http://dx.doi.org/10.1090/surv/133
Traces of Hecke Operators Traces of Hecke Operators
Andrew Knightly Charles Li
American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair
2000 Mathematics Subject Classification. Primary 11-XX.
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Library of Congress Cataloging-in-Publication Data 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
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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 CONTENTS
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/fcrSfe)! 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
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TABLE 2. Various uses of UJ and uo'
J on (Z/NZ)* a fixed character
~ ujf(m mod N) if(ra,JV) = l uof on Z uf(m) — 0 otherwise.
J on r0(AO -'((: 5))-«-(«o
J on Ao(A^)
or T(n) C A0(AO -'<(: s))=^«)- Use strong approximation: CJ on A* ^(
u on Z(A) ^V xP=U^
CJ on Z(Afin) Uj(Zftn) = Cj(loo X Zfa)
W d (L dJ)=^( N), CJ on if0(-AO where d^v = idele agreeing with d at places p\N and 1 at all other places
W CJ on T0(N) <(c J))=«(^)=u/(d)
CJ on M(n, N) (in Section 13) Agrees with cc;(loo x Zfin) on Z(Afin) D M(n, iV).
For de Z, gcd(d,iV) = 1 w(dfl[) = uo'(d)
368 TABLES OF NOTATION 369
TABLE 3. Local factors of /
Type of prime Support of fp Formula for /*
1 p\ nN Z(QP)KP %(zk)=u>p(z)-
1 p|n Z(Qp)M(n, N)p f»(zm)=up(z)-
l 1 p\N Z(QP)KQ(N)P f$(zk)=u>p(z)- u>p(k)- MN) k 2 k &\ k-1 det(5) / (2z) G(R)+ /.((: k J) — OO d)' ~ 4TT (-b + c + (a + d)i)
Here M(n,N)p = {ge M2(ZP)| det# G nZ;} when p|n. 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.
72 1 1 tr(TB) = ^KtfVV )- **-
"2 ^ p-B LM-^r-M*,m,n)
- i ]£ min(d, n/df-1 J2 ¥>fe<**fr N/r))J{y)-\ where:
• VW - [SL2(Z) : T0(N)] = N TT (1 + -) primes p\N • ^/(n1/2)-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 f n • hw( ^2 ) 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).
u where = • /i(£,ra,n) = j(N/N x X^ '^ *' ^ gcd(iV,m), and c runs through all elements of (Z/7VZ)* which lift to solutions of 2 c - tc + n = 0 mod NNm. • d runs through all positive divisors of n
370 STATEMENT OF THE 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 Z such that y = d mod rZ and • y is the unique integer mod cd/^N/r) y = §mod*Z. 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 Bruhat decomposition, 69 -<, 49 \A\ (A a linear map), 129 C(G) (cont. functions on G), 109 Co(X) (cont. functions vanishing at oo), 109 MA, 86 C (G(A),UJ), 5 MP, 52 C G (G,x), 134 <, Ak(N,u>), 201 Cauchy-Schwarz inequality, 131 A, 55 central character, 138 A^ (idelesof E), 311 character value for 7Tk, 302 A5, 55 Chebyshev polynomial, 355 £ e a.e. (almost every), 55 x( ii 2,si,S2) (character of B(R)), 157 A*, 57 X7rk, 190 A* , 57 class group, 306 A1, 87 class number, 306 Afin, 55 closed graph theorem, 131 an(S), 16, 199 closed linear transformation, 131 absolute value, 52 commensurable, 20 Ad(p), 153, 167 compact neighborhood, 49 adpO(y) = [X,Y], 152, 162 complementary series, 186 adeles, 55 conductor adjoint, 131 of a character, 11, 196 of a function, 144 of an order, 303 representation on 0c> 153 congruence subgroup, 8 admissible representation, 153 conjugate-linear functional, 131 antiholomorphic discrete series, 190 constant term, 4 archimedean, 52 countable additivity, 70 arc