LONDON MATHEMATICAL SOCIETY STUDENT TEXTS Managing editor: Dr CM. Series, Mathematics Institute University of Warwick, Coventry CV4 7AL, United Kingdom 1 Introduction to combinators and A,-calculus, J.R. HINDLEY & J.P. SELDIN 2 Building models by games, WILFRID HODGES 3 Local fields, J.W.S. CASSELS 4 An introduction to twistor theory, S.A. HUGGETT & K.P. TOD 5 Introduction to general relativity, L.P. HUGHSTON & K.P. TOD 6 Lectures on stochastic analysis: diffusion theory, DANIEL W. STROOCK 7 The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND 8 Summing and nuclear norms in Banach space theory, GJ.O. JAMESON 9 Automorphisms of surfaces after Nielsen and Thurston, A.CASSON & S. BLEILER 10 Nonstandard analysis and its applications, N.CUTLAND (ed) 11 Spacetime and singularities, G. NABER 12 Undergraduate algebraic geometry, MILES REID 13 An introduction to Hankel operators, J.R. PARTINGTON 14 Combinatorial group theory: a topological approach, DANIEL E. COHEN 15 Presentations of groups, D.L. JOHNSON 16 An introduction to noncommutative Noetherian rings, K.R.GOODEARL & R.B. WARFIELDJR. 17 Aspects of quantum field theory in curved spacetime, S.A. FULLING 18 Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN 19 Steps in commutative algebra, R.Y. SHARP 21 Representations of finite groups of Lie type, FRANCOIS DIGNE & JEAN MICHEL 22 Designs, codes and graphs and their linkages, P. CAMERON & J. VAN LINT 23 Complex algebraic curves, F. KIR WAN 24 Lectures on elliptic curves, J.W.S. CASSELS 25 Hyperbolic geometry, B. IVERSEN 26 Elementary theory of L-functions and Eisenstein series. H. HIDA 27 Hilbert space: compact operators and the trace theorem, J.R. RETHERFORD London Mathematical Society Student Texts 26 Elementary Theory of L-functions and Eisenstein Series Haruzo Hida Department of Mathematics, University of California at Los Angeles H CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521434119 © Cambridge University Press 1993 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 A catalogue record for this publication is available from the British Library ISBN-13 978-0-521 -43411 -9 hardback ISBN-10 0-521 -43411 -4 hardback ISBN-13 978-0-521-43569-7 paperback ISBN-10 0-521-43569-2 paperback Transferred to digital printing 2006 Contents Suggestions to the reader xi Chapter 1. Algebraic number theory 1 §1.1. Linear algebra over rings 1 §1.2. Algebraic number fields 5 §1.3. p-adic numbers 17 Chapter 2. Classical L-functions and Eisenstein series 25 §2.1. Euler's method 25 §2.2. Analytic continuation and the functional equation 33 §2.3. Hurwitz and Dirichlet L-functions 40 §2.4. Shintani L-functions 47 §2.5. L-functions of real quadratic field 54 §2.6. L-functions of imaginary quadratic fields 63 §2.7. Hecke L-functions of number fields 66 Chapter 3. p-adic Hecke L-functions 73 §3.1. Interpolation series 73 §3.2. Interpolation series in p-adic fields 75 §3.3. p-adic measures on Zp 78 §3.4. The p-adic measure of the Riemann zeta function 80 §3.5. p-adic Dirichlet L-functions 82 §3.6. Group schemes and formal group schemes 89 §3.7. Toroidal formal groups and p-adic measures 96 §3.8. p-adic Shintani L-functions of totally real fields 99 §3.9. p-adic Hecke L-functions of totally real fields 102 Chapter 4. Homological interpretation 107 §4.1. Cohomology groups on Gm(C) 107 §4.2. Cohomological interpretation of Dirichlet L-values 117 §4.3. p-adic measures and locally constant functions 118 §4.4. Another construction of p-adic Dirichlet L-functions 120 Chapter 5. Elliptic modular forms and their L-functions 125 §5.1. Classical Eisenstein series of GL(2)/Q 125 §5.2. Rationality of modular forms 131 §5.3. Hecke operators 139 §5.4. The Petersson inner product and the Rankin product 150 §5.5. Standard L-functions of holomorphic modular forms 157 Chapter 6. Modular forms and cohomology groups 160 §6.1. Cohomology of modular groups 160 §6.2. Eichler-Shimura isomorphisms 167 §6.3. Hecke operators on cohomology groups 175 §6.4. Algebraicity theorem for standard L-functions of GL(2) 186 §6.5. Mazur's p-adic Mellin transforms 189 vi Contents Chapter 7. Ordinary A-adic forms, two variable p-adic Rankin products and Galois representations 194 §7.1. p-Adic families of Eisenstein series 195 §7.2. The