Traces of Hecke Operators

http://dx.doi.org/10.1090/surv/133http://dx.doi.org/10.1090/surv/133

Traces of Hecke Operators

Andrew Knightly Charles Li

American Mathematical Society

EDITORIAL COMMITTEE Jer ry L. Bona Pe te r S. Landweber Michael G. Eas twood Michael P. Loss

J. T . Stafford, Chair

2000 Mathematics Subject Classification. P r i m a r y 11-XX.

For addi t ional information and upda t e s on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 1 3 3

Library of Congress Cataloging-in-Publicat ion D a t a Knightly, Andrew, 1972-

Traces of Hecke operators / Andrew Knightly, Charles Li. p. cm. — (Mathematical surveys and monographs ; v. 133)

Includes bibliographical references and index. ISBN-13: 978-0-8218-3739-9 (alk. paper) ISBN-10: 0-8218-3739-7 (alk. paper) 1. Hecke operators. 2. Trace formulas. 3. Number theory. I. Li, Charles, 1973- II. Title.

QA243.K54 2006 515/.7246-dc22 2006047814

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to reprint-permission@ams.org.

© 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights

except those granted to the United States Government. Printed in the United States of America.

@ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Visit the AMS home page at http:/ /www.ams.org/

10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06

Dedicated to: Dwight, Leah, Mom, Dad -A.K.

Mom, Dad -C.L.

Contents

Traces of Hecke Operators 1. Introduction 1 2. The Arthur-Selberg trace formula for GL(2) 3 3. Cusp forms and Hecke operators 7

3.1. Congruence subgroups of SL2(Z) 7 3.2. Weak modular forms 10 3.3. Cusps and Fourier expansions of modular forms 12 3.4. Hecke rings 20 3.5. The level N Hecke ring 24 3.6. The elements T(n) 29 3.7. Hecke operators 34 3.8. The Petersson inner product 37 3.9. Adjoints of Hecke operators 42 3.10. Traces of the Hecke operators 45

Odds and Ends 4. Topological groups 49 5. Adeles and ideles 52

5.1. p-adic Numbers 52 5.2. Adeles and ideles 54

6. Structure theorems and strong approximation for GL2(A) 59 6.1. Topology of GL2(A) 59 6.2. The Iwasawa decomposition 61 6.3. Strong approximation for GL2(A) 63 6.4. The Cart an decomposition 66 6.5. The Bruhat decomposition 69

7. Haar measure 69 7.1. Basic properties of Haar measure 69 7.2. Invariant measure on a quotient space 74 7.3. Haar measure on a restricted direct product 82 7.4. Haar measure on the adeles and ideles 85 7.5. Haar measure on B 87 7.6. Haar measure on GL(2) 90 7.7. Haar measure on SL2(R) 92 7.8. Haar measure on GL(2) 94 7.9. Discrete subgroups and fundamental domains 95 7.10. Haar measure on Q \ A and Q*\A* 102 7.11. Quotient measure on GL2(Q)\GL2(A) 103 7.12. Quotient measure on 5 (Q) \G(A) 105

viii C O N T E N T S

8. The Poisson summation formula 108 9. Tate zeta functions 119

9.1. Definition and meromorphic continuation 119 9.2. Functional equation and behavior at s = 1 124

10. Intertwining operators and matrix coefficients 128 10.1. Linear algebra 129 10.2. Representation theory 134 10.3. Orthogonality of matrix coefficients 140

11. The discrete series of GL2(R) 151 11.1. if-type decompositions 151 11.2. Representations of 0(2) 155 11.3. K-type decomposition of an induced representation 156 11.4. The (s, If)-modules (VX)K 162 11.5. Classification of irreducible (g, K)-modules for GL2(R) 165 11.6. A detailed look at the Lie algebra action 181 11.7. Characterization of the weight k discrete series of GL2(R) 186

Groundwork 12. Cusp forms as elements of LQ(UJ) 195

12.1. From Dirichlet characters to Hecke characters 195 12.2. From cusp forms to functions on G(A) 197 12.3. Comparison of classical and adelic Fourier coefficients 198 12.4. Characterizing the image of Sk(iV,u/) in LQ(U) 201

13. Construction of the test function / 206 13.1. The non-archimedean component of / 206 13.2. Spectral properties of R(f) 213

14. Explicit computations for / k and /<*> 221

The Trace Formula 15. Introduction to the trace formula for R(f) 227 16. Terms that contribute to K(x, y) 230 17. Truncation of the kernel 231 18. Bounds for £ 7 l / f c rSfe ) ! 240 19. Integrability of kj(g) 248 20. The hyperbolic terms as weighted orbital integrals 259 21. Simplifying the unipotent term 270 22. The trace formula 276

Computation of the Trace 23. The identity term 279 24. The hyperbolic terms 280

24.1. A lemma about orbital integrals 280 24.2. The archimedean orbital integral and weighted orbital integral 281 24.3. Simplification of the hyperbolic term 283 24.4. Calculation of the local orbital integrals 283 24.5. The global hyperbolic result 286

25. The unipotent term 288 25.1. Explicit evaluation of the zeta integral at oo 288 25.2. Computation of the non-archimedean local zeta functions 291

