Algebraic number theory, serge lang

Graduate Texts in Mathematics 11 O Editorial Board S Axler F.W Gehring Springer Science+Business Media, LLC K.A Ribet BOOKS OF RELATED lNTEREST BY SERGE LANG Linear Algebra, Third Edition 1987, ISBN 96412-6 Undergraduate Algebra, Second Edition 1990, ISBN 97279-X Complex Analysis, Fourth Edition 1999, ISBN 98592-1 Real and Functional Analysis, Third Edition 1993, ISBN 94001-4 Introduction to Algebraic and Abelian Functions, Second Edition 1982, ISBN 90710-6 Cyclotomic Fields and II 1990, ISBN 96671-4 0THER BOOKS BY LANG PUBLISHED BY SPRINGER-VERLAG Introduction to Arakelov Theory • Riemann-Roch Algebra (with William Fulton) • Complex Multiplication • Introduction to Modular Forms • Modular Units (with Daniel Kubert) • Fundamentals of Diophantine Geometry • Introduction to Complex Hyperbolic Spaces • Elliptic Functions • Number Theory III • Algebraic Number Theory • SL,(R) • Abelian Varieties • Differential and Riemannian Manifolds • Undergraduate Analysis • Elliptic Curves: Diophantine Analysis • lntroduction to Linear Algebra • Calculus of Severa! Variables • First Course in Calculus • Basic Mathematics • Geometry: A High School Course (with Gene Murrow) • Math! Encounters with High School Students • The Beauty of Doing Mathematics • THE FILE Serge Lang Alge braic N um ber Theory Second Edition 'Springer Serge Lang Department of Mathematics Yale University New Haven, CT 06520 USA Editorial Board S Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA F.W Gehring Mathematics Department East Hali University of Michigan Ann Arbor, MI 48109 USA K.A Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject C!assifications (1991): llRxx, llSxx, llTxx With Illustrations Library of Congress Cataloging-in-Publication Data Lang, Serge, 1927Algebraic number theory f Serge Lang - 2nd ed p cm.- (Graduate texts in mathematics; 110) Includes bibliographical references and index ISBN 978-1-4612-6922-9 ISBN 978-1-4612-0853-2 (eBook) DOI 10.1007/978-1-4612-0853-2 Algebraic number theory QA247.L29 1994 512'.74 -dc20 I Title II Series 93-50625 Originally published in 1970 © by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts © 1994, 1986 by Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 1986 Softcover reprint of the hardcover 2nd edition 1986 All rights reserved This wark may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the forrner are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone (Corrected third printing ) ISBN 978-1-4612-6922-9 SPIN 10772455 Foreword The present book gives an exposition of the classical basic algebraic and analytic number theory and supersedes my Algebraic Numbers, including much more material, e.g the class field theory on which make further comments at the appropriate place later For different points of view, the reader is encouraged to read the collection of papers from the Brighton Symposium (edited by Cassels-Frohlich), the Artin-Tate notes on class field theory, Weil's book on Basic Number Theory, Borevich-Shafarevich's Number Theory, and also older books like those of W eber, Hasse, Hecke, and Hilbert's Zahlbericht It seems that over the years, everything that has been done has proved useful, theoretically or as examples, for the further development of the theory Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more The point of view taken here is principally global, and we deal with local fields only incidentally For a more complete treatment of these, cf Serre's book Corps Locaux There is much to be said for a direct global approach to number fields Stylistically, have intermingled the ideal and idelic approaches without prejudice for either also include two proofs of the functional equation for the zeta function, to acquaint the reader with different techniques (in some sense equivalent, but in another sense, suggestive of very different moods) Even though a reader will prefer some techniques over alternative ones, it is important at least that he should be aware of all the possibilities New York June 1970 SERGE LANG V Preface for the Second Edition The principal change in this new edition is a complete rewriting of Chapter XVII on the Explicit Formulas Otherwise, I have made a few additions, and a number of corrections The need