Spectral Methods of Automorphic Forms

American Mathematical Society

Revista Matemática Iberoamericana

Graduate Studies in Mathematics

Volume 53

Henryk Iwaniec

SECOND EDITION

Spectral Methods ofAutomorphic Forms

http://dx.doi.org/10.1090/gsm/053

Henryk Iwaniec

American Mathematical SocietyProvidence, Rhode Island

Graduate Studies in Mathematics

Volume 53

Spectral Methods ofAutomorphic FormsSecond Edition

Revista Matemática Iberoamericana

Editorial Board

Walter CraigNikolai Ivanov

Steven G. KrantzDavid Saltman (Chair)

2000 Mathematics Subject Classification. Primary 11F12, 11F30, 11F72.

Library of Congress Cataloging-in-Publication Data

Iwaniec, Henryk.Spectral methods of automorphic forms / Henryk Iwaniec.—2nd ed.

p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 53)First ed. published in Revista matematica iberoamericana in 1995.Includes bibliographical references and index.ISBN 0-8218-3160-7 (acid-free paper)1. Automorphic functions. 2. Automorphic forms. 3. Spectral theory (Mathematics)

I. Title. II. Series.

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Contents

Preface ix

Preface to the AMS Edition xi

Introduction 1

Chapter 0. Harmonic Analysis on the Euclidean Plane 3

Chapter 1. Harmonic Analysis on the Hyperbolic Plane 7

1.1. The upper half-plane 7

1.2. H as a homogeneous space 12

1.3. The geodesic polar coordinates 15

1.4. Group decompositions 16

1.5. The classification of motions 17

1.6. The Laplace operator 19

1.7. Eigenfunctions of Δ 19

1.8. The invariant integral operators 26

1.9. The Green function on H 32

Chapter 2. Fuchsian Groups 37

2.1. Definitions 37

2.2. Fundamental domains 38

2.3. Basic examples 41

2.4. The double coset decomposition 45

v

vi Contents

2.5. Kloosterman sums 47

2.6. Basic estimates 49

Chapter 3. Automorphic Forms 53

3.1. Introduction 53

3.2. The Eisenstein series 56

3.3. Cusp forms 57

3.4. Fourier expansion of the Eisenstein series 59

Chapter 4. The Spectral Theorem. Discrete Part 63

4.1. The automorphic Laplacian 63

4.2. Invariant integral operators on C(Γ\H) 64

4.3. Spectral resolution of Δ in C(Γ\H) 68

Chapter 5. The Automorphic Green Function 71

5.1. Introduction 71

5.2. The Fourier expansion 72

5.3. An estimate for the automorphic Green function 75

5.4. Evaluation of some integrals 76

Chapter 6. Analytic Continuation of the Eisenstein Series 81

6.1. The Fredholm equation for the Eisenstein series 81

6.2. The analytic continuation of Ea(z, s) 84

6.3. The functional equations 86

6.4. Poles and residues of the Eisenstein series 88

Chapter 7. The Spectral Theorem. Continuous Part 95

7.1. The Eisenstein transform 96

7.2. Bessel’s inequality 98

7.3. Spectral decomposition of E(Γ\H) 101

7.4. Spectral expansion of automorphic kernels 104

Chapter 8. Estimates for the Fourier Coefficients of Maass Forms 107

8.1. Introduction 107

8.2. The Rankin–Selberg L-function 109

8.3. Bounds for linear forms 110

8.4. Spectral mean-value estimates 113

Contents vii

8.5. The case of congruence groups 116

Chapter 9. Spectral Theory of Kloosterman Sums 121

9.1. Introduction 121

9.2. Analytic continuation of Zs(m,n) 122

9.3. Bruggeman–Kuznetsov formula 125

9.4. Kloosterman sums formula 128

9.5. Petersson’s formulas 131

Chapter 10. The Trace Formula 135

10.1. Introduction 135

10.2. Computing the spectral trace 139

10.3. Computing the trace for parabolic classes 142

10.4. Computing the trace for the identity motion 145

10.5. Computing the trace for hyperbolic classes 146

10.6. Computing the trace for elliptic classes 147

10.7. Trace formulas 150

10.8. The Selberg zeta-function 152

10.9. Asymptotic law for the length of closed geodesics 154

Chapter 11. The Distribution of Eigenvalues 157

11.1. Weyl’s law 157

11.2. The residual spectrum and the scattering matrix 162

11.3. Small eigenvalues 164

11.4. Density theorems 168

Chapter 12. Hyperbolic Lattice-Point Problems 171

Chapter 13. Spectral Bounds for Cusp Forms 177