CONTENTS ix

25.3. The global unipotent result 293 26. The elliptic terms 294

26.1. Properties of the elliptic orbital integrals 295 26.2. The archimedean elliptic orbital integral 300 26.3. Orders and lattices in an imaginary quadratic field 302 26.4. Local-global theory for lattices 307 26.5. From G(Afin) to lattices in E 310 26.6. The non-archimedean orbital integral for N — 1 314 26.7. The case of level N 318

Applications 27. Dimension formulas 333

27.1. The elliptic terms 333 27.2. The unipotent term 337 27.3. The dimension of 5k(7V, a/) and some examples 338

28. Computing Hecke eigenvalues 340 28.1. Obtaining eigenvalues from knowledge of the traces 340 28.2. Integrality of Hecke eigenvalues 343 28.3. The r-function 344 28.4. An example with nontrivial character 347

29. The distribution of Hecke eigenvalues 351 29.1. Bounds for the Eichler-Selberg trace formula 352 29.2. Chebyshev polynomials 355 29.3. Distribution of eigenvalues 356 29.4. Further applications and generalizations 359

30. A recursion relation for traces of Hecke operators 360

Bibliography 363

Tables of notation 368

Statement of the final result 370

Index 373

Bibliography

[Arl] Arthur, J., A trace formula for reductive groups I. Duke Math J., 45, (1978), 911-952.

[Ar2] , The invariant trace formula, I & II, J. Amer. Math. Soc. 1 (1988), no. 2,3.

[Ar3] , The L2-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), 257-290.

[Ar4] , An introduction to the trace formula, in "Harmonic analysis, the trace formula, and Shimura varieties", Clay Math. Proa, 4, Amer. Math. Soc, Providence, RI, (2005), 1-263.

[Ax] Axler, S., Linear algebra done right, Second edition, Springer-Verlag, New York, 1997.

[Ba] Baldoni, M. W., General representation theory of real reductive Lie groups, in [Rep].

[Bar] Bargmann, V., Irreducible unitary representations of the Lorentz group, Annals of Math., 48, (1947), 568-640.

[Bo] Bourbaki, N., Integration I, II, translation by S. Berberian, Springer-Verlag, Heidelberg, 2004.

[Bu] Bump, D., Automorphic Forms and Representations, Cambridge Stud­ies in Advanced Mathematics 55, Cambridge University Press 1998.

[CD] Clozel, L., and Delorme, P., Pseudo-coefficients et cohomologie des groupes de Lie reductifs reels, C. R. Acad. Sci. Paris, 300, Serie I, no. 12, 1985.

[CKM] Cogdell, J., Kim, H., and Murty, R., Lectures on automorphic L-functions. Fields Institute Monographs 20, American Mathematical So­ciety, Providence, RI, 2004.

[Coh] Cohen, H., Trace des operateurs de Hecke sur To(N), Seminaire de Theorie des Nombres (1976-1977), Exp. No. 4, 9 pp. Lab. Theorie des Nombres, CNRS, Talence, 1977.

[CDF] Conrey, B., Duke, W., and Farmer, D., The distribution of the eigen­values of Hecke operators, Acta Arith. 78 (1997), no. 4, 405-409.

[Cox] Cox, D., Primes of the form x2 + ny2, John Wiley & Sons, Inc., New York, 1989.

[Da] Darmon, H., A proof of the full Shimura-Taniyama-Weil conjecture is announced, Notices Amer. Math. Soc. 46 (1999), no. 11, 1397-1401.

363

364 BIBLIOGRAPHY

[De] Deligne, P., Formes modularies et representations l-adiques, Seminaire Bourbaki 1968/69, expose 355, Lecture Notes in Math., Vol. 179, p. 139-186, Springer-Verlag, 1971.

[DL] Duflo, M., and Labesse, J.-P., Sur la formule des traces de Selberg, Ann. Scient. Ec. Norm. Sup., series 4, 1971, pp. 193-284.

[DS] Diamond, F., and Shurman, J., A first course in modular forms, Springer-Verlag, New York, 2005.

[E] Eichler, M., Fine Verallgemeinerung der Abelschen Integrale, Math. Z. 67 (1957), 267-298.

[Fl] Flath, D., Decomposition of representations into tensor products, Proc. Symp. Pure Math., 33, Part 1, Amer. Math. Soc, Providence, RI, 1979, 179-183.

[Fol] Folland, G., Real analysis: modern techniques and their applications, Second edition, John Wiley Sz Sons, Inc., New York, 1999.

[Fr] Freitag, E., Hilbert modular forms, Springer-Verlag, Berlin, 1990. [Gl] Gelbart, S., Automorphic forms on adele groups, Princeton University

Press, Princeton, NJ, 1975. [G2] , Lectures on the Arthur-Selberg trace formula, Amer. Math. Soc,

Providence, RI, 1996. [GJ] Gelbart, S., and Jacquet, H., Forms of GL(2) from the analytic point of

view, Proc. Symp. Pure Math., 33, Part 1, Amer. Math. Soc, Providence, RI, 1979, 213-251.

[GGPS] Gelfand, Graev, and Piatetski-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., 1969.

