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 Cartan 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
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