projection to the ordinary part 200 §7.3. Ordinary A-adic forms 208 §7.4. Two variable p-adic Rankin product 221 §7.5. Ordinary Galois representations into GL2(Zp[[X]]) 228 §7.6. Examples of A-adic forms 234 Chapter 8. Functional equations of Hecke L-functions 239 §8.1. Adelic interpretation of algebraic number theory 239 §8.2. Hecke characters as continuous idele characters 245 §8.3. Self-duality of local fields 249 §8.4. Haar measures and the Poisson summation formula 253 §8.5. Adelic Haar measures 257 §8.6. Functional equations of Hecke L-functions 261 Chapter 9. Adelic Eisenstein series and Rankin products 272 §9.1. Modular forms on GL?(FA) 272 §9.2. Fourier expansion of Eisenstein series 282 §9.3. Functional equation of Eisenstein series 292 §9.4. Analytic continuation of Rankin products 298 §9.5. Functional equations of Rankin products 306 Chapter 10. Three variable p-adic Rankin products 310 § 10.1. Differential operators of Maass and Shimura 310 § 10.2. The algebraicity theorem of Rankin products 317 §10.3. Two variable A-adic Eisenstein series 326 §10.4. Three variable p-adic Rankin products 331 §10.5. Relation to two variable p-adic Rankin products 339 §10.6. Concluding remarks 343 Appendix: Summary of homology and cohomology theory 345 References 365 Answers to selected exercises 371 Index 383 Preface Number theory is very rich with surprising interactions of fundamentally different objects. A typical example which springs first to mind is the special values of L-functions, in particular, the Riemann zeta function £(k) (which will be explained in detail in Chapter 2). For each integer k, we sum up all positive integers n, raising to a negative power n"k C(k)=£ n"k n=l Since n"k =—r gets smaller and smaller as n grows, we really get this number n (if k > 1) sitting somewhere on the positive real coordinate line. Number theorists are supposed to study numbers, and in particular, integers. Thus this kind of sum of all integers should be interesting. Of course, one hopes to sum positive integer powers nk. Obviously, even squares n2 get larger and larger as n grows, and there seems to be no way of summing up all squares of integers. Nevertheless, in the mid-18th century, Euler computed the sum of positive integer powers. He introduced an auxiliary variable t and looked at k 2 k 3 k 4 C(-k) = t+2 t +3 t +4 t +---|t=1. For example: 2 3 £(0) = t+t +t +- • • = y- | t=i (the geometric series). Knowing that the derivative (with respect to t) of tn = -r- = nt11 ~\ we differ- entiate the right-hand side of the above formula and get l+2t+3t2+4t3+- • * which is quite near to t+2t2+3t3+4t4+—. Thus we get t=l and similarly, taking the derivatives k times, we get Still one cannot get the answer, because we cannot replace t by 1 in T—. NOW, when number theorists get to this point, in a hunt, they start to smell something interesting and will never give up the chance. Let's separate the sum into two parts, that is, the sum over even integers and the sum over odd integers. Then we see that viii Preface k+1 k k 3 k 4 k k 2 k 3 k 4 (l-2 )C(-k) = (t+2 t*+3 t +4 t +--- |t=1)-2(2 t+4 t +6 t +8 t +--- |t=i) This time we have won, because we can really put 1 in place of t and get a number, which (after dividing by (l-2k+1)) Euler declared to be the value £(-k). It is then easy to see that £(-k) = 0 for even integers k > 0; thus we may concentrate on the odd negative ^-values £(l-2m) for positive integer m. The most remarkable fact Euler discovered in this context is the following relation (whose proof will be given in §2.2 and in Chapter 8): The left-hand side is the actual sum of negative powers n"2m over positive in- tegers and the right-hand side is the value of the ratio of suitable polynomials in t. This is a remarkable interaction of an infinite sum and derivatives of a polynomial created by Euler. There is another example of this kind. Let us fix one prime, say 5. Suppose you live in a country under a very crazy dictator, who decreed that two points are 'near' if the distance between them measured by meters is divisible by very high power of 5; so if you sit 53 = 125 meters distance from a friend, you are 'near' to him. If you sit 55 = 3125 meters distance away, then you are 'very close' to him, and so on.