13.1. Introduction 177

13.2. Standard bounds 178

13.3. Applying the Hecke operator 180

13.4. Constructing an amplifier 181

13.5. The ergodicity conjecture 183

Appendix A. Classical Analysis 185

A.1. Self-adjoint operators 185

viii Contents

A.2. Matrix analysis 187

A.3. The Hilbert–Schmidt integral operators 189

A.4. The Fredholm integral equations 190

A.5. Green function of a differential equation 194

Appendix B. Special Functions 197

B.1. The gamma function 197

B.2. The hypergeometric functions 199

B.3. The Legendre functions 200

B.4. The Bessel functions 202

B.5. Inversion formulas 205

References 209

Subject Index 215

Notation Index 219

Preface

I was captivated by a group of enthusiastic Spanish mathematicians whosedesire for cultivating modern number theory I enjoyed recently during twomemorable events, the first at the summer school in Santander, 1992, andthe second while visiting the Universidad Autonoma in Madrid in June 1993.These notes are an expanded version of a series of eleven lectures I deliveredin Madrid1. They are more than a survey of favorite topics, since proofsare given for all important results. However, there is a lot of basic materialwhich should have been included for completeness, but was not because oftime and space limitations. Instead, to make a comprehensive expositionwe focus on issues closely related to the the spectral aspects of automorphicforms (as opposed to the arithmetical aspects, to which I intend to returnon another occasion).

Primarily, the lectures are addressed to advanced graduate students. Ihope the student will get inspiration for his own adventures in the field.This is a goal which Professor Antonio Cordoba has a vision of pursuingthroughout the new volumes to be published by the Revista MatematicaIberoamericana. I am pleased to contribute to part of his plan.

Many people helped me prepare these notes for publication. In particularI am grateful to Fernando Chamizo, Jose Luis Fernandez, Charles Mozzochi,Antonio Sanchez-Calle and Nigel Pitt for reading and correcting an earlydraft. I also acknowledge the substantial work in the technical preparationof this text by Domingo Pestana and Marıa Victoria Melian, without whichthis book would not exist.

New Brunswick, October 1994 Henryk Iwaniec

1I would like to thank the participants and the Mathematics Department for their warmhospitality and support.

ix

Preface to the AMS

Edition

Since the initial publication by the Revista Matematica Iberoamericana Ihave used this book in my graduate courses at Rutgers several times. I haveheard from my colleagues that they also had used it in their teaching. Onseveral occasions I have been told that the book would be a good source ofideas, results and references for graduate students, if it were easier to obtain.The American Mathematical Society suggested publishing an edition, whichwould increase its availability. Sergei Gelfand should be given special creditfor his persistence during the past five years to convince me of the need for asecond publication. I am also very indebted to the editors of the Revista forthe original edition and its distribution and for the release of the copyrightto me, so this revised edition could be realized.

I eliminated all the misprints and mistakes which were kindly called tomy attention by several people. I also changed some words and slightlyexpanded some arguments for better clarity. In a few places I mentionedrecent results. I did not try to include all of the best achievements, sincethis had not been my intention in the first edition either. Such a task wouldrequire several volumes, and that could discourage beginners. I hope that asubject as great as the analytic theory of automorphic forms will eventuallybe presented more substantially, perhaps by several authors. Until thathappens, I am recommending that newcomers read the recent survey articlesand new books which I have listed at the end of the regular references.