[Go] Godement, Sur les relations d'orthogonalite de V. Bargman, C. R. Acad. Sci. Paris Ser. A-B 225 (1947), 521-523, 657-659.

[Gld] Goldstein, L., Analytic number theory, Prentice-Hall, Inc., Englewood Cliffs, N . J , 1971.

[H] Hijikata, H , Explicit formula of the traces of Hecke operators for To(N), J. Math. Soc. Jap. 26 (1974), pp 56-82.

[Ha] Haar, A , Der Maassbergriff in der Theorie der kontinuierlichen Grup-pen, Ann. of Math. (2), 34 (1933), pp. 147-169.

[Hal] Halmos, P , Measure theory, Van Nostrand, New York, 1950. [HC1] Harish-Chandra, Representations of semi-simple Lie groups. I, Trans.

Amer. Math. Soc, 75 (1953), pp. 185-243. [HC2] Harish-Chandra, The Plancherel formula for complex semi-simple Lie

groups, Trans. Amer. Math. Soc, 76 (1954), pp. 485-528. [Hi] Hida, H., Modular forms and Galois cohomology, Cambridge studies

in advanced mathematics 69, Cambridge University Press, Cambridge, 2000.

[Hu] Humphreys, J., Arithmetic groups, Springer Lecture Notes 789, Springer, Berlin, 1980.

[HR] Hewitt, E., and Ross, K., Abstract harmonic analysis I & II, Springer-Verlag, Berlin, 1963 and 1970.

BIBLIOGRAPHY 365

[HT] Howe, R., and Tan, E., Nonabelian harmonic analysis: Applications of SL(2,R), Springer-Verlag, New York, 1992.

[Iwl] Iwaniec, H., Topics in classical automorphic forms, Graduate Studies in Mathematics, 17, American Mathematical Society, 1997.

[Iw2] Iwaniec, H., Spectral methods of automorphic forms, Graduate Studies in Mathematics, 53, American Mathematical Society, 2002.

[JL] Jacquet, H., and Langlands, R., Automorphic forms on GL{2), Springer Lecture Notes 114, Springer-Verlag, 1970.

[Knl] Knapp, A., Theoretical aspects of the trace formula for GL(2), in [Rep].

[Kn2] , Representation theory of semisimple groups, an overview based on examples, Princeton University Press, Princeton, NJ, 1986.

[Kn3] , Lie groups beyond an introduction, second edition, Birkhauser, Boston, MA, 2002.

[KL] Knightly, A., and Li, C., A relative trace formula proof of the Petersson trace formula, Acta Arith., 122 (2006), No. 3, 297-313.

[Lai] Lang, S., SL2(R), Springer-Verlag, New York, 1985. [La2] , Real and functional analysis, Springer-Verlag, New York, 1993. [La3] , Elliptic functions, Springer-Verlag, New York, 1987. [La4] , Algebraic number theory, Springer-Verlag, New York, 1994. [La5] , Algebra, Second edition, Addison-Wesley, Reading, MA, 1984. [LI] Langlands, R., Shimura varieties and the Selberg trace formula, Cana­

dian journal of Mathematics, 29 (1977). [L2] , On the zeta-functions of some simple Shimura varieties, Canadian

journal of Mathematics, 31 (1979). [Le] Lepowsky, J., Algebraic results on representations of semisimple Lie

groups, Trans. Amer. Math. Soc. 176 (1973), 1-44. [Levi] LeVeque, W. J., Topics in number theory, Vol. I, Addison-Wesley

Publishing Co., Inc., Reading, Mass., 1956. [Lev2] LeVeque, W. J., Topics in number theory, Vol. II, Addison-Wesley

Publishing Co., Inc., Reading, Mass., 1956. [LL] Lai, K. F., and Lan, Y. Z., Representations and automorphic forms of

second-order matrix groups, (Chinese), Beijing University Mathematics Series, Beijing, 1990.

[LR] Langlands, R., Ramakrishnan, D., (eds.) Zeta Functions of Picard Modular Surfaces, U. Montreal Press, 1992.

[Mi] Miyake, T., Modular forms, Springer-Verlag, Berlin, 1989. [Mu] Murty, R., Applications of symmetric power L-functions, in [CKM],

203-283. [Mun] Munkres, J., Topology, 2nd ed., Prentice-Hall, Upper Saddle River,

NJ, 2000. [Oe] Oesterle, J. Sur la trace des operateurs de Hecke, These, L'Universite

de Paris-Sud Centre d'Orsay, 1977.

366 BIBLIOGRAPHY

[Os] Osborne, M. S., Spectral theory and uniform lattices, Lecture notes in representation theory, University of Chicago Dept. of Mathematics, 1977.

[PS] Piatetski-Shapiro, L, Multiplicity one theorems, Proc. Symp. Pure Math., 33, Part 1, Amer. Math. Soc, Providence, RI, 1979, 209-212.

[RR] Ramakrishnan, D., and Rogawski, J., Average values of modular L-series via the relative trace formula, Pure and Applied Mathematics Quarterly, Vol 1, Number 4, 2005, 701-735.

[RV] Ramakrishnan, D., and Valenza, R., Fourier analysis on number fields, Springer-Verlag, New York, 1999.

