of an the maximal abelian extension of F∞ with the prop- arbitrary profinite abelian group G by erties that it is unramified outside p and that its ^ Galois group over F∞ is a pro-p-group. Put (G) = lim Zp[G/U], X∞ = G(M∞/F∞) for the Galois group of M∞ over Z U F∞. Clearly X∞ is a compact p-module. In addi- tion, X∞ has a natural action of G given by where U runs over all open subgroups of G, Z p x = x˜ ˜ 1 denotes the ring of p-adic integers, and Zp[G/U] denotes the ordinary group ring of the finite abelian for in G and x in X∞, where ˜ denotes any lift- group G/U with coefficients in Zp. The Iwasawa al- ing of to the Galois group of M∞ over Q (the max- gebra is particularly useful because it has both an imality of M∞ guarantees that it is Galois over Q). 1222 NOTICES OF THE AMS VOLUME 46, NUMBER 10 mem-iwasawa.qxp 9/23/99 4:02 PM Page 1223 By our general remarks above, X∞ then has a nat- For the sake of completeness, the precise defi- uralV structure as a module over the Iwasawa alge- nition of Y∞ is as follows. Let Fn be the maximal bra (G). It canV easily be shown that X∞ is finitely real subfield of the field generated over Q by the generated over (G). Moreover, as G is a p-adic Lie pn+1-th roots of unity (n =0, 1, ). The group of group of dimension 1 with no element of order p, cyclotomic units Cn of Fn is defined to be the in- a simple structureV theory is known for finitely tersection with the unit group of the ring of inte- generated (G)-modules. By combining arithmetic gers of Fn of the multiplicative group generated by arguments with this structure theory, Iwasawa all conjugates of 1 n, where n is a primitive > n+1 proved that there existV an integer p 1 and ele- p -th root of unity.

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