New Brunswick, June 2002 Henryk Iwaniec

xi

References

[At-Le] A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m). Math. Ann. 185 (1970),134-160.

[Br1] R. W. Bruggeman, Fourier coefficients of cusp forms. Invent. Math. 45 (1978),1-18.

[Br2] R. W. Bruggeman, Automorphic forms, in Elementary and Analytic Theory ofNumbers, Banach Center Pub. 17, PWN, Warsaw, 1985, 31-74.

[Bu-Du-Ho-Iw] D. Bump, W. Duke, J. Hoffstein and H. Iwaniec, An estimate for the Heckeeigenvalues of Maass forms. Duke Math. J. Research Notices 4 (1992), 75-81.

[Ch] F. Chamizo, Temas de Teorıa Analıtica de los Numeros. Doctoral Thesis, Univer-sidad Autonoma de Madrid, 1994.

[Co-Pi] J. W. Cogdell and I. Piatetski-Shapiro, The Arithmetic and Spectral Analysis ofPoincare Series. Perspectives in Math. 13, Academic Press, 1990.

[De-Iw] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients ofcusp forms. Invent. Math. 70 (1982), 219-288.

[El] J. Elstrodt, Die Resolvente zum Eigenwertproblem der automorphen Formen inder hyperbolischen Ebene, I, II, III. Math. Ann. 203 (1973), 295-330; Math. Z.132 (1973), 99-134; Math. Ann. 208 (1974), 99-132.

[Fa] J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group. J. ReineAngew. Math. 293/294 (1977), 143-203.

[Fi] J. Fischer, An Approach to the Selberg Trace Formula via the Selberg Zeta-Function, Springer Lecture Notes in Math. 1253 (1980).

[Foc] V. A. Fock, On the representation of an arbitrary function by an integral involvingLegendre’s functions with a complex index. C. R. (Dokl.) Acad. Sci. URSS 39(1943), 253-256.

[For] L. R. Ford, Automorphic Functions. McGraw-Hill, New York, 1929.

[Fr-Kl] R. Fricke and F. Klein, Vorlesungen uber die Theorie der automorphen FunktionenI, II. Teubner, Leipzig, 1897 and 1912.

209

210 References

[Go] A. Good, Local Analysis of Selberg’s Trace Formula. Springer Lecture Notes inMath. 1040, 1983.

[Go-Sa] D. Goldfeld and P. Sarnak, Sums of Kloosterman sums. Invent. Math. 71 (1983),243-250.

[Gr-Ry] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products. Aca-demic Press, London, 1965.

[Ha-Ra] G. H. Hardy and S. Ramanujan, Asymptotic formulae in combinatory analysis.Proc. London Math. Soc. 17 (1918), 75-115.

[Ha-Ti] G. H. Hardy and E. C. Titchmarsh, Solutions of some integral equations consid-ered by Bateman, Kapteyn, Littlewood, and Milne. Proc. London Math. Soc. 23(1924), 1-26.

[He1] D. A. Hejhal, The Selberg Trace Formula for PSL(2,R). Springer Lecture Notesin Math. 548 (1976) and 1001 (1983).

[He2] D. A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups. Memoirs ofthe Amer. Math. Soc. 469 (1992).

[Ho-Lo] J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero.Ann. of Math. 140 (1994), 161-176.

[Hu1] M. N. Huxley, Scattering matrices for congruence subgroups, in Modular forms,Ellis Horwood Series of Halsted Press, New York, 1984, 141-156.

[Hu2] M. N. Huxley, Introduction to Kloostermania, in Elementary and Analytic Theoryof Numbers. Banach Center Publ. 17, Warsaw, 1985, 2217-306.