[Rep] Representation theory and automorphic forms, Proc. Sympos. Pure Math., 61, Amer. Math. Soc, Providence, RI, 1997.

[Rol] Rogawski, J., Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence, appendix in "Discrete Groups, Ex­panding Graphs and Invariant Measures," by A. Lubotzky, Birkhauser, Basel, 1994, pp. 135-176.

[Ro2] , Analytic expression for the number of points mod p, in [LR]. [Ross] Ross, S. A simplified trace formula for Heche operators for To(N),

Trans. Amer. Math. Soc, 331, no. 1, 1992, pp. 425-447. [Roy] Royden, Real analysis, third edition, Macmillan Publishing, New

York, 1988. [Rul] Rudin, W., Real and complex analysis, McGraw-Hill, New York, 1986. [Ru2] , Fourier analysis on groups, Interscience Publishers, New York-

London, 1962. [Sa] Saito, H., On Eichler's trace formula, J. Math. Soc. Japan, 24 (1972),

333-340. [S] Selberg, A., Harmonic analysis and discontinuous groups in weakly sym­

metric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc (N. S.) 20 (1956) 47-87.

[Sel] Serre, J.-P., A course in arithmetic, Springer-Verlag, New York, 1973. [Se2] , Repartition asymptotique des valeurs propres de Voperateur de

Hecke Tp, J. Amer. Math. Soc, 10 (1997), no. 1, pp. 75-102. [Se3] , Abelian l-adic representations and elliptic curves. W. A. Ben­

jamin, Inc., New York-Amsterdam, 1968. [Se4] , Linear representations of finite groups, Springer-Verlag, New

York, 1977. [Sh] Shimizu, H., On traces of Hecke operators, J. Fac Sci. Univ. Tokyo, 10,

pp. 1-19, 1963. [Shim] Shimura, G., Introduction to the arithmetic theory of automorphic

functions, Princeton University Press, 1971. [Shok] Shokranian, S., The Selberg-Arthur trace formula, Springer Lecture

Notes 1503, Springer-Verlag, Berlin, 1992. [T] Tate, J., Fourier analysis in number fields, and Hecke's zeta-functions,

Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965) pp. 305-347, Thompson, Washington, D.C., 1967.

BIBLIOGRAPHY 367

[Tay] Taylor, R., Automorphy for some l-adic lifts of automorphic mod I Galois representations II, preprint, May 24, 2006.

[Ter] Terras, A., Harmonic analysis on symmetric spaces and applications I & II, Springer-Verlag, New York, 1985 and 1988.

[VDB] van den Ban, E. P., Induced representations and the Langlands clas­sification, in [Rep].

[V] Varadarajan, V., Harmonic analysis on real reductive groups, Springer Lecture Notes 576, Springer, NY, 1977.

[VN] von Neumann, J., The uniqueness of Haar's measure, Mat. Sbornik, 1/43, (1936), p. 721.

[Wal] Wallach, N., Representations of reductive Lie groups, Proc. Sympos. Pure Math., 33, Amer. Math. Soc, Providence, RI, 1979.

[Wa2] Wallach, N., Real reductive groups I, Pure and Applied Mathematics, Vol. 132, Academic Press Inc., Boston, 1988.

[We] Weil, A., Basic number theory, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[Zl] Zagier, D., The Eichler-Selberg trace formula on SL2(Z); appendix in "Introduction to modular forms" by S. Lang, Springer, Berlin, 1976.

[Z2] , Correction to: "The Eichler-Selberg trace formula on SL2(Z)/ ;

Springer Lecture Notes 627, Springer, Berlin, 1977.

http://dx.doi.org/10.1090/surv/133/29

Tables of notation

TABLE 2. Various uses of UJ and uo'

J on (Z/NZ)*

uof on Z

J on r0(AO

J on Ao(A^)

or T(n) C A0(AO

CJ on A*

u on Z ( A )

CJ on Z(A f i n )

CJ on if0(-AO

CJ on T0(N)

CJ on M(n , N) (in Section 13)

For de Z, gcd(d,iV) = 1

a fixed character

uf(m) — ~ ujf(m mod N) if(ra,JV) = l

0 otherwise.

-'((: 5))-«-(«o

-'<(: s))=^«)-Use strong approximation:

^(<?(£oo x ^fin)) = ^'(zfin mod TVZ) for g G Q*, Xoo € R ^ ,

^V xP=U^ Uj(Zftn) = Cj(loo X Zfa)

W(L dJ)=^(dN), where d̂ v = idele agreeing with d at places p\N

and 1 at all other places

W<(c J))=«(^)=u/(d)

Agrees with cc;(loo x Zfin) on Z(Afin) D M(n, iV).

w(dfl[) = uo'(d)

368

TABLES O F NOTATION

TABLE 3. Local factors of /

369

Type of prime

p\ nN

p|n

p\N

J) — OO

Support of fp

Z(QP)KP

Z(Qp)M(n, N)p

Z(QP)KQ(N)P

G(R)+

Formula for /*

%(zk)=u>p(z)-1

f»(zm)=up(z)-1

f$(zk)=u>p(z)-lu>p(k)-1MN)

