Introduction to cyclotomic fields, lawrence c washington 1

Graduate Texts in Mathematics 83 Editorial Board F W Gehring P R Halmos (Managing Editor) c C Moore Lawrence C Washington Introduction to Cyclotomic Fields Springer-Verlag New York Heidelberg Berlin Lawrence C Washington Department of Mathematics University of Maryland College Park, MD 20742 U.S.A Editorial Board P R Halmos F W Gehring' C C Moore Managing Editor Indiana University Department of Mathematics Bloomington, IN 47401 U.S.A University of Michigan Department of Mathematics Ann Arbor, MI 48104 U.S.A University of California at Berkeley Department of Mathematics Berkeley, CA 94720 U.S.A AMS Subject Classifications (1980): 12-01 Library of Congress Cataloging in Publication Data Washington, Lawrence C Introduction to cyclotomic fields (Graduate texts in mathematics; 83) Bibliography: p Includes index Fields, Algebraic Cyclotomy L Title II Series 512'.3 QA247.w35 82-755 AACR2 © 1982 by Springer-Verlag New York Inc All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A Softcover reprint of the hardcover 1st edition 1982 98765432 ISBN-l3:978-1-4684-0l35-6 DOl: lO lO07/978-1-4684-0 l3 -2 e-ISBN-l3:978-1-4684-0 l33-2 To My Parents Preface This book grew out of lectures given at the University of Maryland in 1979/1980 The purpose was to give a treatment of p-adic L-functions and cyclotomic fields, including Iwasawa's theory of Zp-extensions, which was accessible to mathematicians of varying backgrounds The reader is assumed to have had at least one semester of algebraic number theory (though one of my students took such a course concurrently) In particular, the following terms should be familiar: Dedekind domain, class number, discriminant, units, ramification, local field Occasionally one needs the fact that ramification can be computed locally However, one who has a good background in algebra should be able to survive by talking to the local algebraic number theorist I have not assumed class field theory; the basic facts are summarized in an appendix For most of the book, one only needs the fact that the Galois group of the maximal unramified abelian extension is isomorphic to the ideal class group, and variants of this statement The chapters are intended to be read consecutively, but it should be possible to vary the order considerably The first four chapters are basic After that, the reader willing to believe occasional facts could probably read the remaining chapters randomly For example, the reader might skip directly to Chapter 13 to learn about Zp-extensions The last chapter, on the Kronecker-Weber theorem, can be read after Chapter The notations used in the boo,k are fairly standard; Z, C, Zp, and Cp denote the integers, the rationals, the p-adic integers, and the p-adic rationals, respectively If A is a ring (commutative with identity), then A x denotes its group of units At Serge Lang's urging I have let the first Bernoulli number be Bl = - rather than +t This disagrees with Iwasawa [23] and several of my papers, but conforms to what is becoming standard usage t vii viii Preface Throughout the preparation of this book I have found Serge Lang's two volumes on cyclotomic fields very helpful The reader is urged to look at them for different viewpoints on several of the topics discussed in the present volume and for a different selection of topics The second half of his second volume gives a nice self-contained (independent of the remaining one and a half volumes) proof of the Gross-Koblitz relation between Gauss sums and the p-adic gamma function, and the related formula of Ferrero and Greenberg for the derivative of the p-adic L-function at 0, neither of which I have included here I have also omitted a discussion of explicit reciprocity laws For these the reader can consult Lang [4], Hasse [2], Henniart, Ireland-Rosen, Tate [3], or Wiles [1] Perhaps it is worthwhile to give a very brief history of cyclotomic fields The subject got its real start in the 1840s and 1850s with Kummer's work on Fermat's Last Theorem and reciprocity laws The basic foundations laid by Kummer remained the main part of the theory for around a century Then in 1958, Iwasawa introduced his theory of Zp-extensions, and a few years later Kubota and Leopoldt invented p-adic L-functions In a major paper (lwasawa [18]), Iwasawa interpreted these p-adic L-functions in terms of Zp-extensions In 1979, Mazur and Wiles proved the Main Conjecture, showing that p-adic L-functions are essentially the cha!acteristic power series of certain Galois actions arising in the theory of Zp-extensions What remains? Most ofthe universally accepted conjectures, in particular those derived from analogy with function fields, have been proved, at least