[Hu3] M. N. Huxley, Exponential sums and lattice points, II. Proc. London Math. Soc.66 (1993), 279-301; Corrigenda, ibid. 68 (1994), 264.

[Iw1] H. Iwaniec, Non-holomorphic modular forms and their applications, in ModularForms, Ellis Horwood Series of Halsted Press, New York, 1984, 157-196.

[Iw2] H. Iwaniec, Spectral theory of automorphic functions and recent developments inanalytic number theory. Proc. I.C.M., Berkeley, 1986, 444-456.

[Iw3] H. Iwaniec, Prime geodesic theorem. J. Reine Angew. Math. 349 (1984), 136-159.

[Iw-Sa] H. Iwaniec and P. Sarnak, L∞-norms of eigenfuntions of arithmetic surfaces. Ann.of Math. 141 (1995), 301-320.

[Kac] M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73 (1966),1-23.

[Kat] S. Katok, Fuchsian Groups. Univ. of Chicago Press, 1992.

[Ki-Sa] H. Kim and P. Sarnak, Functoriality for the exterior square of GL2 (by HenryKim); with the appendix, Refined estimates towards the Ramanujan and Selbergconjectures (by Henry Kim and Peter Sarnak), to appear in J.Amer.Math.Soc.

[Kl1] H. D. Kloosterman, On the representation of numbers in the form ax2 + by2 +

cz2 + dt2. Acta Math. 49 (1926), 407-464.

[Kl2] H. D. Kloosterman, Asymptotische Formel fur die Fourier-Koeffizienten ganzerModulformen. Abh. Math. Sem. Univ. Hamburg 5 (1927), 338-352.

[Ko-Le] M. J. Kontorovitch and N. N. Lebedev, J. Exper. Theor. Phys. USSR 8 (1938),1192-1206, and 9 (1939), 729-741.

References 211

[Ku] N. V. Kuznetsov, Petersson’s conjecture for cusp forms of weight zero and Linnik’sconjecture. Sums of Kloosterman sums. Math. USSR Sbornik 29 (1981), 299-342.

[L] E. Landau, Uber die Klassenzahl imaginar-quadratischer Zahlkorper, Gott. Nachr.(1918), 285-295.

[La] S. Lang, SL2(R). Addison-Wesley, Reading, MA, 1973.

[Le] N. N. Lebedev, Special Functions and their Applications. Dover Publications, NewYork, 1972.

[Lu] W. Luo, On the non-vanishing of Rankin-Selberg L-functions, Duke Math. J. 69(1993), 411-427.

[Lu-Ru-Sa] W. Luo, Z. Rudnick and P. Sarnak, On Selberg’s eigenvalue conjecture, Geom.Funct. Anal. 5 (1995), 384-401. See also; On the generalized Ramanujan conjecturefor GL(n), Proc. Symp. Pure Math. 66, part 2 (1999), 301-310.

[Lu-Sa] W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on SL2(Z)\H2.IHES Publ. Math. 81 (1995), 207-237.

[Ma] H. Maass, Uber eine neue Art von nichtanalytschen automorphen Funktionen unddie Bestimmung Dirichletscher Reihen durch Funktionalgleichungen. Math. Ann.121 (1949), 141-183.

[Me] F. G. Mehler, Uber eine mit den Kugel- und Cylinderfunctionen verwandte Funk-tion und ihre Anwendung in der Theorie der Elektricitatsvertheilung. Math. Ann.18 (1881), 161-194.

[Mi-Wa] R. Miatello and N. R. Wallach, Kuznetsov formulas for rank one groups. J. Funct.Anal. 93 (1990), 171-207.

[Ni] J. Nielsen, Uber Gruppen linearer Transfromationen. Mitt. Math. Ges. Hamburg8 (1940), 82-104.

[Pe1] H. Petersson, Uber die Entwicklungskoeffizienten der automorphen Formen. ActaMath. 58 (1932), 169-215.

[Pe2] H. Petersson, Uber einen einfachen Typus von Untergruppen der Modulgruppe.Archiv der Math. 4 (1953), 308-315.

