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Class fields of abelian extensions of . (English) Zbl 0545.12005

It has been a frequent hope to relate the zeros of various zeta functions to the eigenvalues of appropriate operators. For example, this was accomplished for the zeta functions of curves over finite fields by A. Weil in the 1940’s. The authors solve this problem for the Kubota-Leopoldt p-adic L-functions by relating these functions to the characteristic polynomials obtained from Galois actions on certain Iwasawa modules (inverse limits of ideal class groups). This relationship had been conjectured by K. Iwasawa and came to be known as the ”Main Conjecture” of Iwasawa’s theory of p -extensions.

The initial step towards the present result perhaps was that of K. A. Ribet [ibid. 34, 151–162 (1976; Zbl 0338.12003)], who used modular curves to prove the converse of Herbrand’s theorem. Let p be an odd prime and consider the pth cyclotomic field K p . The action of Gal(K p /) breaks the p-part of the ideal class group of K p into various eigenspaces. Herbrand showed that if an eigenspace is non-trivial then a corresponding Bernoulli number is divisible by p, and Ribet proved the converse, which amounted to constructing certain unramified extensions of K p . The second author strengthened and extended Ribet’s results in [ibid. 58, 1–35 (1980; Zbl 0436.12004)], which finally led to the present paper.

It would be pointless to say more about the result and its proof. The authors include an excellent 8-page introduction at the beginning of their paper, outlining the result, the ideas of the proof, and the consequences (see also pp. 214–225). There are also the Bourbaki talk of J. Coates [Sémin. Bourbaki, 33e année, Vol. 1980/81, Exp. No. 575, Lect. Notes Math. 901, 220–241 (1981; Zbl 0506.12001)] and the AMS talk of S. Lang [Bull. Am. Math. Soc., New Ser. 6, 253–316 (1982; Zbl 0482.12002)], both of which can be consulted profitably by the interested reader.