for abelian extensions of O Many of the conjectures that remain are probably better classified as open questions," since the evidence for them is not very overwhelming, and there not seem to be any compelling reasons to believe or not to believe them The most notable are Vandiver's conjecture, the weaker statement that the p-Sylow subgroup of the ideal class group of the pth cyclotomic field is cyclic over the group ring of the Galois group, and the question of whether or not A = for totally real fields In other words, we know a lot about imaginary things, but it is not clear what to expect in the real case Whether or not there exists a fruitful theory remains to be seen Other possible directions for future developments could be a theory of Z-extensions (Z = nZp; some progress has recently been made by Friedman [1 ]), and the analogues oflwasawa's theory in the elliptic case (Coates-Wiles [4]) I would like to thank Gary Cornell for much help and many excellent suggestions during the writing of this book I would also like to thank John Coates for many helpful conversations concerning Chapter 13 This chapter also profited greatly from the beautiful courses of my teacher, Kenkichi Iwasawa, at Princeton University Finally, I would like to thank N.S.F and the Sloan Foundation for their financial support and I.H.E.S and the University of Maryland for their academic support during the writing of this book Contents CHAPTER I Fermat's Last Theorem CHAPTER Basic Results CHAPTER Dirichlet Characters 19 CHAPTER Dirichlet L-series and Class Number Formulas 29 CHAPTER p-adic L-functions and Bernoulli Numbers 5.1 5.2 5.3 5.4 5.5 5.6 p-adic functions p-adic L-functions Congruences The value at s = I The p-adic regulator Applications of the class number formula 47 47 54 59 63 70 77 ix Contents x CHAPTER Stickelberger's Theorem 87 6.1 6.2 6.3 6.4 6.5 87 93 100 102 107 Gauss sums Stickelberger's theorem Herbrand's theorem The index of the Stickel berger ideal Fermat's Last Theorem CHAPTER Iwasawa's Construction of p-adic L-functions 7.1 7.2 7.3 7.4 7.5 Group rings and power series p-adic L-functions Applications Function fields fJ = 113 113 117 125 128 130 CHAPTER Cyclotomic Units 8.1 8.2 8.3 8.4 Cyclotomic units Proof of the p-adic class number formula Units ofGJ«(p) and Vandiver's conjecture p-adic expansions 143 143 151 153 160 CHAPTER The Second Case of Fermat's Last Theorem 9.1 The basic argument 9.2 The theorems 167 167 173 CHAPTER 10 Galois Groups Acting on Ideal Class Groups 10.1 Some theorems on class groups 10.2 Reflection theorems 10.3 Consequences of Vandiver's conjecture 184 184 187 195 CHAPTER 11 Cyclotomic Fields of Class Number One 11.1 11.2 11.3 11.4 11.5 The estimate for even characters The estimate for all characters The estimate for h;;' Odlyzko's bounds on discriminants Calculation of h,~ 204 205 210 217 221 228 Contents Xl CHAPTER 12 Measures and Distributions 12.1 Distributions 12.2 Measures 12.3 Universal distributions 231 231 236 251 CHAPTER 13 Iwasawa's Theory of Zp-extensions 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Basic facts The structure of A-modules Iwasawa's theorem Consequences The maximal abelian p-extension unramified outside p The main conjecture Logarithmic derivatives Local units modulo cyclotomic units 263 264 268 276 284 290 295 299 310 CHAPTER 14 The Kronecker-Weber Theorem 319 Appendix 331 Inverse limits Infinite Galois theory and ramification theory Class field theory Tables I Bernoulli numbers Irregular primes Class numbers 331 332 336 347 347 350 352 Bibliography 361 List of Symbols 386 Index 388 Bibliography 377 Masley, J and Montgomery, H Cyclotomic fields with unique factorization J reine angew Math., 286/287 (1976), 248-256 Mazur, B Rational points of abelian varieties with values in towers of number fields Invent math., 18 (1972), 183-266 Review of E E Kummer's Collected Papers Bull Amer Math Soc., 83 (1977), 976-988 On the arithmetic of special values of L-functions Invent math., 55 (1979), 207-240 Mazur, B and Swinnerton-Dyer, H Arithmetic of Wei1curves Invent math., 18 (1972),183-266 Mazur, B and Wiles, A Class fields of abelian extensions ofO Preprint McCarthy, P Algebraic Extensions of Fields Blaisdell; Ginn: Boston, 1966 McCulloh, L I A Stickel berger condition on Galois module structure for Kummer extensions of prime degree Algebraic Number Fields (Durham Symposium, 1975; ed by A Frohlich), 561-588 Academic Press: London, 1977 A class number formula for elementary-abelian-group rings J Algebra, 68 (1981), 443-452 Metsankyla, T Ober den ersten Faktor der Klassenzahl des Kreiskorpers Ann Acad Sci Fenn., Ser AI, No 416 (1967), 48 pp Ober die Teilbarkeit des ersten Faktors der Klassenzahl des Kreiskorpers Ann Univ Turku., Ser AI, No 124 (1968),6 pp On prime factors of the relative class numbers of cyclotomic fields Ann Univ Turku., Ser AI, No 149 (1971), pp On the growth of the first factor of the cyclotomic class number Ann Univ Turku., Ser AI, No 155 (1972), 12 pp A class number congruence for cyclotomic fields and their subfields Acta Arith., 23 (1973), 107-116 Class numbers and Jl-invariants of cyclotomic fields Proc Amer Math Soc., 43 (1974),299-300 On the Iwasawa invariants of imaginary abelian fields Ann Acad Sci Fenn., Ser AI, Math., (1975), no 2, 343-353 On the cyclotomic invariants oflwasawa Math Scand., 37 (1975), 61-75 Distribution of irregular prime numbers J reine angew Math., 282 (1976), 126- 130 10 Iwasawa invariants and Kummer congruences J Number Theory, 10 (1978), 510-522 11 Note on certain congruences for generalized Bernoulli numbers Arch Math (Basel), 30 (1978),595-598 12 An upper bound for the A-invariant of imaginary abelian fields (to appear) Miki,H On Zp-extensions of complete p-adic power series fields and function fields J Fac Sci Univ Tokyo, Sec lA, 21 (1974), 377-393 On unramified abelian extensions of a complete field under a discrete valuation with arbitrary residue field of characteristic p oft and its application to wildly ramified Zp-extensions J Math Soc Japan, 29 (1977),363-371 A relation between Bernoulli numbers J Number Theory, 10 (1978), 297-302 378 Bibliogra-phy On the maximal abelian I-extension of a finite algebraic number field with given ramification Nagoya Math J., 70 (1978), 183-202 Milgram, R J I Odd index subgroups of units in cyclotomic fields and applications Algebraic K-theory, Evanston 1980, Springer Lecture Notes in Mathematics, vol 854 (1981), 269-298 Milnor, J I Introduction to Algebraic K- Theory Ann of Math Studies, no 72 Princeton Univ Press: Princeton, 1971 Monsky, P I Onp-adic power series Math Ann., 255 (1981),217-227 Some invariants of £:~-extensions Math Ann., 255 (1981),229-233 Morita Y I Ap-adic analogue of the r-function J Fac Sci Univ Tokyo, Sec lA, 22 (1975), 255-266 On the Hurwitz-Lerch L-functions J Fac Sci Univ Tokyo, Sec lA, 24 (1977), 29-43 A p-adic integral representation of the p-adic L-function J reine angew Math., 302 (1978), 71-95 On the radius of convergence of the p-adic L-function Nagoya Math J., 75 (1979), 177-193 The integral forms of p-adic L-functions (Japanese) Research on microlocal analysis Proc Symp RIMS, Kyoto 1977, 30-37 Zentralblatt436: 12015 Examples of p-adic arithmetic functions A~qebraic Number Theory (Kyoto conference, 1976; ed by Iyanaga), Jap Soc Promotion Sci.: Tokyo, 1977, 143-148 Moser, C Representation de - comme somme de carrees dans un corps cyclotomique queIconque J Number Theory, (1973), 139-141 Nombre de classes d'une extension cyclique reelle de Q, de degre ou et de conducteur premier (to appear) Moser, C and Payan, J Majoration du nombre de classes d'un corps cubique cyclique de conducteur premier (to appear) Nakazato, H A remark on Ribet's theorem Proc Japan A cad , Ser A, Math Sci., 56 (1980), no 4, 192-195 Narkiewicz, W Elementary and Analytic Theory of Algebraic Numbers Monografie Matematyczne, No 57 Polish Scientific Publishers (pWN): Warsaw 1974 Neukirch J Klassenkorpertheorie Bibliographisches Institut: Mannheim-Wien-Zurich, 1969 Neumann, O Two proofs of the Kronecker-Weber theorem "according to Kronecker, and Weber," J reine angew Math., 323 (1981), 105-126 Newman,M A table of the first factor for prime cyclotomic fields Math Comp., 24 (1970), 215219 Units in cyclotomic number fields J reine angew Math., 250 (1972), 3-11 (see Loxton [1], Ennola [3]) Diophantine equations in cyclotomic fields J reine angew Math., 265 (1974), 84-89 Bibliography 379 Nielsen, N l Traite Etementaire des Nombres de Bernoulli Gauthier-Villars: Paris, 1923 Northcott, D l Finite Free Resolutions Cambridge Tracts in Maths., no 71, Cambridge Univ Press: Cambridge, 1976 Odlyzko, A l Some analytic estimates of class numbers and discriminants Invent math., 29 0975),275-286 Lower bounds for discriminants of number fields Acta Arith., 29 (1976), 275-297 Lower bounds for discriminants of number fields, II Tohoku Math J., 29 (1977), 209-216 On conductors and discriminants A~qebraic Number Fields (Durham Symposium, 1975; ed by A Frohlich), 377-407 Academic Press: London, 1977 Oesterle, J l Travaux de Ferrero et Washington sur Ie nombre de classes d'ideaux des corps cyclotomiques Sem Bourbaki, 1978/1979, Exp no 535 Springer Lecture Notes in Mathematics, vol 770 (1980), 170-182 Une nouvelle formulation de la conjecture d'Iwasawa Sem Delange-PisotPoitou, Theorie des Nombres, 1980/1981 (to appear with Birkhiiuser) Ojala, T l Euclid's algorithm in the cyclotomic field ~(( 16)' Math Comp., 31 (1977),268-273 