[Ph-Ru] R. S. Phillips and Z. Rudnick, The circle problem in the hyperbolic plane, J.Funct. Anal. 121 (1994), 78-116.

[Ph-Sa] R. S. Phillips and P. Sarnak, On cusp forms for cofinite subgroups of PSL(2,R).Invent. Math. 80 (1985), 339-364.

[Pr] N. V. Proskurin, Summation formulas for general Kloosterman sums. Zap. Nauchn.Sem. LOMI 82 (1979), 103-135; English transl., J. Soviet Math. 18 (1982), 925-950.

[Rad] H. Rademacher, A convergence series for the partition function p(n). Proc. Nat.Acad. Sci. U.S.A. 23 (1937), 78-84.

[Ran] R. A. Rankin, Modular Forms of Negative Dimensions. Dissertation. Clare College,Cambridge, 1940.

[Ro] W. Roelcke, Uber die Wellengleichung be Grenzkreisgruppen erster Art. S.-B.Heidelberger Akad. Wiss. Math. Nat. Kl. 1953/55, Abh. 4, 159-267 (1956).

212 References

[Ru-Sa] Z. Rudnick and P. Sarnak, The behavior of eigenstates of arithmetic hyperbolicmanifolds, Comm. Math. Phys. 161 (1991), 195-213.

[Sa1] P. Sarnak, Determinants of Laplacians. Comm. Math. Phys. 110 (1987), 113-120.

[Sa2] P. Sarnak, Arithmetic Quantum Chaos. R. A. Blyth Lectures, Univ. of Toronto,1993.

[Sa3] P. Sarnak, Some Applications of Modular Forms. Cambridge Tracts in Math. 99,Cambridge University Press, 1990.

[Sa4] P. Sarnak, Class numbers of indefinite binary quadratic forms. J. Number Theory15 (1982), 229-247.

[Se-Ti] D. B. Sears and E. C. Titchmarsh, Some eigenfunction formulae, Quart J. Math.Oxford Ser. (2) 1 (1950), 165-175.

[Se1] A. Selberg, On the estimation of Fourier coefficients of modular forms. Proc. Symp.Pure Math. 7 (1965), 1-15.

[Se2] A. Selberg, Collected Papers, vol. I. Springer, 1989.

[Se3] A. Selberg, Uber die Fourierkoeffizienten elliptischer Modulformen negativer Di-mension. C. R. Neuvieme Congres Math. Scandinaves (Helsingfors, 1938), Mer-catorin Kirjapaino, Helsinki, 1939, 320-322.

[Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions.Princeton Univ. Press, 1971.

[Si1] C. L. Siegel, Bemerkung zu einem Satze von Jakob Nielsen. Mat. Tidsskr. B 1950,66-70.

[Si2] C. L. Siegel, Discontinuous groups. Ann. of Math. 44 (1943), 674-689.

[Te] A. Terras, Fourier Analysis on Symmetric Spaces and Applications, I, II. Springer-Verlag, 1985, 1988.

[Ti1] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals. 2nd ed., Claren-don Press, Oxford, 1948.

[Ti2] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Dif-ferential Equations. Vols. I (rev. ed.), II, Clarendon Press, Oxford, 1962, 1958.

[Ve] A. B. Venkov, Spectral Theory of Automorphic Functions and its Applications.Math. and Its Applications (Soviet Series), 51, Kluwer, Dordrecht, 1990.

[Vi] N. Ya. Vilenkin, Special Functions and Theory of Group Representations. Transl.Math. Monographs 22, Amer. Math. Soc., 1968.

[Vig1] M.-F. Vigneras, L’equation fonctionelle de la fonction zeta de Selberg du groupemodulaire PSL(2,Z). Asterisque 61 (1979), 235-249.

[Vig2] M.-F. Vigneras, Varietes riemanniennes isospectrales et non isometriques. Ann. ofMath. 112 (1980), 21-32.