/ . ( ( : &\ k - 1 det(5)k/2(2z)k

d)' ~ 4TT (-b + c + (a + d)i)k

Here

when p|n. M(n,N)p = {ge M2(ZP)| det# G nZ;}

Statement of the final result

The Eichler-Selberg trace formula

We include here a statement of the main result, which is the following formula for the trace of Tn on £k(7V,a/) for (n, N) = 1, k > 2. For the statement when N — 1, see Proposition 28.5 on page 344. Although we have assumed (n, N) = 1 throughout, according to [Coh] and [Oe] the same formula holds for all n. (In this case, any instance of uof(x)~l in the formula below would have to be interpreted as 0 if gcd(x, N) > 1.) The modification needed in the k = 2 case is given afterwards.

tr(TB) = ^ K t f V V 7 2 ) - 1 * * - 1

" 2 ^ p-B L M - ^ r - M * , m , n )

- i ] £ min(d, n/df-1 J2 ¥>fe<**fr N/r))J{y)-\

where:

• V W - [SL2(Z) : T0(N)] = N TT (1 + - ) primes p\N

• ^ / (n 1 / 2 ) - 1 = 0 if n is not a perfect square • t runs through all integers which satisfy — 2 ^ < t < 2v/n • p and p are the roots of the polynomial X2 — tX + n • m runs through integers > 1 such that m2 divides t2 — 4n, and

£=£* = 0 , 1 mod4 • hw(f ^2n) is the weighted class number of the order in Q[p] which

has discriminant l ~fr. This is the usual class number, divided by 2 (resp. 3) if the discriminant is —4 (resp. —3) (see Theorem 26.11 for an explicit formula for these class numbers).

c runs through all elements of (Z/7VZ)* which lift to solutions of c2 - tc + n = 0 mod NNm.

• d runs through all positive divisors of n

• /i(£,ra,n) = j(N/N x X^ u ' ^ *' w h e r e ^ = gcd(iV,m), and

370

STATEMENT O F T H E FINAL RESULT 371

T runs through all positive divisors of N such that gcd(r, N/r) divides both -iff- and d 2, where N^ is the conductor of u/

• (p is the Euler (/^-function • y is the unique integer mod c d / ^

y = §mod*Z. N/r) Z such that y = d mod rZ and

It is useful to remember that the terms corresponding to t — to and t = —to coincide, and that the terms corresponding to d — do and d = n/do coincide (for any to and do). If N — 1, then /i(£,ra,n) = 1. If, in the definition of /i(t, 77i, n), there are no such x, then //(£, ra, n) — 0. (An empty sum is 0; an empty product is 1.)

When k = 2, the formula is also valid with the addition of the following extra term:

0 if k > 2 or if J + 1 A def

E c if k — 2 and u/ = 1. c|n,c>0,

gcd(AT,n/c) = l

Index

© (Hilbert space direct sum), 5 -<, 49 \A\ (A a linear map), 129 M A , 86 MP, 52 < , < n , 20 ll(c d)llp, ll(c d)llA , 88 |[a]k (slash operator), 34 | r « r (slash operator), 34, 43

A, 68 A+, 68 Ak(N,u>), 201 A, 55 A ^ (idelesof E), 311 A 5 , 55 a.e. (almost every), 55 A*, 57 A* , 57 A 1 , 87 Afin, 55 an(S), 16, 199 absolute value, 52 Ad(p), 153, 167 a d p O ( y ) = [X,Y], 152, 162 adeles, 55 adjoint, 131

of a function, 144 representation on 0c> 153

admissible representation, 153 antiholomorphic discrete series, 190 archimedean, 52 arc tan, 63

B (Borel subgroup), 3, 60 BG (Borel subsets of G), 70 BT, 252 B, 123 BT, 252 B(R) + , 14, 63 Banach space, 131 Beta function B(n,m), 289 Borel measure, 70 Borel subgroup, 60

Bruhat decomposition, 69

C(G) (cont. functions on G), 109 Co(X) (cont. functions vanishing at oo), 109 CC(G(A),UJ), 5 G c (G,x) , 134 CC(X), 49 Cartan decomposition

p-adic, 208 real, 66, 68, 93

Cartan subalgebra, 162 Casimir element, 167 Cauchy-Schwarz inequality, 131 central character, 138 character value for 7Tk, 302 Chebyshev polynomial, 355 x(£iie2,si,S2) (character of B(R) ) , 157 X7rk, 190 class group, 306 class number, 306 closed graph theorem, 131 closed linear transformation, 131 commensurable, 20 compact neighborhood, 49 complementary series, 186 conductor

of a character, 11, 196 of an order, 303

congruence subgroup, 8 conjugate-linear functional, 131 constant term, 4 countable additivity, 70 critical strip, 124 cusp

holomorphic at, 16 of T, 12 vanishing at, 17

cusp form, 18, 19 cuspidal function, 4 cuspidal representation, 5

D (subset of SL 2 (R)) , 99 (subset of H) , 19, 99

Do, 105, 261

373

374 INDEX

DN, 104 Dx, 164 D T o ( / ) , 2 3 6 d*x, 86 dlx, 87, 103 dN, 195 dk = ^ (formal degree of 7rk), 214, 224 d(iV) (number of positive factors of N), 352 £>, 105 £>o, 241 A (Casimir element, Laplace operator), 166 A(z) £ 512(1), 338 A C <3(Q)+ (defining a Hecke ring), 21 Ao(iV), 24 A G ( P ) (modular function), 73 dimension formulas, 338 Dirichlet character, 11 discrete series, 139

for GL 2 (R) , 151 for SL 2 (R) , 187

divorce theorem, 101 d/j,oo (Sato-Tate measure), 352 dominated convergence theorem, 71 dual group, 108