Oriat, B l Relations entre les 2-groupes des classes d'ideaux des extensions quadratiques k(Jd) et k(~) Ann Inst Fourier, Grenoble, 27 (1977), fasc 2, 37-59 Generalisation du "Spiegelungsatz." ASlf!risque, 61 (1979), 169-175 Annulation de groupes de classes reelles Nagoya Math J., 81 (1981), 45-56 Oriat, B and Satge, Ph l Un essai de generalisation du "Spiegelungsatz." J reine angew Math., 307/308 (1979), 134-159 Osipov, Ju l p-adic zeta functions (Russian) Uspehi Mat Nauk, 34 (1979), 209-210; English trans.: Russian Math Surveys, 34 (1979), 213-214 p-adic zeta functions and Bernoulli numbers (Russian) Studies in Number Theory 6, Zap Nauen Sem Leningrad Otdel Mat Inst Steklov (LOM!), 93 (1980).192-203 Pajunen, S Computations on the growth of the first factor for prime cyclotomic fields Nordisk Tidskr lriformationsbehandling (BIT), 16 (1976), no I, 85-87; 17 (1977), no I, 113-114 MR 53:5533, MR 55:10425 Pei, Ding Yi and Feng, Ke Qin l A note on the independence of units of cyclotomic fields (Chinese) Acta Math Sinica, 23 (1980), no 5, 773-778 Plymen, R l Cyclotomic integers and the inner invariant of Connes J London Math Soc., (2), 22 (1980), 14-20 Poitou, G l Sur les petits discriminants Sem Delange-Pisot-Poitou, Theorie des Nombres, 18e annee, 1976/1977 Exp no 6,17 pp Minorations de discriminants (d'apn':s A M Odlyzko) Sem Bourbaki, 1975/1976 Exp no 479 Springer Lecture Notes in Mathematics, vol 567 (1977), 136-153 Pollaczek, F l Uber die irregularen Kreiskorper der I-ten und f2-ten Einheitswurzeln Math Zeit., 21 (1924), 1-38 380 Bibliography Queen, C I The existence of p-adic abelian L-functions Number Theory and Algebra, 263-288 Academic Press: New York, 1977 Ramachandra, K I On the units ofcycIotomic fields Acta Arith., 12 (1966),165-173 Ribenboim, P I 13 Lectures on Fermat's Last Theorem, Springer-Verlag: New York, 1979 Algebraic Numbers Wiley-Interscience: New York, 1972 Ribet, K p-adic interpolation via Hilbert modular forms Algebraic Geometry (proc Sympos Pure Math., vol 29; Arcata), 581-592 Amer Math Soc.: Providence, 1975 A modular construction of unramified p-extensions of Q{Jlp} Invent math., 34 (1976), 151-162 Sur la recherche des p-extensions non-ramifiees de Q{Jlp) Groupe d 'Etude d' Algebre (Marie-Paule Malliavin), Ire annee, 1975/1976, Exp no 2, pp MR 80f: 12005 p-adic L-functions attached to characters of p-power order Sem Delange-PisotPoitou, Theorie des Nombres, 1ge annre, 1977/1978, Exp no 9, pp Fonctions L p-adiques et tMorie d'Iwasawa Cours redige par Ph Satge Publ Math.: Orsay, 1979 Report on p-adic L-functions over totally real fields Asu!risque, 61 (1979), 177-192 Rideout, D I On a generalization of a theorem of Stickelberger, Ph.D Thesis, McGill Univ., 1970 (see Dissertation Abstracts International, vol 32B, No.1 (1971) 438-B) Robert, G I Unites elliptiques Bull Soc Math France, Mem 36 (1973), 77 pp Nombres de Hurwitz et regularite des ideaux premiers Sem Delange-PisotPoitou, Theorie des Nombres, 16e annee, 1974/1975, Exp no 21, pp Nombres de Hurwitz et unites elliptiques Ann Scient Ec Norm Sup., 11 (1978), 297-389 Rosen, M I The asymptotic behavior of the class group of a function field over a finite field, Arch Math (Basel), 24 (1973), 287-296 An elementary proof of the local Kronecker-Weber theorem, Trans Amer Math Soc., 265 (1981),599-605 Rubin, K I On the arithmetic of CM elliptic curves in Zp-extensions, Ph.D thesis, Harvard Univ., 1981 Sarkisjan, Ju I Profinitely generated r-modules (Russian) Zap Naucn Sem Leningrad Otdel Mat Inst Steklov (LaM!), 86 (1979), 157-161 Translation: J Soviet Math., 17 (1981), No.4, 2058-2061 Sarkisjan, Ju and Jakovlev, A Homological determination of r -modules (Russian) Zop Naucn, Sem Leningrad Otdel Mat Inst Steklov (LaM!), 64 (1976), 104-126 Translation: J Soviet Math., 17 (1981), No.2, 1783-1801 Schaffstein, K Tafel der Klassenzahlen der reellen quadratischen Zahlkorper mit Primzahldiskriminante unter 12000 und zwischen 100000-101000 und 1000000-1001000 Math Ann., 98 (1928), 745-748 Bibliography 381 Schertz, R I Dber die analytische Klassenzahlformel fUr reeIIe abelsche Zahlkorper J reine angew Math., 307/308 (1979), 424-430 Schmidt, c.