[Wa] G. N. Watson, A Treatise on the Theory of Bessel Functions. Cambridge Univer-sity Press, 1962.

[We] A. Weil, On some exponential sums. Proc. Nat. Acad. Sci. U.S.A. 34 (1948),204-207.

References 213

[Wi] A. Winkler, Cusp forms and Hecke groups. J. Reine Angew. Math. 386 (1988),181-204.

[Wo] S. Wolpert, Disappearence of cusp forms in families. Ann. of Math. 139 (1994),239-291.

[Zo] P. Zograf, Fuchsian groups and small eigenvalues of the Laplace operator. Zap.Nauchn. Sem. LOMI 122 (1982), 24-29; English transl., J. Soviet Math. 26 (1984),1618-1621.

Recommended new books:

[Br3] R. W. Bruggeman, Families of Automorphic Forms. Birkhauser Monographs inMathematics 88, Basel, 1994.

[Bu] D. Bump, Automorphic Forms and Representations. Cambridge University Press,1997.

[Mo] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function. Cambridge Univer-sity Press, 1997.

Recommended survey articles:

[Iw4] H. Iwaniec, Harmonic analysis in number theory, in Prospects in Mathematics(Hugo Rossi, editor), AMS, Providence, RI, 1999, 51-68.

[Iwa-Sar] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, inVisions in Mathematics Towards 2000, Birkhauser, Basel, 2000, 705-741.

Subject Index

Acting discontinuously, 37

amplifying linear form, 182

Atkin-Lehner theory of newforms, 116

automorphic form, 54

of weight k, 131

automorphic function, 53

automorphic Green function, 71-72

automorphic kernel, 64

Barnes G-function, 153

Bessel functions, 4, 20, 202

Bessel’s inequality, 98

Bruggeman-Kuznetsov formula, 127

Bruhat decomposition, 17

Cartan’s decomposition, 15

central part F (Y ), 40

centralizer, 136

co-compact group, 39

column vector of the Eisenstein series, 87

compact part of K(z, w), 66

congruence group of level N , 43

conjugacy classes, 17, 136

continuous spectrum, 186

critical line, 88, 93

cusp, 39

cusp forms, 58, 132

cuspidal zones, 40

Deformation of g, 8

density theorem, 168

dilation, 18

Dirichlet divisor problem, 171-172

discrete subgroup, 37

distance function, 7

double cosets, 46

Eigenfunction of Δ, 19

eigenpacket, 103

Eisenstein series, 57

Eisenstein transform, 96

elliptic element, 18

energy integral, 166

Epstein zeta-functions, 163

equivalent points, 38

exceptional eigenvalue, 164

Finite volume group, 38-39

first kind group, 40

Fourier expansion

of automorphic forms, 54, 132

of Eisenstein series, 60

Fourier transform, 3-4

Fredholm equation, 190

free space, 32

frequencies of a Kloosterman sum, 48

Fuchsian group, 38

functional equation

of Eisenstein series, 62

of the scattering matrix, 88

fundamental domain, 38

G-invariant, 4

gamma function, 197

Gauss circle problem, 6, 171

Gauss defect, 10

Gauss hypergeometric function, 32

Gauss-Bonnet formula, 43

geodesic polar coordinates, 16

215

216 Subject Index

geometric side of K, 135-136Green function, 25, 69

of a differential equation, 76, 194Green’s formula, 35

Hankel inversion, 130, 206Hankel type transforms, 5, 130Hecke group, 42Hecke operator, 116Hilbert’s formula for the iterated resolvent,

69, 187Hilbert-Polya dream, 154homogeneous-space model, 12

horocycles, 18hyperbolic circle problem, 172hyperbolic element, 18hyperbolic group, 38hyperbolic plane, 7hypercycles, 18hypergeometric function, 24, 199