Reviewer: L.Washington

MSC:
11R23Iwasawa theory
11G05Elliptic curves over global fields
References:
[1]Artin, M.: Algebraization of formal moduli I. Global Analysis. Papers in honor of K. Kodaira Spencer, D.C., Iyanaga, S., (eds.) Univ. of Tokyo Press and Princeton Univ. Press, 21-71 1969
[2]Atkin, A.O.L., Lehner, J.: Hecke operators on? 0(m). Math. Ann.185, 134-160 (1970) · Zbl 0185.15502 · doi:10.1007/BF01359701
[3]Atkin, A.O.L., Li, W.: Twists of newforms and pseudo-eigenvalues ofW-operators. Invent. Math.43, 221-244 (1978) · Zbl 0377.10017 · doi:10.1007/BF01390245
[4]Auslander, M., Buchsbaum, D.: Groups, Rings, Modules. New York, Evanston, San Francisco, London: Harper & Row 1974
[5]Bayer, P., Neukirch, J.: On values of zeta functions andl-adic Euler characteristics. Invent Math.50, 35-64 (1978) · Zbl 0409.12018 · doi:10.1007/BF01406467
[6]Casselman, W.: On representations ofGL 2 and the arithmetic of modular curves. (International Summer School on Modular functions, Antwerp 1972) Modular functions of one variable II. Lecture Notes in Mathematics Vol. 349, pp. 109-141. Berlin-Heidelberg-New York: Springer 1973
[7]Cassels, J.W.S., Fröhlich, A.: Algebraic number theory. London-New York: Academic Press 1967
[8]Coates, J.:p-adicL-functions and Iwasawa’s theory. In: Algebraic Number Fields, Fröhlich, A., (ed.) London-New York: Academic Press 1977
[9]Coates, J.: The Work of Mazur and Wiles on Cyclotomic Fields. Séminaire Bourbaki No. 575, Lecture Notes in Mathematics Vol. 901. Berlin-Heidelberg-New York: Springer 1981
[10]Coates, J.:K-theory and Iwasawa’s analogue of the Jacobian, in AlgebraicK-theory II. Lecture Notes in Mathematics Vol. 342. Berlin-Heidelberg-New York: Springer 1973
[11]Coates, J., Lichtenbaum, S.: Onl-adic zeta functions. Ann. of Math.98, 498-550 (1973) · Zbl 0279.12005 · doi:10.2307/1970916
[12]Coates, J., Sinnott, W.: An analogue of Stickelberger’s theorem for the higherK-groups. Invent. Math.24, 149-161 (1974) · Zbl 0282.12006 · doi:10.1007/BF01404303
[13]Deligne, P.: Formes modulaires et représentationsl-adiques, Séminaire Bourbaki 68/69 no. 355. Lecture Notes in Mathematics Vol. 179, pp. 136-172. Berlin-Heidelberg-New York: Springer 1971
[14]Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publications Mathématiques I.H.E.S.,36, 75-109 (1969) · Zbl 0181.48803 · doi:10.1007/BF02684599
[15]Deligne, P., Rapoport, M.: Schémas de modules de courbes elliptiques. Lecture Note in Mathematics Vol. 349. Berlin-Heidelberg-New York: Springer 1973
[16]Demazure, M.: Lectures onp-divisible groups. Lecture Notes in Mathematics Vol. 302. Berlin-Heidelberg-New York: Springer 1972
[17]Federer, L., Gross, B.: Regulators and Iwasawa Modules. Invent. Math.62, 443-457 (1981) · Zbl 0468.12005 · doi:10.1007/BF01394254
[18]Ferrero, B., Greenberg, R.: On the behavior of thep-adicL-function ats=0. Invent. Math.50, 91-102 (1978) · Zbl 0441.12003 · doi:10.1007/BF01406470
[19]Ferrero, B., Washington, L.: The Iwasawa invariantμ p vanishes for abelian number fields. Ann. of Math.109, 377-396 (1979) · Zbl 0443.12001 · doi:10.2307/1971116
[20]Fontaine, J.-M.: Groupes finis commutatifs sur les vecteurs de Witt. C. R. Acad. Sc. Paris t.280, (serie A) 1423-1425 (1979)
[21]Gras, G.: Classes d’idéaux des corps abéliens et nombres de Bernoulli généralisés. Ann. Inst. Fourier27, 1-66 (1977)
[22]Greenberg, R.: On a certainl-adic representation. Invent. Math.,21, 198-205 (1973)
[23]Greenberg, R.: Onp-adicL-functions and cyclotomic fields. Nagoya Math. J.56 61-77 (1974)
[24]Greenberg, R.: Onp-adicL-functions and cyclotomic fields II. Nagoya Math. J.67, 139-158 (1977)
[25]Greenberg, R.: On the structure of certain Galois groups. Invent. Math.47, 85-99 (1978) · Zbl 0403.12004 · doi:10.1007/BF01609481
[26]Grothendieck, A.: Modéles de Néron et monodromie. SGA 7 I exposé IX. Lecture Notes in Mathematics, Vol. 288. Berlin-Heidelberg-New York: Springer 1972
[27]Hartshore, R.: Algebraic Geometry. Berlin-Heidelberg-New York: Springer 1977
[28]Igusa, J.: Kroneckerian model of fields of elliptic modular functions. Amer. J. Math.81, 561-577 (1959) · Zbl 0093.04502 · doi:10.2307/2372914
[29]Igusa, J.: On the algebraic theory of elliptic modular functions. J. Math. Soc. Japan20, 96-106 (1968) · Zbl 0164.21101 · doi:10.2969/jmsj/02010096
[30]Iwasawa, K.: Onp-adicL-functions. Ann. Math.89, 198-205 (1969) · Zbl 0186.09201 · doi:10.2307/1970817
[31]Iwasawa, K.: Lectures onp-adicL-functions. Princeton: Princeton Univ. Press and Univ. of Tokyo Press 1972
[32]Iwasawa, K.: OnZ l -extensions of algebraic number fields. Ann. of Math.98, 246-326 (1973) · Zbl 0285.12008 · doi:10.2307/1970784
[33]Kamienny, S.: OnJ 1(p) and the arithmetic of the kernel of the Eisenstein ideal. Harvard Ph.D. Thesis, 1980