-G I Die Relationen von Gausschen Summen und Kreiseinheiten Arch Math (Basel), 31 (1978/1979),457-463 Grossencharaktere und Relativklassenzahl abelscher Zahlkorper J Number Theory, 11 (1979), 128-159 Dber die Fuhrer von Gausschen Summen als Grossencharaktere J Number Theory, 12 (1980), 283-310 Die Relationenfaktorgruppen von Stickelberger-Elementen und Kreiszahlen J reine angew Math., 315 (1980), 60-72 Gauss sums and the classical r-function Bull London Math Soc., 12 (1980), 344346 Schmidt, H Zur Theorie und Anwendung BernouIIi-Noriundscher Polynome und gewissen VeraIIgemeinerungen der Bernoullischen und der Stirlingschen Zahlen Arch Math (Basel), 33 (1979/1980), 364-374 Schneider, P Dber die Werte der Riemannschen Zetafunktion an den ganzzahligen SteIIen J reine angew Math., 313 (1980), 189-194 Scholz, A Dber die Beziehung der Klassenzahlen quadratischer Korper zueinander J reine angew Math., 166 (1932), 201-203 Schrutka von Rechtenstamm, G TabeIIe der (Relativ)-Klassenzahlen der Kreiskorper, deren ¢-Funktion des Wurzelexponenten (Grad) nicht grosser als 256 ist Abh Deutschen Akad Wiss Berlin, Kl Math Phys., no (1964),1-64 Sen, S On explicit reciprocity laws J reine angew Math., 313 (1980), 1-26; 323 (1981), 68-87 Serre, J.-P Classes des corps cyclotomiques (d'apn!s K Iwasawa) Sem Bourbaki, 1958, Exp no 174, II pp Formes modulaires et fonctions zeta p-adiques Modular functions of one variable, III (Antwerp 1972), 191-268 Springer Lecture Notes in Mathematics, Vol 350 (1973); correction: Modular functions, IV 149-150, Springer Lecture Notes in Mathematics, Vol 476 (1975) Sur Ie residu de la fonction zeta p-adique d'un corps de nombres C R Acad Sci Paris, Ser A, 287 (1978), AI83-AI88 Shafarevich, I A new proof of the Kronecker-Weber theorem (Russian) Trudy Mat Insf Steklov, 38 (1951), 382-387 (see Narkiewicz [1]) Shanks, D The simplest cubic fields Math Comp., 28 (1974), 1137-1152 Shatz, S Profinite Groups, Arithmetic, and Geometry Ann of Math Studies, no 67 Princeton Univ Press: Princeton, 1972 Shimura, G Introduction to the Arithmetic Theory of Automorphic Functions Iwanami Shoten and Princeton Univ Press: Princeton, 1971 382 Bibliography Shintani, T On evaluation of zeta functions of totally real algebraic number fields at nonpositive integers J Fac Sci Univ Tokyo, Sec lA, 23 (1976),393-417 Shirai, S On the central ideal class group of cyclotomic fields Nagoya Math J., 75 (1979), 133-143 Shiratani, K A generalization of Vandiver's congruence Mem Fac Sci Kyushu Univ., Ser A, 25 (1971), 144-151 Kummer's congruence for generalized Bernoulli numbers and its application Mem Fac Sci Kyushu Univ., Ser A, 26 (1972), ll9-138 On certain values of p-adic L-functions Mem Fac Sci Kyushu Univ., Ser A, 28 (1974),59-82 On a kind of p-adic zeta functions Algebraic Number Theory (Kyoto conference, 1976; ed by Iyanaga), Jap Soc Promotion Sci.: Tokyo, 1977,213-217 On a formula for p-adic L-functions J Fac Sci Univ Tokyo, Sec lA, 24 (1977), 45-53 Siegel, C I Zu zwei Bermerkungen Kummers Nachr Akad Wiss Gottingen, Math.-phys Kl (1964), no 6, 51-57; Gesammelte Abhandlungen Springer-Verlag: Berlin, 1966, vol III, 436-442 Sinnott, W On the Stickelberger ideal and the circular units of a cyclotomic field Ann of Math (2), 108 (1978), 107-134 On the Stickelberger ideal and the circular units of an abelian field Invent math., 62 (1980), 181-234 On the Stickelberger ideal and the circular units of an abelian field Sem de Theorie des Nombres, Paris 1979-1980( Sem Delange-Pisot-Poitou), 277-286 Birkhiiuser: Boston-Basel-Stuttgart, 1981 Skula, L Non-possibility to prove infinity of regular primes from some theorems J reine angew Math., 291 (1977), 162-181 On certain ideals of the group ring Z[G] Arch Math (Brno), 15 (1979), no 1,53-66 Index of irregularity of a prime J reine angew Math., 315 (1980), 92-106 Another proof of Iwasawa's class number formula (to appear) Slavutskii, I I Local properties of Bernoulli numbers and a generalization of the Kummer-Vandiver theorem (Russian) 1zl' Vyss Ucebn Zaved Matematika, 1972, no (ll8), 61-69 MR 46: 151 Generalized Bernoulli numbers that belong to unequal characters, and an extension of Vandiver's theorem (Russian) Leningrad Gos Ped lnst Ucen Zap., 496 (1972), Cast' 1,61-68 MR·46:7194 Snyder, C I A concept of Bernoulli numbers in algebraic function fields J reine angew Math., 307/308 (1979), 295-308 A concept of Bernoulli numbers in algebraic function fields (II), Manuscripta Math., 35 (1981), 69-89 Soule, C On higher p-adic regulators, Alg K-theory, Evanston 1980, Springer Lecture Notes in Mathematics, vol 854 (1981), 372-401 Speiser, A Zerlegungsgruppe J reine angew Math., 149 (1919), 174-188 Bibliography 383 Stepanov, S Proof of the Davenport-Hasse relations Mat Zametki, 27 (1980), 3-6; English trans.: Math Notes A cad Sci USSR, 27 (1980),3-4 Stichtenoth, H Zur Divisorklassengruppe eines Kongruenzfunktionenkorpers Arch Math (Basel), 32 (1979), 336-340 Stickelberger, L Uber eine Verallgemeinerung der Kreistheilung Math Ann., 37 (1890),321-367 Sunseri, R Zeros of p-adic L-functions and densities relating to Bernoulli numbers Ph.D Thesis, Univ of Illinois, 1979 Sze, A On the values of L-functions at negative integers, Ph.D thesis, Cornell Univ., 1976 (see Dissertation Abstracts International, vol 37B, No 10 (1977), 