163

E = Q[7] = Q[X]/{X2 - tX + n), 302 1 - O

-i - l y

R(E~), 185 Ep = E®QP, 310 B]H(N,LJ') (basis of eigenforms), 352 £k(A0» 3 5 2

Eichler-Selberg trace formula, 47 eigenform, 45

normalized, 46 elementary divisors theorem, 26, 319 elliptic, 6, 232 en(ke) = ein0, 155 equidistribution, 351

r]= f"1 J €<3(R), 155, 166

exponential map, 152

FG(g,T), 260 Fg^(t), 253 / n , 211 / n ( L , L ' ) , 322 /£ , 210 / (Fourier transform), 112 / + , / ~ (basis elements for r n ) , 155 , (a) _ (k-i) de t ( g ) k / 2 (2 i ) k

/o (lowest weight vector for 7Tk), 190 / + (lowest weight vector for TT£), 188 /o = /o/H/oll , 191 /k(p) (matrix coeff. for / Q ) , 192, 221 f4p. (finite part), 125

214,225

formal degree, 142 Fourier inversion formula, 109 Fourier transform, 109 Fubini's theorem, 72 functional equation for Z(f>(s)1 126 fundamental domain, 19, 37, 96

strict, 96

G(R) for a ring R, 3 <3(A)S, 59 G1 (centralizer of 7), 232 G = Z\G, 4 G (dual group), 108 (g, if )-module, 153

for GL 2 (R) , 164 infinitesimal equivalence, 154 underlying, 154

g, 152 g-module, 152 flc, 153 G-invariant subspace, 134 G-stable subspace, 134 T(A) (graph of A), 131 r(7V), 8 r ( s ) (Gamma function), 289 r 0 W , 8 T^(N) (determinant ±1), 25 ri(JV), 8 Too, 14 ^E (Euler constant), 124 geometric side, 230 Godement's theorem, 139 G ( R ) + , 1 0 graph, 131

H (height function), 88, 236

H=(^ ^ G g o 166

R(H), 184 H (complex upper half-plane), 10 h(0), 306 hs(z)=j(6,z)-*h(6(z)), 11, 16 h\[sh(z)=u;/(S)hs(z),S4, Haar functional, 73 Haar measure, 72

on A, A*, 85 on A 1 , 103 on GL2 , 90 o n G L 2 ( R ) , 95 on 3 , 94 on ~K, 94 on Q \ A , 102 on Q*\A*, 102 on Q ^ A 1 , 103 on B , 87, 90 on Koo, 91 on Kp, 91

Hausdorff, 49

INDEX 375

Hecke character, 4 Hecke eigenform, 45 Hecke operator, 36

eigenvalues of, 340 integrality of, 343

Hecke ring, 21 of level N, 24

height function, 88 Hilbert space, 131 Hilbert space direct sum, 5 holomorphic vector, 164 HoniG (TTO , 7r), 136

(d),hw(0), 318 hyperbolic, 6, 232

-( ' 0 GflC, 166

ideles, 57 induced representation, 157 infinitesimal equivalence, 154 infinity type, 204 inner product space, 130 integrable

function on G, 70, 138 vector-valued function, 132

integrality of Hecke eigenvalues, intertwining operator, 136 invariant measure, 74 J o , 305 isometry, 130 isotypic component, 152 Iwasawa coordinates, 62 Iwasawa decomposition, 3, 62

343

JTU), 259 j(g,z) = (cz + d)det(g)-Jacobi identity, 152 Jell, 294

V2, 10

K, 62 JC-finite vectors, 151 K(x,y), 228 K = 0(2) (Section 11 only), K° = SO(2) (in Section 11), Koo = SO(2), 4, 61

155 156

0 < 6 < t } , 98 K[o,t) = {ke ^fin, 62 K, 151 *5i(fl0, 250 ^ y p ( 3 ) , 250 *uniG>)> 250 kT(g,cf>), 239 i _ ( cos 9 s i n # \ * ~ V-sin6> cos 6>J

K-types, 152 K(N), 60 Xo(-Af), 100, 197, 206 K±(N), 200

61

KBA9U92)> 249 KB,o(9i,92), 260 kernel, 227 tf0 (01,02), 248 K > , 252 ^(<?), 249 kjn 252 ^ ( p ) , 260

79

138

LHG/K), L2(LU), 4 L p function, LP(G), 72 LP(G,X) , 134 LP(X,/z), 71 L _ 1 (inverse lattice), 305 LgM, 4 L p , 307 £ (set of lattices in Q 2 ) , 308 £p (set of Zp-lattices in Q 2 ) , 308 £Ep (Zp-lattices in Ep), 311 £ A f i n , 3 0 8 AT<p (truncation), 235 A', 308 A : G(A f i n) - • £, 310 AT(p), 266

