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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 in the tirement in 1986. In 1987 he and his wife returned second half of the twentieth century, died in 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 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 , 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

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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, , 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).

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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. There is a unique prime vn

ments f1,...,fp of (G) that are not divisorsV of of Fn above p, and we write Un for the group of zero, such that we have an exact sequence of (G)- local units in the completion of Fn at vn that are modules congruent to 1 mod vn. Finally, we define Cn to be Mp ^ ^ the completion in the vn-adic topology of Cn ∩ Un. (2) 0 → X∞ → (G) fi (G) → D → 0, ∞ C i=1 Then Y is the projective limit of the Un/ n taken V with respect to the norm maps. where D is a (G)-module of finite cardinality. Via The beauty of (5) is that, via (1), it makes the val- a remarkable series of papers in the 1960s, Iwa- ues of the Riemann zeta function at the odd neg- sawa was led to the startling idea that there should ative integers appear naturally in the arithmetic of be a simple relation between the analytic object cyclotomic fields. If we assume that the class num- given by (1) and the algebraic object defined by (2). ber of the maximal real subfield of Q(p) is prime The precise statement, which became known as the to p, Iwasawa showed that ϕ is an isomorphism, main conjecture on cyclotomic fields, is the asser- and hence (5) implies (3). In 1990 K. Rubin showed tion that that one could use V. Kolyvagin’s ideas on Euler ^ ^ systems derived from cyclotomic units to prove (3) (G) = f f (G), p 0 1 p just enough about the structure of the kernel and V V → Z cokernel of the map ϕ appearing in (4) to be able where (G)0 =V ker ( (G) p) is the augmenta- tion ideal of (G). Although it is not obvious, to derive the main conjecture (3) from Iwasawa’s Kummer’s criterion mentioned above is a conse- theorem (5) without any restriction on the prime quence of (3) (the starting point for proving this p. is the classical argument showing that p is irreg- Returning to the module X∞ defined above, ular if and only if the maximal real subfield of Iwasawa also conjectured that X∞ is a finitely gen- erated Z -module, and this was proven in 1979 by Q(p) possesses a cyclic extension of degree p p that is unramified outside p and that is distinct B. Ferrero and L. Washington. It then follows from Z from the maximal real subfield of the field gener- (2) that X∞ is a free p-module of finite rank, say ated by the p2 roots of unity). Much deeper results tp. It is perhaps of interest to note that the largest 6 due to J. Herbrand and K. Ribet about the eigen- value of tp for p<4 10 is tp =7, and this is spaces for the action of the Galois group of Q(p) achieved for the single prime p =3, 238, 481 (see over Q on the p-primary subgroup of the ideal class the paper by Buhler, Crandall, Ernvall, and Met- group of Q(p) are also corollaries of (3). sankla in volume 61 (1993) of Mathematics of Com- The honour of giving the first proof of (3) for putation). It should also be noted that (3) says all primes p fell in 1984 to B. Mazur and A. Wiles nothing about the value of the integer p appear- using modular curves, but there is still great in- ing in (2). However, if the class number of the terest in studying the evolution of Iwasawa’s ideas maximal real subfield of Q(p) is prime to p, then leading to (3), especially in his wonderful paper Iwasawa’s theorem (5) and the fact that ϕ is then published in 1964 in [2]. In this paper IwasawaV used an isomorphism show that we can take p =1. No cyclotomic units to construct a compactV (G)- theoretical approach is known for showing that the module Y∞ together with a natural (G)-homo- class number of the maximal real subfield of Q(p) morphism is prime to p, but it has been verified numerically for p<4 106 (see the above paper). → (4) ϕ : Y∞ X∞. These brief remarks highlight only several facets of Iwasawa’s varied work on cyclotomic fields. For He then proved, by a very ingenious use of an ex- example, another theme of his research, growing plicit reciprocity law going back to E. Artin and out of his proof of (5), was his discovery of a beau- H. Hasse, that we have an isomorphism ^ ^ tiful explicit formula for the Hilbert norm residue symbol in the field generated over Qp by the (5) Y∞ → (G) p (G)0; pn-th roots of unity for all n > 1. His paper [3] has here we are implicitly using the construction of p inspired a large body of work on explicit reci- given in the later paper [1]. procity laws in general over the last twenty years.