5141-B) Takeuchi, H On the class number of the maximal real subfield of a cyclotomic field, Canad J Math., 33 (1981),55-58 Tate, J I Letter from Tate to lwasawa on a relation between K2 and Galois cohomology AZqebraic K-theory II (Seattle 1972), 524-527 Springer Lecture Notes in Mathematics, Vol 342 (1973) Relations between K2 and Galois cohomology Invent math., 36 (1976), 257-274 Problem 9: The general reciprocity law Mathematical Developments Arising from Hilbert Problems (Proc Sympos Pure Math., vol 28), 311-322 Amer Math Soc.: Providence, 1976 Sur la conjecture de Stark Cours redige par D Bernardi et N Schappacher (to appear with Birkhiiuser: Boston-Basel-Stuttgart) Topunov, V A connection of cyclotomic fields with the ring of cyclic matrices of prime and of primary order (Russian) Moskov Gos Ped Inst Ucen Zap., No 375 (1971), 215-223 MR 48:2110 Uchida, K Class numbers of imaginary abelian number fields T6hoku Math J (2),23 (1971), 97-104, 335-348, 573-580 Imaginary abelian number fields with class number one T6hoku Math J (2), 24 (1972),487-499 On a cubic cyclic field with discriminant 163 • J Number Theory, (1976), 346-349 (see Shanks [1]) Class numbers of cubic cyclic fields J Math Soc Japan, 26 (1974), 447-453 Uehara, T Vandiver's congruence for the relative class number of an imaginary abelian field Mem Fac Kyushu Univ., Ser A, 29 (1975), 249-254 Fermat's Conjecture and Bernoulli numbers Rep Fac Sci Engrg Saga Univ Math., No.6 (1978), 9-14 MR 80a: 12008 Ullom, S Class groups of cyclotomic fields and group rings J London Math Soc (2), 17 (1978),231-239 Upper bounds for p-divisibility of sets of Bernoulli numbers J Number Theory, 12 (1980), 197-200 Vandiver, H Fermat's Last Theorem: Its history and the nature of the known results concerning it Amer Math Monthly, 53 (1946), 555-578; 60 (1953),164-167 384 Bibliography Visik,M Non-archimedean measures connected with Dirichlet series Mat Sbornik (N.S.), 99 (141) (1976), 248-260 English trans.: Math USSR-Sb., 28 (1976), 216-228 The p-adic zeta function of an imaginary quadratic field and the Leopoldt regulator Mat Sbornik (N.S.), 102 (144) (1977),173-181; English trans.: Math USSR-Sb., 31 (1977), 151-158 (1978) Volkenborn, A On generalized p-adic integration Bull Soc Math France, Mem no 39-40 (1974), 375-384 Wagstaff, S The irregular primes to 125,000 Math Comp., 32 (1978),583-591 Zeros ofp-adic L-functions Math Comp., 29 (1975),1138-1143 p-Divisibility of certain sets of Bernoulli numbers Math Comp., 34 (1980), 467-649 Waldschmidt, M Transcendance et exponentielles en plusieurs variables Invent math., 63 (1981), 97-127 Washington, L Class numbers and Zp-extensions Math Ann., 214 (1975),177-193 A note on p-adic L-functions J Number Theory, (1976), 245-250 The class number of the field of 5nth roots of unity Proc Amer Math Soc., 61 (1976), 205-208 The calculation of Lp(l, X) J Number Theory, (1977),175-178 Euler factors for p-adic L-functions Mathematika, 25 (1978), 68-75 Kummer's calculation of Lil, X) J reine angew Math., 305 (1979), 1-8 The non-p-part of the class number in a cyclotomic Zp-extension Invent math., 49 (1979),87-97 Units of irregular cyclotomic fields Ill J Math., 23 (1979),635-647 The derivative of p-adic L-functions Acta Arith., 40 (1980), 109-115 10 Class numbers and cyclotomic Zp-extensions Proc Queen's Number Theory Conf., 1979 (Kingston, Ontario; ed by P Ribenboim) Queen's Papers in Pure and Applied Math., no 54 (1980), 119-127 11 p-adic L-functions at s = and s = Springer Lecture Notes in Mathematics (Grosswald Symposium, Philadelphia, 1980) (to appear) 12 Zeroes of p-adic L-functions Sem Delange-Pisot Poitou, Theorie des Nombres, 1980/1981 (to appear with Birkhauser: Boston-Basel-Stuttgart) Watabe, M On class numbers of some cyclotomic fields J reine angew Math., 301 (1978), 212-215; correction: 329 (1981),176 Weber, H Theorie der Abel'schen Zahlkarper Acta Math., (1886), 193-263 Weil, A Number of solutions of equations in finite fields Bull Amer Math Soc., 55 (1949), 497-508 Collected Papers, vol I, 399-410 Jacobi sums as" Grassencharaktere." Trans Amer Math Soc., 73 (1952), 487-495 Collected Papers, vol II, 63-71 Springer-Verlag: New York, 1979 La cyclotomie jadis et naguere Sem Bourbaki, 1973/1974, Exp no 452, Springer Lecture Notes in Mathematics, Vol 431 (1975), 318-338; I'Enseignement Math., 20 (1974),247-263 Collected Papers, vol III, 311-327 Sommes de Jacobi et caracteres de Heeke, Gatt Nachr 1974, Nr I, 14pp Collected Papers, vol III, 329-342 Courbes Algebriques et Varietes Abeliennes Hermann: Paris, 1971 Bibliography 385 Basic Number Theory, 3rd ed Springer-Verlag: New York, 1974 Whittaker, E and Watson, G A Course of Modern Analysis, 4th ed Cambridge Univ Press: Cambridge, 1958 