Incomplete Eisenstein series, 57inner product, 55, 132invariant height, 52invariant integral operator, 4, 26invariant linear operator, 19inverse Selberg/Harish-Chandra transform,

32isometric circle, 8isoperimetric inequality, 11iterated resolvent, 68-69, 138

Iwasawa decomposition, 13

Kernel of L, 26Kloosterman sums, 48

formula, 128-129zeta-function, 75

Kontorovich-Lebedev inversion, 22, 206

Laplace operator, 19Laplace-Beltrami operator, 3, 19lattice-point problem, 171Legendre function, 23, 200length of closed geodesics, 146Lie algebra, 33

Lie derivative, 34

Maass automorphic form, 54Maass-Selberg relations, 89, 91mean-value operator, 28Mehler-Fock inversion formula, 24method of averaging images, 20Mobius transformations, 8modular function of a locally compact

group, 15modular group, 43

modulus of a Kloosterman sum, 48

Newforms of level N , 118

Neumann coefficients, 206

Neumann series, 131, 190, 206

norm of g, 18

norm of P , 146

norm in M2(R), 37

normal matrix, 188

normal polygon, 39

Oldform of level N , 118

Parabolic element, 18

parabolic vertex, 39

Petersson’s formulas, 131

Petersson-Rankin-Selberg formula, 132

Poincare area, 10

Poincare differential, 7

Poincare series, 53, 132

point-pair invariant, 26

point spectrum, 186

Poisson summation formula, 5

primitive motion, 38

principal congruence group of level N , 43

principal parts of an automorphic kernel,65

Quaternion group, 43

quotient space Γ\H, 41

Radial function at w, 28

Ramanujan sum, 48

Ramanujan-Petersson conjecture, 117

Rankin-Selberg function, 109

reflection, 9

residual eigenvalues, 162

resolvent of K, 191

resolvent operator, 68, 72, 186

resolvent trace formula, 151

Riemann surface, 41

riemannian measure, 10

rotation, 18

Scaling matrix, 40

scattering matrix, 87

Sears-Titchmarsh inversion, 134, 207

Selberg conjecture, 165

Selberg Trace Formula, 152

Selberg zeta-function, 152

Selberg/Harish-Chandra transform, 31

signature of Γ, 43

space of the Eisenstein series, 98

spectral trace of K, 135

spherical functions, 24, 200

Subject Index 217

spherical harmonics, 201stability group, 28, 37standard polygon, 41

Titchmarsh coefficients, 207Titchmarsh integral, 207total principal part of K(z, w), 66trace

of a conjugate class, 18of a kernel, 135of L, 4-5

trace class operator, 135translation, 18

triangular group of type (α, β, γ), 41truncated Eisenstein series, 89twisted Eisenstein series, 114twisted Maass forms, 113

Unimodular, 14unique ergodicity conjecture, 184upper half-plane, 7

Variational principle, 189

Weighted Eisenstein series, 56weighted Poincare series, 56Weil bound for Kloosterman sums, 48Weyl’s law, 159Whittaker function, 20-21Wronskian, 77, 196

