**Mathematical Surveys and Monographs**

Volume: 133;
2006;
378 pp;
Hardcover

MSC: Primary 11;

Print ISBN: 978-0-8218-3739-9

Product Code: SURV/133

List Price: $103.00

Individual Member Price: $82.40

**Electronic ISBN: 978-1-4704-1360-6
Product Code: SURV/133.E**

List Price: $103.00

Individual Member Price: $82.40

#### Supplemental Materials

# Traces of Hecke Operators

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*Andrew Knightly; Charles Li*

The Fourier coefficients of modular forms are
of widespread interest as an important source of arithmetic
information. In many cases, these coefficients can be recovered from
explicit knowledge of the traces of Hecke operators. The original
trace formula for Hecke operators was given by Selberg in 1956. Many
improvements were made in subsequent years, notably by Eichler and
Hijikata.

This book provides a comprehensive modern treatment of the
Eichler–Selberg/Hijikata trace formula for the traces of Hecke
operators on spaces of holomorphic cusp forms of weight
\(\mathtt{k}>2\) for congruence subgroups of
\(\operatorname{SL}_2(\mathbf{Z})\). The first half of the
text brings together the background from number theory and
representation theory required for the computation. This includes
detailed discussions of modular forms, Hecke operators, adeles and
ideles, structure theory for \(\operatorname{GL}_2(\mathbf{A})\), strong
approximation, integration on locally compact groups, the Poisson
summation formula, adelic zeta functions, basic representation theory
for locally compact groups, the unitary representations of
\(\operatorname{GL}_2(\mathbf{R})\), and the connection between
classical cusp forms and their adelic counterparts on
\(\operatorname{GL}_2(\mathbf{A})\).

The second half begins with a full development of the geometric
side of the Arthur–Selberg trace formula for the group
\(\operatorname{GL}_2(\mathbf{A})\). This leads to an
expression for the trace of a Hecke operator, which is then computed
explicitly. The exposition is virtually self-contained, with complete
references for the occasional use of auxiliary results. The book
concludes with several applications of the final formula.

#### Readership

Graduate students and research mathematicians interested in number theory, particularly modular forms, Hecke operators, and trace formulas.

#### Reviews & Endorsements

Clearly this book should be in any institutional library, and the text is warmly recommended to any serious student of modular forms and representation theory.

-- Zentralblatt MATH

While this book is certainly of interest to experts in the area, it is accessible to graduate students and researchers from other disciplines that are interested in studying automorphic forms and trace formulas.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Traces of Hecke Operators

- Contents vii8 free
- Traces of Hecke Operators 112 free
- 1. Introduction 112
- 2. The Arthur-Selberg trace formula for GL(2) 314
- 3. Cusp forms and Hecke operators 718
- 3.1. Congruence subgroups of SL[sub(2)](Z) 718
- 3.2. Weak modular forms 1021
- 3.3. Cusps and Fourier expansions of modular forms 1223
- 3.4. Hecke rings 2031
- 3.5. The level N Hecke ring 2435
- 3.6. The elements T(n) 2940
- 3.7. Hecke operators 3445
- 3.8. The Petersson inner product 3748
- 3.9. Adjoints of Hecke operators 4253
- 3.10. Traces of the Hecke operators 4556

- Odds and Ends 4960
- 4. Topological groups 4960
- 5. Adeles and ideles 5263
- 6. Structure theorems and strong approximation for GL[sub(2)](A) 5970
- 7. Haar measure 6980
- 7.1. Basic properties of Haar measure 6980
- 7.2. Invariant measure on a quotient space 7485
- 7.3. Haar measure on a restricted direct product 8293
- 7.4. Haar measure on the adeles and ideles 8596
- 7.5. Haar measure on B 8798
- 7.6. Haar measure on GL(2) 90101
- 7.7. Haar measure on SL[sub(2)](R) 92103
- 7.8. Haar measure on GL(2) 94105
- 7.9. Discrete subgroups and fundamental domains 95106
- 7.10. Haar measure on Q\ A and Q*\ A* 102113
- 7.11. Quotient measure on GL[sub(2)](Q)\GL[sub(2)](A) 103114
- 7.12. Quotient measure on B(Q)\G(A) 105116

- 8. The Poisson summation formula 108119
- 9. Tate zeta functions 119130
- 10. Intertwining operators and matrix coefficients 128139
- 11. The discrete series of GL[sub(2)](R) 151162
- 11.1. K-type decompositions 151162
- 11.2. Representations of O(2) 155166
- 11.3. K-type decomposition of an induced representation 156167
- 11.4. The (g,K)-modules (V[sub(χ)])[sub(K)] 162173
- 11.5. Classification of irreducible (g,K)-modules for GL[sub(2)](R) 165176
- 11.6. A detailed look at the Lie algebra action 181192
- 11.7. Characterization of the weight k discrete series of GL[sub(2)](R) 186197

- Groundwork 195206
- The Trace Formula 227238
- 15. Introduction to the trace formula for R(f) 227238
- 16. Terms that contribute to K(x,y) 230241
- 17. Truncation of the kernel 231242
- 18. Bounds for ∑[sub(γ)]|f(g[sup(…1)][sub(1)]γg[sub(2)]| 240251
- 19. Integrability of k[sup(T)][sub(o)](g) 248259
- 20. The hyperbolic terms as weighted orbital integrals 259270
- 21. Simplifying the unipotent term 270281
- 22. The trace formula 276287

- Computation of the Trace 279290
- 23. The identity term 279290
- 24. The hyperbolic terms 280291
- 25. The unipotent term 288299
- 26. The elliptic terms 294305
- 26.1. Properties of the elliptic orbital integrals 295306
- 26.2. The archimedean elliptic orbital integral 300311
- 26.3. Orders and lattices in an imaginary quadratic field 302313
- 26.4. Local-global theory for lattices 307318
- 26.5. From G(A[sub(fin)]) to lattices in E 310321
- 26.6. The non-archimedean orbital integral for N =1 314325
- 26.7. The case of level N 318329

- Applications 333344
- Bibliography 363374
- Tables of notation 368379
- Statement of the final result 370381
- Index 373384