Wiles, A I Higher explicit reciprocity laws Ann of Math (2), 107 (1978), 235-254 Modular curves and the class group of Q(Cp )' Invent math., 58 (1980), 1-35 Woodcock, C A note on some congruences for the Bernoulli numbers Bm J London Math Soc (2), 11 (1975), 256 Yahagi, O Construction of number fields with prescribed I-class groups Tokyo J Math., (1978), no 2,275-283 Yamaguchi, I On a Bernoulli numbers conjecture J reine angew Math., 288 (1976), 168-175 MR 54: 12628 Yamamoto, K On a conjecture of Hasse concerning multiplicative relations of Gaussian sums J Combin Theory, (1966),476-489 The gap group of mUltiplicative relationships of Gaussian sums Symp Math., 15 (1975),427-440 Yamamoto, S On the rank of the p-divisor class group of Galois extensions of algebraic number fields Kumamoto J Sci (Math.), (1972),33-40 MR 46: 1757 (note: Theorem listed in the review applies only to O(Cp ), not O(Cpn+ I»~' List of Symbols (n ~ H.l L(s, x) Lis, x) rex) Bn Bn,x BneX) (s, b) K+ h+ hQ RK RK,p Cp exp logp q w(a) (a) nth root of unity, conductor, 19 character group, 21 annihilator, 22 L-series, 29 p-adic L-function, 57 Gauss sum, 29 Bernoulli number, 30 generalized Bernoulli number, 30 Bernoulli polynomial, 31 Hurwitz zeta function, 30 maximal real subfield, 38 class number of K+, 38 relative class number, 38 unit index, 39 regulator, 41 p-adic regulator, 70 completion of algebraic closure of Qp, 48 p-adic exponential, 49 p-adic logarithm, 50 or p, 51 Teichmiiller character, 51 51 (~) 52 g(x) Gauss sum, 88 386 List of Symbols ex' ej Aj A- A, /1, v Koo A A-B r Jacobi sum, 88 Stickelberger element, 93 fractional part, 93 idempotents, 100 ith component of class group, 101 minus component, 101, 192 Iwasawa invariants, 127 Zp-extension, 264 Zp[[T]], 268 pseudo-isomorphism, 271 276 278 280 291 387 Index Adams, J c., 86 Ankeny-Artin-Chowla, 81, 85 Artin map, 338, 342 Baker-Brumer theorem, 74 Bass' theorem, 151,260 Bernoulli distribution, 233, 238 numbers, 6, 30, 347 polynomials, 31 Brauer-Siegel theorem, 42 Capitulation of ideal classes, 40,185,286,317 Carlitz, L., 86 Class field theory, 3361f towers, 222 Class number formulas, 371f., 71, 771f., 1511f eM-field, 381f., 185, 192, 193 Coates-Wiles homomorphism, 307 Conductor, 19, 338 Conductor-discriminant formula, 27, 34 Cyclotomic polynomial, 12, 18 units, 2, 1431f., 313 Zp-extension, 128,286 Davenport-Hasse relation, 112 Dirichlet characters, 191f Dirichlet's theorem, 13, 34 Discriminant, 388 Distinguished polynomial, 115 Distributions, 2311f., 2511f Eichler, M., 107 Ennola, V., 262 Even character, 19 Exponential function, 49 Fermat curve, 90 Fermat's Last Theorem, 1, 107, 1671f First factor, 38 Fitting ideal, 297 Frobenius automorphism, 14, 337 Function fields, 128, 129, 296 Functional equation, 29, 34, 86 Gamma transform, 241 r -extension, 127 Gauss sum, 29, 35, 36, 871f Generalized Bernoulli numbers, 30 Herbrand's theorem, 102 Hurwitz zeta function, 30, 55 Ideles,344 Imprimitive characters, 205 Index of Stickel berger ideal, 103 Infinite Galois theory, 3321f Integration, 2371f Index Inverse limits, 331 Irregular primes, 7, 62, 63, 165, 193, 350 Iwasawa algebra ( = A), 268 function, 69, 246, 261 invariants, 127, 276 theorem, 103,276 Jacobi sums, 88 Krasner's lemma, 48 Kronecker-Weber theorem, 319ff., 341 Kubert's theorem (= 12.18),260 Kummer congruences, 61, 141,241 homomorphism, 300 lemma (= 5.36), 79, 162 pairing, 188ff., 292 A, 127, 141,201,276 A-modules, 268ff L-functions, 29ff., 57ff Lenstra, R W., 18 Leopoldt's conjecture, 71ff., 265, 291 Local units, 163, 310ff Logarithm, 50 Logarithmic derivative, 299 Mahler's theorem, 52 Main conjecture, 146, 198, 199, 295ff Masley, J., 204 Maximal real subfield, 38 Measures, 236ff Mellin transform, 242 Minkowski bound, 17,320 unit, 72 Montgomery, R., 204 j)., 127, 130,276,284, 286 Nakayama's lemma, 279 Normal numbers, 136, 142 Odd character, 19 Odlyzko, A., 221 Ordinary distribution, 234 p-adic class number formula, 71, 77ff p-adic L-functions, 57ff., 117ff., 199, 239, 251, 295,314 389 p-adic regulator, 70ff., 77, 78, 85, 86 Parity of class numbers, 184, 193 Partial zeta function, 30, 95 Periods, 16 Polya-Vinogradov inequality, 214 Primitive character, 19,28 Probability, 62, 86, 108, 112, 159 Pseudo-isomorphic, 271 Punctured distribution, 233 Quadratic fields, 17,45,46, 81ff., 111, 190,337 reciprocity, 18, 341 Ramachandra units, 147 Rank, 186-193 Reflection theorems, 187ff Regular prime, 7, 62, 63 Regulator, 40, 70, 77, 78, 85, 86 Relative class number, 38 Residue formula, 37, 71, 165 Ribet's theorem, 102 Scholz's theorem, 83, 190 Second factor, 38 Sinnott, W., 103, 147 Spiegelungsatz (= reflection theorem), 187ff Splitting laws, 14 Stickel berger element, 93, 119 ideal,94, 195, 298 theorem, 94 