Zero-th term, 22, 54

Notation Index

a, b, c, 39

‖a‖, 111

a⊗ uj , 113

A, 12

a(y), 13

A(Γ\H), 53

As(Γ\H), 54

B, 45

Bν(x), 129

B(Γ\H), 57

Bμ(Γ\H), 72

cab , 49

ca , 49

c(a, b), 49

Cg , 8

C(Γ\H), 58

Cs(Γ\H), 58

Cab , 48

Cnew(Γ0(N)\H), 118

Cold(Γ0(N)\H), 118

dμz, 10

dμj(z), 183

dνt(z), 183

D , 3

D(w), 39

D(Γ\H), 63

Dab , 74

D(λ), 84

Dλ(z, z′), 84

Δ, 19

δab , 47

e(z) = e2πiz , 1

E(z, s), 87

E(Γ\H), 57

Ea(Γ\H), 98

Ea , 96

Eaf , 96

Ea(z|p), 53

Ea(z, s), 57

EYa (z, s), 89

Ea(z|w), 56

Eam(z|ψ), 56

ηac(n, t), 108

ηt(n), 62

〈f, g〉, 55, 96

|F |, 38

F (α, β, γ; z), 199

f0(x), 130

f∞, 129

f∞(x), 129

fa(y), 54

fa(n), 54

fajk(m), 132

fY (z), 92

Fa(Y ), 40

Fs(u), 24

F∞ , 41

ϕab(s), 60

ϕab(n, s), 60

ϕ(s) = detΦ(s), 140

Φ(s), 87

{g}, 17

G = SL2(R), 8

Gs(z/z′), 71

GYab(z/z

′), 82

Gs(u(z,w)), 25, 32

219

220 Notation Index

[γ], 136

Γ(s), 197

Γq , 42

Γn , 116

Γz , 38

Γ(n, p), 43

Γ(N), 43

Γ0(N), 44

Γ1(N), 44

H , 7

h−(x), 126

h+(x), 126

Ha(z,w), 65

Hf (x), 130

Hs(z,w), 33-34

Iν(y), 20, 203

jg(z), 9

J0(z), 4

Jν(y), 202

K, 12

KC(z,w), 136

Kf (t), 128

k(ϕ), 12

Kν(y), 20, 203

K(z,w), 64

K(z,w), 66

L, 66

Laj(s), 109

Lf (t), 206

Ls(m,n), 121

L(Γ\H), 55

λ = s(1− s), 20, 64

λab , 82

Λ(λ, k), 30

MΓ(R), 142

Mk(Γ) 131

N , 13

n(x), 13

Nf (�), 131

NΓ(T ), 101

νab , 82

νaj(n), 108

ν2 , ν3 , 45

ω(z,w), 29

ωd/c , 46

ω∞ , 46

Ωd/c , 46

Ωd/c(m), 47

Ω∞ , 45

p = NP , 146

P (Y ), 40

Pam(z), 132

Pab(m,n), 133

Pn(y, y′), 73

Pn,m(y, y′), 73

PSL2(R), 8

P (X), 172

Pn(X), 174

Pm−s(v), 23, 201

ψajk(m), 133

ψ(s), 144

ψ(s), 57

(r, ϕ), 16

r(�), 5

Rs(f), 32, 72

R(Γ\H), 103

Rs(Γ\H), 103

ρ(z,w), 7

ρaj(n), 107

S(m,n; c), 48

Sab(m,n; c), 48

SL2(R), 8

SL2(Z), 43

σa , 40

Tf (t), 129

Tnf , 116

TrK, 135

τ(c), 48

u(z,w), 8

uaj(z), 102

Ums (z), 24

Vs(z), 22

Ws(z), 20

yΓ, 52

yΓ(z), 52

Z(γ), 136

ZΓ(s), 152

Zs(m,n), 122

www.ams.orgAMS on the WebGSM/53

From a review of the First Edition:

The material and exposition are well-suited for second-year or higher graduate students . . . This clear and comprehensive book concerning the spectral theory of GL(2) automorphic forms belongs on many a bookshelf.

—Mathematical Reviews

Automorphic forms are one of the central topics of analytic number theory. In fact, they sit at the confl uence of analysis, algebra, geometry, and number theory. In this book, Henryk Iwaniec once again displays his penetrating insight, powerful analytic techniques, and lucid writing style.

The fi rst edition of this book was an underground classic, both as a textbook and as a respected source for results, ideas, and references.

Iwaniec treats the spectral theory of automorphic forms as the study of the space of L2 functions on the upper half plane modulo a discrete subgroup. Key topics include Eisenstein series, estimates of Fourier coeffi cients, Kloosterman sums, the Selberg trace formula and the theory of small eigenvalues.

Henryk Iwaniec was awarded the 2002 Cole Prize for his fundamental contri-butions to number theory.