tan, 63 critical strip, 124 cusp B (Borel subgroup), 3, 60 holomorphic at, 16 BG (Borel subsets of G), 70 of T, 12 BT, 252 vanishing at, 17 B, 123 cusp form, 18, 19 BT, 252 cuspidal function, 4 + B(R) , 14, 63 cuspidal representation, 5 Banach space, 131 Beta function B(n,m), 289 D (subset of SL2(R)), 99 Borel measure, 70 (subset of H), 19, 99 Borel subgroup, 60 Do, 105, 261 373 374 INDEX DN, 104 formal degree, 142 Dx, 164 Fourier inversion formula, 109 DTo(/),236 Fourier transform, 109 d*x, 86 Fubini's theorem, 72 l d x, 87, 103 functional equation for Z(f>(s)1 126 dN, 195 fundamental domain, 19, 37, 96 dk = ^ (formal degree of 7rk), 214, 224 strict, 96 d(iV) (number of positive factors of N), 352 £>, 105 G(R) for a ring R, 3 £>o, 241 <3(A)S, 59 A (Casimir element, Laplace operator), 166 G1 (centralizer of 7), 232 A(z) £ 512(1), 338 G = Z\G, 4 A C <3(Q)+ (defining a Hecke ring), 21 G (dual group), 108 Ao(iV), 24 (g, if )-module, 153 AG(P) (modular function), 73 for GL2(R), 164 dimension formulas, 338 infinitesimal equivalence, 154 Dirichlet character, 11 underlying, 154 discrete series, 139 g, 152 for GL2(R), 151 g-module, 152 for SL2(R), 187 flc, 153 divorce theorem, 101 G-invariant subspace, 134 d/j,oo (Sato-Tate measure), 352 G-stable subspace, 134 dominated convergence theorem, 71 T(A) (graph of A), 131 dual group, 108 r(7V), 8 2 r(s) (Gamma function), 289 E = Q[7] = Q[X]/{X - tX + n), 302 r W, 8 1 -O 0 163 T^(N) (determinant ±1), 25 -i -l y ri(JV), 8 R(E~), 185 Too, 14 E = E®Q , 310 p P ^E (Euler constant), 124 B]H(N,LJ') (basis of eigenforms), 352 geometric side, 230 352 £k(A0» Godement's theorem, 139 Eichler-Selberg trace formula, 47 G(R)+,10 eigenform, 45 graph, 131 normalized, 46 H (height function), 88, 236 elementary divisors theorem, 26, 319 elliptic, 6, 232 H=(^ ^G go 166 e (k ) = ein0, 155 n e R(H), 184 equidistribution, 351 H (complex upper half-plane), 10 1 r]= f" J €<3(R), 155, 166 h(0), 306 exponential map, 152 hs(z)=j(6,z)-*h(6(z)), 11, 16 / h\[sh(z)=u; (S)hs(z),S4, G F (g,T), 260 Haar functional, 73 Fg^(t), 253 Haar measure, 72 /n, 211 on A, A*, 85 /n(L,L'), 322 on A1, 103 /£, 210 on GL2, 90 / (Fourier transform), 112 onGL2(R), 95 + / ,/~ (basis elements for rn), 155 on 3, 94 , _ (k-i) det( )k/2(2i)k on ~K, 94 (a) g 214,225 on Q\A, 102 /o (lowest weight vector for 7Tk), 190 on Q*\A*, 102 /+ (lowest weight vector for TT£), 188 on Q^A1, 103 /o = /o/H/oll , 191 on B, 87, 90 on Koo, 91 /k(p) (matrix coeff. for /Q), 192, 221 on Kp, 91 f4p. (finite part), 125 Hausdorff, 49 INDEX 375 Hecke character, 4 KBA9U92)> 249 Hecke eigenform, 45 KB,o(9i,92), 260 Hecke operator, 36 kernel, 227 eigenvalues of, 340 tf0 (01,02), 248 integrality of, 343 K>, 252 Hecke ring, 21 ^( (d),hw(0), 318 LP(G), 72 hyperbolic, 6, 232 LP(G,X), 134 LP(X,/z), 71 _1 GflC, 166 L (inverse lattice), 305 LgM, 4 ideles-(,' 570 Lp, 307 induced representation, 157 £ (set of lattices in Q2), 308 infinitesimal equivalence, 154 £p (set of Zp-lattices in Q2), 308 infinity type, 204 £ (Zp-lattices in E ), 311 inner product space, 130 Ep p £ ,308 integrable Afin AT Zp isometry, 130 y ^N,p '• 9P ~^ 9P 318 isotypic component, 152 UzpJ^NZP ' Iwasawa coordinates, 62 lattice, 24, 304 Iwasawa decomposition, 3, 62 left regular action, 73 Legendre symbol, 306 JTU), 259 level N j(g,z) = (cz + d)det(g)- V2, 10 congruence subgroup, 8 Jacobi identity, 152 modular form, 12 Lie algebra, 152 Jell, 294 representation, 152 K, 62 linear extension theorem, 130, 137 JC-finite vectors, 151 linear functional, 131 K(x,y), 228 local-global principle for lattices, 310 K = 0(2) (Section 11 only), 155 locally compact, 49 K° = SO(2) (in Section 11), 156 lowest weight vector, 190, 193 Koo = SO(2), 4, 61 M (diagonal matrices), 60 K[o,t) = {ke 0 < 6 < t}, 98 M(n,iV), 207 ^fin, 62 M5 = MS(T)1 15 K, 151 MkfiV.w'), 12 *5i(fl0, 250 Mk(r), n ^ p(3), 250 M2(R), 7 y m {X) (characteristic polynomial of 7), 296 *uniG>)> 250 1 matrix coefficient, 138 kT(g,cf>), 239 M (G,x), 147 i _ ( cos 9 sin#\ c 61 measurable function, 70 *~ V-sin6> cos 6>J K-types, 152 measure, 69 K(N), 60 Borel, 70 Radon, 70 Xo(-Af), 100, 197, 206 trivial, 70 K±(N), 200 376 INDEX measure space, 70 positive functional, 71 minimal K-type, 190 principal O-lattice, 306 modular form, 11, 18 principal congruence subgroup, 8 modular function, 73 principal part of x G Qp, 110 of B, 90 principal series, 154, 160 monotone convergence theorem, 71 product formula, 87 £t(t,m,n), 327 product measure, 83 product topology, 54 N (unipotent matrices), 60 proper O-lattice, 304 Nn, 169, 176 pseudo-coefficient, 214 Nw (conductor of a;), 196 *, 308 Np (=vp(N)), 9, 52 *JV, 319 N^/ (conductor of a;'), 11 N>1 nebentypus, 11 a + p-) , 8, 203, ^nP N = 1 neighborhood, 49 207 Newton-Girard formula, 341 (P+I)PNP p\N IP (N) = 206 non-archimedean, 52 P i p\N norm (of a linear transformation), 129 normal operator, 42, 44, 132 Q*2 (squares in QJ), 230 Q2(R), 118 0(2) (orthogonal group), 61 quadratic decay, 117 O , 303 E quadratic form, 66 O , 304 L quotient measure, 76 O = OE®Z , 311 p P for a discrete subgroup, 97 (9-lattice, 304 onG(Q)\5(A), 105 o«' -= onB(Q), 251 on S(Q)\G(A), 106 o" = o- o', 251 on 5, 94 n INDEX 377 Sk(7V,u/), 18 Arthur-Selberg, 5 sk(r), is Eichler-Selberg, 370 Sato-Tate conjecture, 351 triv (trivial representation), 155 Schur orthogonality relations, 142 truncated kernel function, 239 Schur's lemma, 137 truncation, 235 Schwartz function, 116 TychonofFs theorem, 55 Schwartz-Bruhat function, 113 U(g), 166 Z (center), 4 Z(R) + , 11, 189 Z(g), 166 Zoo = Z(R), 241 Z+ (positive integers), 200 Z, 55 Z*, 57 z(Riemann zeta function), 121 s CP(s) = (l-p- )-\l21 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 Jin 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 William 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 Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William 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. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck's FGA explained, 2005 122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005 121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma 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 William G. Dwyer, Philip 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 Bennett Chow and Dan 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 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel 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 Christian 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 William 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 titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.