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V algebra, since ( ) is isomorphic to the ring Zp[[T ]] of formal power series in an indeterminate T with coefficients in Zp. Iwasawa wrote a whole series of papers (see, in particular, [4]) in which he exploits this structure theory to study various compact - modules attached to an arbitary Zp-extension F∞/F, which are important for arithmetic ques- tions. The assertion (2) is typical of the rather gen- eral results he obtains in this direction. Iwasawa himself has said that the discovery of the asymptotic formula (7) suggested to him that there might be analogies between the p-primary subgroup of the ideal class group of a Zp-exten- sion F∞ of F and the p-primary subgroup of the

Photograph © Springer-Verlag Tokyo. group of points on the Jacobian variety of a curve Left to right: Shigeru Iitaka, Ichiro Satake, André Weil, and over a finite field. Such an analogy would suggest Kenkichi Iwasawa. November 1, 1994. that the invariant appearing in (7) is zero, and Iwasawa conjectured that this should always be the case if F∞ is the cyclotomic Zp-extension of F. This Zp-Extensions conjecture was proven by B. Ferrero and L. Wash- As we have already stressed, the deepest aspect of ington (and a rather different proof was given by Iwasawa’s work on cyclotomic fields was the mar- W. Sinnott) when F is an abelian extension of Q, riage of algebraic and analytic objects expressed but it remains open in general. In the early 1970s by the main conjecture (3). However, already from Iwasawa discovered the first examples of non- his papers published in the late 1950s, Iwasawa saw cyclotomic Z -extensions where >0 (see [6]). that many of the algebraic aspects of his theory p Today there remain many interesting open ques- were not special to cyclotomic fields and could be tions about Iwasawa’s invariants and , includ- established in the more general setting of Z - p ing the following two conjectures about the cy- extensions of number fields. Let F be a finite ex- clotomic Z -extension F∞ of F. If F is totally real, tension of Q. A Z -extension of F is an infinite p p R. Greenberg has conjectured that we always have tower of fields = =0. Secondly, if we fix F and vary p over all (6) F = F0 F1 Fn , primes, it is conjectured, by analogy with the Ja- cobian variety of a curve over a finite field, that where, for each n > 0, F is a cyclic extension of n is bounded. This last conjecture is known for no F of degree pn. The reason for this terminology is other number field than F = Q, where we have that if we define F∞ to be the union of all the =0for all p. Fn(n > 0), then F∞ is a Galois extension of F whose Galois group over F is topologically isomorphic to Group Theory the additive group of the p-adic integers Z . It is p Iwasawa’s contributions to group theory, and par- easy to see that every F has a unique Z -extension p ticularly to the theory of locally compact groups, contained in F( ∞ ), which is called the cyclotomic p have also had a lasting impact. His paper [5] gave Z -extension of F. However, if F is not totally real, p an essential step towards the solution of Hilbert’s shows that F admits infinitely fifth problem, which asked whether any locally many non-cyclotomic Z -extensions. Iwasawa’s p Euclidean topological group is necessarily a Lie first general result about arbitrary Z -extensions p group. One of the many new results about topo- was the following asymptotic formula. For each logical groups and Lie groups that are proven in n > 0, let pen denote the order of the p-primary this paper is what is now known as the Iwasawa subgroup of the ideal class group of F . Then there n decomposition of a real semisimple . He exist integers , , and , depending on the Z - p also shows that any connected Lie group is topo- extension F∞/F, such that, for all sufficiently large logically the product of a compact Lie group and n, we have a Euclidean space. In another direction, he proves n G (7) en = n + p + . that if a locally compact group has a closed nor- mal subgroup N such that both N and G/N are Lie What lies behind the proof of (7) is the fact that, groups, then G itself is a Lie group. Iwasawa if denotes the Galois group of F∞ over F, then a recorded many years later his pleasure as a young simple structure theory is known for finitelyV gen- man at receiving a letter from C. Chevalley prais- erated modules over the Iwasawa algebra ( ). ing this paper. He also was one of the pioneers of Iwasawa originally gave an ad hoc proof of this employing methods from the theory of locally structure theory, but J.-P. Serre pointed out that it compact groups to number theory. In particular, followed from known results in commutative independently of J. Tate, he discovered the adelic