[34]Katz, N.:p-adic properties of modular schemes and modular forms, vol. III of the Proceedings of the International Summer School on Modular Functions, Antwerp (1972), Lecture Notes in Mathematics, Vol. 350, pp. 69-190. Berlin-Heidelberg-New York: Springer 1973
[35]Katz, N.: Higher congruences between modular forms. Ann. of Math.101, (no. 2) 332-367 (1975) · Zbl 0356.10020 · doi:10.2307/1970994
[36]Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. To appear in Annals of Math. Studies, Princeton U. Press.
[37]Knutson, D.: Algebraic spaces. Lecture Notes in Mathematics, Vol. 203. Berlin-Heidelberg-New York: Springer 1971
[38]Kubert, D.: Quadratic relations for generators of units in the modular function fields. Math. Ann.225, 1-20 (1977) · Zbl 0331.10011 · doi:10.1007/BF01364888
[39]Kubert, D., Lang, S.: Modular Units. Berlin-Heidelberg-New York: Springer 1981
[40]Lang, S.: Introduction to modular forms. Berlin-Heidelberg-New York: Springer 1976
[41]Lang, S.: Cyclotomic fields. Berlin-Heidelberg-New York: Springer 1978
[42]Lang, S.: Cyclotomic fields II. Berlin-Heidelberg-New York: Springer 1980
[43]Lang, S.: Units and class numbers in Number theory and algebraic Geometry. Lecture notes distributed in conjunction with the colloquium lectures given at the 85-th summer meeting of the A.M.S., University of Pittsburgh, Pittsburgh, Pennsylvania, August 17-20, 1981
[44]Langlands, R.P.: Modular forms andl-adic representations. (International Summer School on Modular Functions, Antwerp, 1972) Modular functions of one variable II, Lecture Notes in Mathematics, Vol. 349, pp. 361-500, Berlin-Heidelberg-New York: Springer 1973
[45]Li, W.: Newforms and functional equations. Math. Ann.212, 285-315 (1975) · Zbl 0286.10016 · doi:10.1007/BF01344466
[46]Lichtenbaum, S.: On the values of zeta andL-functions I. Ann. of Math.96 (no. 2) 338-360 (1972) · Zbl 0251.12002 · doi:10.2307/1970792
[47]Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. I.H.E.S.47, (1948)
[48]Mazur, B., Tate, J.: Points of order 13 on elliptic curves. Invent. Math.22, 41-49 (1973) · Zbl 0268.14009 · doi:10.1007/BF01425572
[49]Mazur, B.: Rational isogenies of prime degree. Invent. Math.44, 129-162 (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348
[50]Northcott, D.G.: Finite free resolutions. Cambridge Univ. Press, Cambridge-New York 1976
[51]Oort, F.: Commutative group schemes. Lecture Notes in Mathematics Vol. 15. Berlin-Heidelberg-New York: Springer 1966
[52]Oort, F., Tate, J.: Group schemes of prime order. Ann. Scient. Ec. Norm. Sup. (serie 4)3, 1-21 (1970)
[53]Raynaud, M.: Passage au quotient par une relation d’équivalence plate. Proc. of a Conference on Local Fields, NUFFIC Summer School held at Driebergen in 1966, pp. 133-157, Berlin-Heidelberg-New York: Springer 1967
[54]Raynaud, M.: Spécialisation du foncteur de Picard. Publ. Math. I.H.E.S.38, 27-76 (1970)
[55]Raynaud, M.: Schémas en groupes de type (p, ... p). Bull. Soc. Math. France102, 241-280 (1974)
[56]Ribet, K.: A modular construction of unramifiedp-extensions ofQ(μ p ). Invent. Math.34, 151-162 (1976) · Zbl 0338.12003 · doi:10.1007/BF01403065
[57]Serre, J.-P.: Sur la topologie des variétés algébriques en caractéristique p. Symp. Int. de Top. Alg., Mexico, 1958
[58]Serre, J-P.: Classes des corps cyclotomiques, Séminaire Bourbaki. Exp.174 (1958-9)
[59]Shafarevitch, I.R.: Lectures on minimal models and birational transformations of two-dimensional schemes. Tata Institute of fundamental research: Bombay 1966
[60]Shimura, G.: Introduction to the arithmetic theory of automorphic forms. Publ. Math. Soc. Japan11, Tokyo-Princeton (1971)
[61]Sinnott, W.: On the Stickelberger ideal and the circular units of a cyclotomic field. Ann. Math.108, 107-134 (1978) · Zbl 0395.12014 · doi:10.2307/1970932
[62]Sinnott, W.: On the Stickelberger ideal and the circular units of an abelian field. Invent. Math.62, 181-234 (1980) · Zbl 0465.12001 · doi:10.1007/BF01389158
[63]Soulé, C.:K-theorie des anneaux d’entiers de corps de nombres et cohomologie étale. Invent. Math.55, 251-295 (1979) · Zbl 0437.12008 · doi:10.1007/BF01406843
[64]Tate, J.:p-divisible groups. Proceedings of a conference on local fields (Driebergen 1966). Berlin-Heidelberg-New York: Springer 1967
[65]Tate, J.: Relations between K2 and Galois cohomology. Invent. Math.36, 257-274 (1976) · Zbl 0359.12011 · doi:10.1007/BF01390012
[66]Tate, J.: Number theoretic background. Proceedings of Symposia in Pure Mathematics33, (part II) 3-26 (1979)
[67]Wiles, A.: Modular curves and the class group ofQ(? p ). Invent. Math.58, 1-35 (1980) · Zbl 0436.12004 · doi:10.1007/BF01402272
[68]Yu, J.: A cuspidal class number formula for the modular curvesX 1(N). Math. Ann.252, 197-216 (1980) · Zbl 0436.12002 · doi:10.1007/BF01420083
[69]Serre, J-P., Tate, J.: Good reduction of abelian varieties. Ann. of Math.88, 492-517 (1968) · Zbl 0172.46101 · doi:10.2307/1970722