A r j • 9 P ^ 9P

^N,p '• 9P y~^ 9P

(z:>308

UzpJ' 318

137

Zp ^NZ P

lattice, 24, 304 left regular action, 73 Legendre symbol, 306 level N

congruence subgroup, 8 modular form, 12

Lie algebra, 152 representation, 152

linear extension theorem, 130, linear functional, 131 local-global principle for lattices, 310 locally compact, 49 lowest weight vector, 190, 193

M (diagonal matrices), 60 M(n,iV), 207 M5 = MS(T)1 15 MkfiV.w'), 12 Mk(r), n M2(R), 7 m1{X) (characteristic polynomial of 7), 296 matrix coefficient, 138 M c (G ,x ) , 147 measurable function, 70 measure, 69

Borel, 70 Radon, 70 trivial, 70

376 INDEX

«' -= o n B ( Q ) , 251 = o - o', 251

measure space, 70 minimal K-type, 190 modular form, 11, 18 modular function, 73

of B, 90 monotone convergence theorem, 71 £t(t,m,n), 327

N (unipotent matrices), 60 Nn, 169, 176 Nw (conductor of a;), 196 Np (=vp(N)), 9, 52 N^/ (conductor of a;'), 11 nebentypus, 11 neighborhood, 49 Newton-Girard formula, 341 non-archimedean, 52 norm (of a linear transformation), 129 normal operator, 42, 44, 132

0(2) (orthogonal group), 61 OE, 303 OL, 304 Op = OE®ZP, 311 (9-lattice, 304 o' o" n<f>(s), 124 w, 195 a;',a; (table of usage), 368 a/, 11, 12, 195 Ljp, 196 u;p, 328 orbital integral, 232, 280 order, 303

Vo, 306 V(iV), 319 p-adic numbers, 52 parabolic element, 12 Peter-Weyl theorem, 142, 151 Petersson inner product, 37 Petersson trace formula, 229 $ ( 7 , / ) (orbital integral), 232, 281, 295, 298 r(9) = tig'1) (adjoint of </>), 144 4>h, 198 <t>n e vx, i6i 0p (char, function of Z p ) , 113 4>v,w(g) (matrix coefficient), 138 </? (Euler (/^-function), 9 y?p (local Euler cp-function), 286 7r(<j))v, 140 TTX, 154, 157 ?rk (discrete series of GL 2 (R)) , 186, 190 7r̂ " (discrete series of SL2(R)), 187 Plancherel theorem, 110 Poisson summation formula, 117 polar decomposition, 67 Pontriagin duality, 109

positive functional, 71 principal O-lattice, 306 principal congruence subgroup, 8 principal part of x G Q p , 110 principal series, 154, 160 product formula, 87 product measure, 83 product topology, 54 proper O-lattice, 304 pseudo-coefficient, 214 * , 308 * J V , 319

207

IPP(N) =

^ n P a + p-)

(P+ I )P N P i

N>1 N = 1

p\N p\N

, 8, 203,

206

Q*2 (squares in QJ) , 230 Q 2 (R) , 118 quadratic decay, 117 quadratic form, 66 quotient measure, 76

for a discrete subgroup, 97 o n G ( Q ) \ 5 ( A ) , 105 on S ( Q ) \ G ( A ) , 106 on 5 , 94 on Q \ A , 102 o n Q * \ A * , 102 on Q * \ A \ 103

quotient topology, 51

R (right regular representation), 4, 135 R(E~), 163, 185, 201 R ^ , 58 K(N), 24 ^ ( r , A ) , 21 rp(x) (principal part of x E Qp), HO R(f), 5, 140 Radon measure, 70 Radon measure space, 70 rapid decay, 115 relative topology, 49, 61 relatively compact, 49 representation, 134

irreducible, 134 square-integrable, 139 unitary, 134

of GL 2 (R) , 186 restricted direct product, 82

examples of, 55, 57, 59 Riemann zeta function, 121 Riesz representation theorem, 71 right regular action, 73 root, 162

space, 162

S*(N), 352

INDEX 377

Sk(7V,u/), 18 sk(r), is Sato-Tate conjecture, 351 Schur orthogonality relations, 142 Schur's lemma, 137 Schwartz function, 116 Schwartz-Bruhat function, 113 <S(Afin), 113 Schwarz inequality, 131 second-countable, 49, 97, 101 cr(N) (number of prime factors of TV), 353 a-algebra, 69 cr-finite space, 72 simple function, 70 SL2(i2), 7 SL±(Z) = GL 2(Z) , 25 SL±(R) (determinant ±1), 189 SO(2) (special orthogonal group), 61 spectral side, 230 square-integrable, 139 standard character

of A, 111 of Q p , 111

strong approximation for A, 56 for A*, 58 for £ ( A ) , 63 for G(A), 65

subrepresentation theorem, 154 Supp( / ) , 49 support of a function, 49 symmetric neighborhood, 52 symmetric polynomial, 340

T ^ n - ^ - 1 ) / 2 ^ , 351 T(n), 29, 30

T(a,d)=T0(N)(a d ) r o ( i \ 0 , 31

Tn (Hecke operator), 36, 45 fn, 356 tangent space, 182 r (char, function of (0,oo)), 235 r(n) (Ramanujan r-function), 338, 344 r n (representation of 0(2)) , 155 ©fa), 63 6 (adele character), 110