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approach to E. Hecke’s L-functions with Grossen- characters, foreshadowing a vast amount of sub- sequent work done in this direction from the point of view of automorphic forms.

Influence and Legacy It is not easy to discern who influenced Iwasawa most in the early stages of his mathematical career. He himself cited S. Iyanaga and Z. Suetuna as having played an important role in assisting his initial research in group theory. When he came to the United States, E. Artin and A. Weil were the dominant influence in the shift of his interests to algebraic number theory. He had relatively few research students of his own, and the best Photograph courtesy of Hideo Wada. Iwasawa on the occasion of his 77th birthday in Tokyo, June 4, known of these are B. Ferrero, R. Greenberg, and 1994, with (left to right) Mrs. Koizumi, Mrs. Satake, and Mrs. L. Washington. Iwasawa. Although nearly all of Iwasawa’s work in num- ber theory was concerned with Zp-extensions of number fields, he was very conscious of the par- References allels with curves over finite fields and was aware [1] K. IWASAWA, On p-adic L-functions, Ann. of Math. 89 (1969), 198–205. that his ideas might be fruitfully applied to a much [2] ——— , On some modules in the theory of cyclo- wider circle of problems. The first step in this di- tomic fields, J. Math. Soc. Japan 16 (1964), 42–82. rection was carried out by B. Mazur, who initiated, [3] ——— , On explicit formulas for the norm residue sym- in the late 1960s, the study of the arithmetic of bol, J. Math. Soc. Japan 20 (1968), 151–165. Z abelian varieties over Zp-extensions of number [4] ——— , On l-extensions of algebraic number fields, fields. Many others followed, and today it is no ex- Ann. of Math. 98 (1973), 246–326. [5] , On some types of topological groups, Ann. aggeration to say that Iwasawa’s ideas have played ——— of Math. 50 (1949), 507–558. a pivotal role in many of the finest achievements Z [6] ——— , On the -invariants of l-extensions, Num- of modern arithmetical algebraic geometry on ber Theory, Algebraic Geometry and Commutative such questions as the conjecture of B. Birch and Algebra, in Honour of Y. Akizuki, Kinokuniya, Tokyo, H. Swinnerton-Dyer on elliptic curves; the conjec- 1973, pp. 1–11. tures 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 mod- ularity of elliptic curves and Fermat’s Last Theo- rem. In all of these problems, values of L-functions play a central role, and a key step in progress on them has been to establish at least some partial analogue of Iwasawa’s main conjecture (3) for the relevant L-function. It would have been anathema to Iwasawa’s great personal modesty to write at too great a length about the way in which his ideas have influenced a whole generation of number theorists in Europe, Japan, and the United States. It is sufficient per- haps to recall Narihira’s beautiful tanka about the flowering influence of the Fujiwara family in Heian Japan Saku hana no Longer than ever before Shita ni kakururu Is the wisteria’s shadow - Hito o oomi So many are those Arishi ni masaru Who find shelter Fuji no kage kamo Beneath its blossoms! and say that it can very aptly be applied to Iwa- sawa’s deep and lasting influence on the whole field of arithmetical algebraic geometry today.

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