for them was pointed out to me by severa! people, but I am especially indebted to Keith Conrad for the list he provided for me as a result of a very careful reading of the book New Haven, 1994 SERGE LANG vi Prerequi sites Chapters I through VII are self-contained, assuming only elementary algebra, say at the level of Galois theory Some of the chapters on analytic number theory assume some analysis Chapter XIV assumes Fourier analysis on locally compact groups Chapters XV through XVII assume only standard analytical facts (we even prove some of them), except for one allusion to the Plancherel formula in Chapter XVII In the course of the Brauer-Siegel theorem, we use the conductordiscriminant formula, for which we refer to Artin-Tate where a detailed proof is given At that point, the use of this theorem is highly technical, and is due to the fact that one does not know that the zeros of the zeta function don't occur in a small interval to the left of If one knew this, the proof would become only a page long, and the L-series 'vould not be needed at all W e give Siegel's original proof for that in Chapter XIII My Algebra gives more than enough background for the present book In fact, Algebra already contains a good part of the theory of integral extensions, and valuation theory, redone here in Chapters I and IL Furthermore, Algebra also contains whatever will be needed of group representation theory, used in a couple of isolated instances for applications of the class field theory, or to the Brauer-Siegel theorem The word ring will always mean commutative ring without zero divisors and with unit element (unless otherwise specified) If K is a field, then K* denotes its multiplicative group, and K its algebraic closure Occasionally, a bar is also used to denote reduction modulo a prime ideal W e use the o and O notation If /, g are two functions of a real variable, and g is always ~ O, we write f = O(g) if there exists a constant C > O such that lf(x)l ~ Cg(x) for all sufficiently large x We writef = o(g) if lim"_ ,f(x)/g(x ) = O We writef ~ g if lim"_ ,f(x)jg(x ) = vii Conten ts PartOne General Basic Theory I CHAPTER Algebraic lntegers Localization Integral closure Prime ideals Chinese remainder theorem Galois extensions Dedekind rings Discrete valuation rings Explicit factorization of a prime Projective modules over Dedekind rings CHAPTER 11 12 18 22 27 29 II Completion s Definitions and completions Polynomials in complete fields Some filtrations Unramified extensions Tamely ramified extensions 31 41 45 48 51 CHAPTER III The Different and Discriminan t Complement ary modules The different and ramification The discriminant 57 62 64 IX CONTENTS X CHAPTER IV Cyclotomic Fields Roots of unity Quadratic fields Gauss sums Relations in ideal classes 71 76 82 96 CHAPTER V Parallelotopes The product formula Lattice points in parallelotopes A volume computation Minkowski's constant CHAPTER 99 110 116 119 VI The Ideal Function Generalized ideal classes Lattice points in homogeneously expanding domains The number of ideals in a given class CHAPTER 123 128 129 VII ldeles and Adeles Restricted direct products Adeles Ideles Generalized ideal class groups; relations with idele classes in the idele classes Embedding of Galois operation on ideles and idele classes k: CHAPTER 137 139 140 145 151 152 VIII Elementary Properties of the Zeta Function and L-series Lemmas on Dirichlet series Zeta function of a number field The L-series Density of primes in arithmetic progressions Faltings' finiteness theorem 155 159 162 166 170 342 [XVII, §3] EXPLICIT FORMULAS W e define the restricted Schartz space to be the space of all functions F(x) = P(x)e~Kx with some real constant K >O and some polynomial P We denote this space by Sch0 (R) Then Sch0 (R) is self dual, and functions in Sch0 (R) satisfy the three Bamer conditions, as is easily verified If F is a restricted Schwartz function, so is the function F above, again by an immediate verification We recall that the Riemann hypothesis states that Re(p) =~for all zeros p of A(s) in the strip We say that corresponds to a function Fif, putting s =a+ it, we have J~oo F(x)eClf2~a)xe-itx dx, (s) = just as the function M 11z] was obtained from F Theorem 3.3 The Riemann hypothesis is equivalent to the