Stirling's series, 58 Teichmiiller character (= w), 51, 57 Twist, 294 Uchida, K., 204 Uniform distribution, 134ff Universal distribution, 251ff Vandiver's conjecture, 78, 157ff., 186, 195ff Von Staudt-Clausen, 55, 141 Wagstaff, S., 181 Weierstrass preparation theorem, 115 Weyl criterion, 135 Zeta function for curves, 92, 128,296 ilp-extension, 127, 263ff Graduate Texts in Mathematics Soft and hard cover editions are available for each volume up to Vol 14, hard cover only from Vol 15 T AKEUTI/ZARING Introduction to Axiomatic Set Theory OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTIIZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory 2nd printing, revised 10 COHEN A Course in Simple Homotopy Theory 11 CONWAY Functions of One Complex Variable 2nd ed 12 BEALS Advanced Mathematical Analysis 13 ANDERSON/FuLLER Rings and Categories of Modules 14 GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities 15 BERBERIAN Lectures in Functional Analysis and Operator Theory 16 WINTER The Structure of Fields 17 ROSENBLATT Random Processes 2nd ed 18 HALMos Measure Theory 19 HALMos A Hilbert Space Problem Book 20 HUSEMOLLER Fibre Bundles 2nd ed 21 HUMPHREYS Linear Algebraic Groups 22 BARNES/MACK An Algebraic Introduction to Mathematical Logic 23 GREUB Linear Algebra 4th ed 24 HOLMES Geometric Functional Analysis and Its Applications 25 HEWITT/STROMBERG Real and Abstract Analysis 4th printing 26 MANES Algebraic Theories 27 KELLEY General Topology 28 ZARISKIISAMUEL Commutative Algebra I 29 ZARISKI/SAMUEL Commutative Algebra II 30 JACOBSON Lectures in Abstract Algebra I: Basic Concepts 31 JACOBSON Lectures in Abstract Algebra II: Linear Algebra 32 JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory 33 HIRSCH Differential Topology 34 SPITZER Principles of Random Walk 2nd ed 35 WERMER Banach Algebras and Several Complex Variables 2nd ed 36 KELLEy/NAMIOKA Linear Topological Spaces 37 MONK Mathematical Logic 38 GRAUERT/FRITZSCHE Several Complex Variables 39 ARVESON An Invitation to C*-Algebras 40 KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed 41 APOSTOL Modular Functions and Dirichlet Series in Number Theory 42 SERRE Linear Representations of Finite Groups 43 GILLMAN/JERISON Rings of Continuous Functions 44 KENDIG Elementary Algebraic Geometry 45 LOEVE Probability Theory 4th ed Vol 46 LOEVE Probability Theory 4th ed Vol 47 MOISE Geometric Topology in Dimensions and 48 SACHS/WU General Relativity for Mathematicians 49 GRUENBERG/WEIR Linear Geometry 2nd ed 50 EDWARDS Fermat's Last Theorem 51 KLINGENBERG A Course in Differential Geometry 52 HARTSHORNE Algebraic Geometry 53 MANIN A Course in Mathematical Logic 54 GRA VERlW ATKINS Combinatorics with Emphasis on the Theory of Graphs 55 BROWN/PEARCY Introduction to Operator Theory Vol 1: Elements of Functional Analysis 56 MASSEY Algebraic Topology: An Introduction 57 CRowELlJFox Introduction to Knot Theory 58 KOBLITZ p-adic Numbers, p-adic Analysis, and Zeta-Functions 59 LANG Cyclotomic Fields 60 ARNOLD Mathematical Methods in Classical Mechanics 61 WHITEHEAD Elements of Homotopy Theory 62 KARGAPOLOV/MERZlJAKOV Fundamentals of the Theory of Groups 63 BOLLOBAS Graph Theory-An Introductory Course 64 EDWARDS Fourier Series 2nd ed Vol 65 WELLS Differential Analysis on Complex Manifolds 66 WATERHOUSE Introduction to Affine Group Schemes 67 SERRE Local Fields 68 W ElDMANN Linear Operators in Hilbert Spaces 69 LANG Cyclotomic Fields II 70 MASSEY Singular Homology Theory 71 FARKAS/KRA Riemann Surfaces 72 STILLWELL Classical Topology and Combinatorial Group Theory 73 HUNGERFORD Algebra 74 DA VENPORT Multiplicative Number Theory 75 HOCHSCHILD Basic Theory of Algebraic Groups and Lie Algebras 76 IITAKA Algebraic Geometry 77 HECKE Lectures on the Theory of Algebraic Numbers 78 BURRIS/SANKAPPANAVAR A Course in Universal Algebra 79 WALTERS An Introduction to Ergodic Theory 80 ROBINSON A Course in the Theory of Groups 81 FORSTER Lectures on Riemann Surfaces 82 BOTT/Tu Differential Forms in Algebraic Topology 83 WASHINGTON Introduction to Cyclotomic Fields 84 IRELAND/RoSEN A Classical Introduction to Modern Number Theory ... L-functions 7 .1 7.2 7.3 7.4 7.5 Group rings and power series p-adic L-functions Applications Function fields fJ = 11 3 11 3 11 7 12 5 12 8 13 0 CHAPTER Cyclotomic Units 8 .1 8.2 8.3 8.4 Cyclotomic units... 11 Cyclotomic Fields of Class Number One 11 .1 11. 2 11 .3 11 .4 11 .5 The estimate for even characters The estimate for all characters The estimate for h;;'' Odlyzko''s bounds on discriminants Calculation... Mathematics 83 Editorial Board F W Gehring P R Halmos (Managing Editor) c C Moore Lawrence C Washington Introduction to Cyclotomic Fields Springer-Verlag New York Heidelberg Berlin Lawrence C Washington