6p(x) =e2lTirp(x\ 111 0x(y)=0{~xy), 112 0fin, H I topological group, 50 topology

on A, 55 on A*, 57 on GL(2) over R, Q p , Z p , 59 on GL 2 (A) , 59

trace class operator, 229 trace formula, 276

Arthur-Selberg, 5 Eichler-Selberg, 370

triv (trivial representation), 155 truncated kernel function, 239 truncation, 235 TychonofFs theorem, 55

U(g), 166 UN (subgroup of Z*), 57 Upn (open subgroup of G(Q P ) ) , 59 uniform continuity, 52 uniform distribution, 351 unimodular, 73 unipotent, 232 unitary

operator, 130 representation, 134

of GL 2 (R) , 186 universal enveloping algebra, 165 unramified

character, 196 Urysohn's lemma, 50

V , 175 V°, 175 Vk, 151 Vx, 157 V°, 157 Vn, 168 Vp (p-adic valuation), 52 v(g) (weight function), 265, 267 vp(gp) (local weight function), 268 valuation, 52 vector-valued integral, 132

W= f ^ Gflc, 162

R(W), 184 Wp, Wpig), 113, 198 W]a(N,UJ') (weak modular forms), 12

Wk(r), n

w = ( - i X ) ' 6 9

weak modular form, 11 weight k vector, 186 weight function, 6, 267 weighted class number, 318 weighted orbital integral, 264 Whittaker function, 198 width (of a cusp), 16

X=(°Q J) e0c, 166 R(X), 185

Xn(x), Chebyshev polynomial, 355 x(<?), 63 4, 351

y(0), 63

http://dx.doi.org/10.1090/surv/133/30

378 INDEX

Z (center), 4 Z(R) + , 11, 189 Z(g), 166 Zoo = Z(R) , 241 Z+ (positive integers), 200 Z, 55 Z*, 57 z(<?), 62 Z^(s) (Tate zeta function), 6, 119-128, 270,

288 £(s) (Riemann zeta function), 121 CP(s) = ( l - p - s ) - \ l 2 1

Titles in This Series

133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J . P . May and J . Sigurdsson, Parametrized homotopy theory, 2006 131 J in Feng and Thomas G. Kurtz , Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in

Euclidean spaces, 2006 129 Wil l iam M. Singer, Steenrod squares in spectral sequences, 2006 128 Athanassios S. Fokas, Alexander R. Its , Andrei A. Kapaev, and Victor Yu.

Novokshenov, Painleve transcendents, 2006 127 Nikolai Chernov and Rober to Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson , and Wil l iam T. Ross , The Cauchy Transform,

2006 124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006 123 Barbara Fantechi, Lothar Gottsche , Luc Illusie, Steven L. Kle iman, Ni t in

Nitsure , and Angelo Vistol i , Fundamental algebraic geometry: Grothendieck's FGA explained, 2005

122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005

121 A n t o n Zettl , Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian M a and Shouhong Wang, Geometric theory of incompressible flows with

applications to fluid dynamics, 2005 118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005 113 Wil l iam G. Dwyer , Phi l ip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith ,

Homotopy limit functors on model categories and homotopical categories, 2004 112 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups

II. Main theorems: The classification of simple QTKE-groups, 2004 111 Michael Aschbacher and Stephen D . Smith , The classification of quasithin groups I.

Structure of strongly quasithin K-groups, 2004 110 Benne t t Chow and D a n Knopf, The Ricci flow: An introduction, 2004 109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups,

2004 108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D . Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with

applications to economics, second edition, 2003 104 Graham Everest , Alf van der Poorten , Igor Shparlinski, and Thomas Ward,

Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre,

Lusternik-Schnirelmann category, 2003 102 Linda Rass and John RadclifFe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004

TITLES IN THIS SERIES

99 Phi l ip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guil lemin, Viktor Ginzburg, and Yael Karshon, Moment maps,

cobordisms, and Hamiltonian group actions, 2002 97 V . A . Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Mart in Markl, Steve Shnider, and J im Stasheff, Operads in algebra, topology and

physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l and Larry Smith , Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:

Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1:

Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery , A tour of subriemannian geometries, their geodesies and

applications, 2002 90 Christ ian Gerard and Izabella Laba, Multiparticle quantum scattering in constant

magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Prenkel and David Ben-Zvi , Vertex algebras and algebraic curves, second

edition, 2004 87 Bruno Poizat , Stable groups, 2001 86 Stanley N . Burr is, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with

corner singularities of solutions to elliptic equations, 2001 84 Laszlo Puchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant

differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential

operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module

theory, 2000 79 Joseph A. Cima and Wil l iam T. Ross , The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt , KP or mKP: Noncommutative mathematics of Lagrangian,

Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz , The semicircle law, free random variables and entropy,

2000 76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmuller theory, 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,

2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems

and differential equations, 1999

For a complete list of t i t les in this series, visit t he AMS Bookstore a t w w w . a m s . o r g / b o o k s t o r e / .