property that for all F0 in the restricted Schwartz space Prooj Suppose the Riemann hypothesis satisfied, and let F0 satisfy only the three weaker conditions of Theorem 3.1 Then F = F0 * F6 also satisfies the three conditions (as is easily verified) Letting correspond to F one verifies directly that the function corresponding to Fis (s) = (s)0 (1- s) In particular, if Re(s) = ~ then (s) ~O and the Weil functional is ~O on Conversely, assume the positivity condition for all functions in the restricted Schwartz space The Fourier transform of such a function is again of the same form Suppose that L(s) has one zero Po such that {30 #- ~ and the functions f3o + iyo Change variables, and put z = i(s- ~- iy ), Then = o(s) = 'l'o(z), 'l'(z) = '1' (z)'l'o(z), (s) = 'l'(z) [XVII, §3] 343 THE WEIL FORMULA are the Fourier transforms (up to a positive factor of functions and J2;,) of the tH 'P(t) We consider functions 'P0 (z) = P(z)e-Kz• with K > O Then F satisfies the three Barner conditions, and so does F Furthermore, (s) = O(e-K't•) with O < K' < K for Iti-+ oo, uniformly in the strip O ~ a ~ Therefore the sum L(p) is absolutely convergent To conclude the proof, it suffices to show that for suitable choice of P and large K, the value W() is negative Corresponding to the change of variables s H z we let p H 11 where 1J = i(p - ~ - iyo) = - (y - Yo) + i(f3 - ~) so 1'/o = i(f3o - ~) If 'P0 (z) = P(z)e-Kz• and P has real coefficients, then a direct computation shows that We let Q(z) = (z -11) with the product taken over 1Re(1J)I ~ The functional equation A(l- 8) = A(s)u with lui= shows that if 11 is not real, then its complex conjugate occurs in the product, so Q has real coefficients Then we let P(z) = zQ(z)Q( -z) Therefore, if m is the order of A(s) at p0 , then we get: W() = L (p) = -2m(f3o _ ~)2 IQ(1Jo)l4e2K(Po-l/2)• + L P(1])2e-2K~· 1Re(~)l>2 But 1]2 = (y- Yo) - (/3- ~) + iy with y real The condition 1Re(1J) 1~ is equivalent with the condition 1y - y0 1~ 2, and je-2K~ ~ e2K((P-l/2l - O such that {3(x) = {3(0) + O(lxl') for x + O Then Iim T +oo rn.:_ -y 2n fT {3A(t)r'fr(a + itfb) dt -T = i oo o J [{3(0) be(l-a)b -{3( -x) e-bx dx X - e-bx The formula of the theorem is Barner's reworking of the Weil functional, as in [Ba 81] and [Ba 90], Theorem 86 Observe that we did not assume that Fis continuou s at O, but in the applicatio n to (3), the integrand is symmetric , {3 = Fx + Y;, which is even, and thus continuou s at O, satisfying the last important condition of Theorem 5.1 Furthermo re, Barner's expression is itself a special case of a Parseval formula concerning a wider class of functions, of which r' ;r is only the most elementar y I reproduce here the statemen t of the result as in [JoL 93a] Suppose we are given: A Borel measure J1 on R+ such that dJ1(X) bounded (Borel) measurabl e function A measurabl e function cp on R+ such that: = ljl(x)dx, where ljJ is some (a) The function cp(x)- 1/x is bounded as x approache s O, x >O (b) Both functions 1/x and cp(x) are in Ll(IJ11) outside a neighborhood of O 350 [XVII, §5] EXPLICIT FORMULAS We call (J1, cp) a special pair Then we can define the regularized functional W 1,,qJ(f3) = -;- dJ1(X) Jorx ( cp(x)f3(x)- {3(0)) and its Fourier transform, which is the function WA such that ţl,lp This functional will be applied to functions f3 satisfying the Barner conditions, namely: Condition f3 E BV(R) n Ll(R) Condition There exists X + > O such that f3(x) = {3(0) + O(lxl') for l Condition f3 is normalized Then the following general theorem is valid Theorem 5.2 Let f3 satisjy the three conditions, and let (J1, cp) bea special pair Then lim A~ao IA P"(t)~~ll'(t) dt = Joloo (cp(x)/3( -x) -A f3(0)) d11 (x) X The reader will find a proof of this theorem in [JoL 93] The proof amounts to relatively refined Fourier analysis Furthermor e, this reference also gives a proof of a generalizati on, whereby instead of the simple polar expression 1/x, the function cp admits a more general polar expression with higher terms, with possibly real exponents instead of integral exponents, and also with logarithmic factors For the application of this chapter, Theorem 5.2 suffices Indeed, as an immediate consequence of the classical Gauss formula for the gamma function, one has - r'fr(a +it) = i oo ( 1) · e